Physics-based algorithms and divergence free finite elements for coupled flow problems

Transcription

1 Pysics-based algoritms and divergence free finite elements for coupled flow problems A Dissertation Presented to te Graduate Scool of Clemson University In Partial Fulfillment of te Requirements for te Degree Doctor of Pilosopy Matematical Science by Nicolas E. Wilson August Accepted by: Dr. Leo Rebolz, Committee Cair Dr. Vincent Ervin Dr. Eleanor Jenkins Dr. Jeong-Rock Yoon

2 Abstract Tis tesis studies novel pysics-based metods for simulating incompressible fluid flow described by te Navier-Stokes equations (NSE and magnetoydrodynamics equations (MHD. It is widely accepted in computational fluid dynamics (CFD tat numerical scemes wic are more pysically accurate lead to more precise flow simulations especially over long time intervals. A prevalent teme trougout will be te inclusion of as muc pysical fidelity in numerical solutions as efficiently possible. In algoritm design, model selection/development, and element coice, subtle canges can provide better pysical accuracy, wic in turn provides better overall accuracy (in any measure. To tis end we develop and study more pysically accurate metods for approximating te NSE, MHD, and related systems. Capter 3 studies extensions of te energy and elicity preserving sceme for te 3D NSE developed in [64], to a more general class of problems. Te sceme is studied togeter wit stabilizations of grad-div type in order to mitigate te effect of te Bernoulli pressure error on te velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments tat demonstrate te advantages of te sceme. In Capter 4, we study a finite element sceme for te 3D NSE tat globally conserves energy and elicity and, troug te use of Scott-Vogelius elements, enforces pointwise te solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also sow te metod can be efficiently computed by exploiting a connection between tis metod, its associated penalty metod, and te metod arising from using grad-div stabilized Taylor-Hood elements. Finally, we give numerical examples wic verify te teory and demonstrate te effectiveness of te sceme. In Capter 5, we extend te work done in [8] tat proved, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to te Scott-Vogelius solution of tat same problem. In [8] even toug te analytical convergence rate was only sown to be γ (were γ is te stabilization parameter, te computational results suggest te rate may be ii

3 improvable γ. We prove erein te analytical rate is indeed γ, and extend te result to oter incompressible flow problems including Leray-α and MHD. Numerical results are given tat verify te teory. Capter 6 studies an efficient finite element metod for te NS-ω model, tat uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove te dependence of te (often large Bernoulli pressure error on te velocity error. We provide a complete numerical analysis of te metod, including wellposedness, unconditional stability, and optimal convergence. Several numerical experiments are given tat demonstrate te performance of te sceme, and ow te use of Scott-Vogelius elements can dramatically improve solutions. Capter 7 extends Leray-α-deconvolution modeling to te incompressible MHD. Te resulting model is sown to be well-posed, and ave attractive limiting beavior bot in its filtering radius and order of deconvolution. Additionally, we present and study a numerical sceme for te model, based on an extrapolated Crank-Nicolson finite element metod. We sow te numerical sceme is unconditionally stable, preserves energy and cross-elicity, and optimally converges to te MHD solution. Numerical experiments are provided tat verify convergence rates, and test te sceme on bencmark problems of cannel flow over a step and te Orszag-Tang vortex problem. iii

4 Acknowledgments I would like to tank all of my family members, professors, and friends wo elped me over te past four years. Your support and encouragement made tis possible. I especially want to tank my advisor Dr. Leo Rebolz. I owe muc of my success to im and te knowledge e sared wit me. I am also tankful for my committee members Dr. Vincent Ervin, Dr. Eleanor Jenkins, and Dr. Jeong-Rock Yoon. I ave learned a lot from teir courses as well as our conversations. I would like to tank te following people for collaborating wit me on te contents of tis dissertation: Dr. Micael Case, Benjamin Cousins, Dr. Vincent Ervin, Dr. Alexander Labovsky, Dr. Alexander Linke, Dr. Carolina Manica, Dr. Monica Neda, Dr. Maxim Olsanskii, and Dr. Leo Rebolz. I am tankful to ave a great family. My moter, fater, and sister ave supported my education and believed in my success even wen I doubted it. Lastly, I would like to tank my fiance Dr. Tara Steuber, wo made te last four years seem to go by quickly. iv

5 Table of Contents Title Page i Abstract ii Acknowledgments iv List of Tables vi List of Figures vii Introduction Preliminaries Stable Computing wit an Enanced Pysics Based Sceme for te 3D Navier- Stokes Equations Algoritms Stability, conservation laws, and existence of solutions Convergence Numerical Experiments Large scale NSE computations witout a pressure space Te Algoritm Numerical analysis of te sceme Improved efficiency troug decoupling Numerical Experiments Convergence Rate of Grad-div Stabilized TH solutions to SV solutions Order of convergence for NSE approximations Extension to turbulence models Extension to magnetoydrodynamic flows Extrapolating to approximate te γ = solution NS-Omega A numerical sceme for NS-ω Numerical experiments Leray-deconvolution for MHD A Numerical Sceme for te Leray-deconvolution model for MHD Numerical Experiments Conclusions Bibliograpy v

6 List of Tables 3. Te u NSE u, errors and convergence rates for eac of te tree elicity sceme of algoritm Te errors and rates for te velocity solution in numerical experiment. Rates appear optimal. Te Total dof column sows te total degrees of freedom required if SV elements were used Te errors and rates for te vorticity in numerical experiment, as well as te errors in te velocity and vorticity divergences Convergence of te grad-div stabilized TH Leray-α solutions toward te SV Leray-α solution, first order as γ Convergence of te grad-div stabilized TH steady MHD solutions toward te SV steady MHD solution, first order as γ Improved mass conservation using linear extrapolation Improved mass conservation using quadratic extrapolation Errors in velocity and divergence for Experiment for SV and TH elements used wit Algoritm Convergence rates for te Leray-deconvolution (N = and te Leray -α (N = models vi

7 List of Figures. A barycenter refined triangle Te velocity solution to te Etier-Steinman problem wit a =.5, d = at t = on te (, 3 domain. Te complex flow structure is seen in te stream ribbons in te box and te velocity streamlines and speed contours on te sides Te plot above sows L error of te velocity vs time for te four elicity scemes of Test. We see in te plot tat te stabilizations add accuracy to te enanced-pysics elicity sceme, and tat te altered grad-div stabilization gives sligtly better results tan te usual grad-div stabilization. It can also be seen tat te enanced-pysics elicity sceme is far more accurate in tis metric tan te usual Crank-Nicolson elicity sceme Te plot above sows elicity error vs time for te four elicity scemes of Test. We see in te plot tat elicity is far more accurate in te enanced-pysics elicity sceme, and even better wit stabilizations, tan te usual Crank-Nicolson elicity sceme Sown above is domain for te 3d cannel flow over a step problem Sown above is te x = 5 sliceplane of speed contours and velocity streamlines at T =. An eddy can be observed to ave moved down te cannel, and a second eddy as formed beind te step Sown above is a plot of te same velocity field as in Figure 4., but zoomed in near te step. Streamribbons are used to visualize te flow, and more clearly sow te eddies SV and TH solutions of te Leray-α model at t = Sown above are te T = 4 SV solutions as velocity streamlines over speed contours for te step problem from Experiment. Sown are te NSE (top wic is somewat underresolved on tis mes as te eddies are not fully detacing, NS-ω wit α = ν t (second from top wic agrees wit te known true solution, NS-ω wit α =.3 (tird from top wic as oscillations present in te speed contours, and NS-ω wit α = =.6 (bottom wic is a poor approximation. All of te solutions are pointwise divergence-free Sown above are te T = 4 SV solutions as velocity streamlines over speed contours for te step problem from Experiment, wit parameter cosen as α = ν t, for varying timesteps Te above pictures sow te velocity fields and speed contours at t = 7 using SV elements wit α = (top and α = ν t (bottom. Te α = solution is underresolved, as it loses resolution of te vortex street, and its speed contours are inaccurate. Te α = ν t solution captures te entire wake, and its speed contours agree well wit te known solution Te above picture sow te t = 7 solution using TH elements, as a velocity vector field and speed contours. Tis solution is incorrect, as it fails to capture any wake beind te cylinder vii

8 6.5 Sown above are te plots of te L norms of te divergence of te velocity solutions versus time, for te SV and TH solutions, bot wit α = ν t = Te domain and boundary conditions for cannel flow problem Te true solution Top: MHD Leray-deconvolution, Bottom: DNS of MHD, at T = 5 on mes giving 8,666 dof Current seets found wit Leray α deconvolution (N= at t= Energy and Cross elicity versus time for Orszat-Tang Vortex problem; tey are conserved viii

9 Capter Introduction Te study of fluid flow is at te forefront of many scientific fields and engineering applications suc as aerodynamics, weater predictions and modeling ocean currents. It is tempting to tink tat after centuries of study and great advances in computational power little, only little knowledge migt remain undiscovered. However, even te most fundamental matematical questions remains unanswered: te existence and uniqueness of solutions in 3d of te equations of fluid motion. Additionally, wile te advances in computer tecnology ave aided te scientific community s understanding of fluid flow, tere are still many flows wic we cannot ope to simulate wit modern computational tools. In fact, te field of computational fluid dynamics (CFD is and will continue to be stifled by current tecnologies for te foreseeable future. Tis tesis is concerned wit more accurately and efficiently predicting fluid flow in computational simulations. A prevalent teme trougout will be te inclusion of as muc pysical fidelity in numerical solutions as efficiently possible. In algoritm design, model selection/development, element coice, etc., subtle canges can provide better pysical accuracy, wic in turn provides better overall accuracy (in any measure. To tis end, we develop and study more pysically accurate metods for approximating te Navier-Stokes equations (NSE, magnetoydrodynamics (MHD, and related models. Te NSE are derived from conservation of momentum and mass, and describe te motion of incompressible viscous Newtonian fluid flows. Te equations in dimensionless form are given by u t ν u + u u + p = f, (.. u =. (..

10 Here u is te fluid velocity, p is te pressure, f is an external body force, and ν is te kinematic viscosity. Te Reynolds number (Re=ν is te only control parameter of te flow. For low Reynolds number, te viscous forces dominate, making te flow smoot and easily predictable. Flows caracterized by tese properties are referred to as laminar. Turbulent flows, in contrast, are unstable and caotic. Suc flows are a result of dominating inertial forces and correspond to ig Reynolds numbers. Coupling te NSE wit Maxwell s equations gives te MHD system, wic describe te motion of electrically conducting non-magnetic fluids, suc as salt water, liquid metals (e.g. mercury and sodium and plasmas. MHD in dimensionless form is given by u t + (uu T ν u + s (B B s BBT + p = f, (..3 u =, (..4 B t + ν m ( B + (B u = g, (..5 B =, (..6 were u, p, ν and f represent te same quantities as in (..-(.., g represents a forcing on te magnetic field B, ν m (=Re m is te inverse of te magnetic Reynolds number, and te coupling number s = Ha ReRe m, were Ha is te Hartmann number (te ratio of Lorentz forces to sear forces. In te absence of viscosity and external force te NSE conserves energy (= Ω u dx. MHD also conserves energy (= Ω ( u + s B dx in te absence of external force, kinematic viscosity and magnetic diffusivity. In addition to energy, under te respective conditions te 3d NSE conserves elicity (= Ω u ( u and te 3d MHD conserves cross elicity (= u B. More Ω recently, scientists ave come to better understand te importance of elicity and cross elicity. In 99 a famous paper of Moffatt and Tsoniber found elicity to be of comparable importance to energy in te NSE [54]. Similarly, te importance of cross-elicity in MHD is also known to be of fundamental importance [49]. Most numerical scemes wic simulate fluid flow conserve energy as tis typically coincides wit algoritm stability. However, numerical scemes often neglect oter fundamental pysical laws or only enforce te pysical laws in a weak sense. Tis is, in part, due to a perceived increase in computational cost to conserve a second integral invariant. However, it is widely accepted in CFD tat pysical fidelity in a numerical sceme produces more accurate predictions, especially over longer time intervals. Tus, numerical scemes wic correctly account for pysical quantities in addition to energy can give solutions wic are not only more accurate, but also more pysically

11 relevant. Examples of suc scemes include Arakawa s energy and enstropy conserving sceme for te D NSE [] and related extensions [], energy and potential enstropy scemes pioneered by Arakawa and Lamb, and Navon, [, 55, 56], an energy and elicity conserving sceme for 3D axisymmetric flow by J.-G. Liu and W. Wang [49], and most recently a 3d sceme for te full NSE wit periodic boundary conditions [64]. A pysical law wic is commonly enforced weakly in te NSE is conservation of mass. Pysically, (.. is interpreted to mean mass is a pointwise conserved quantity in incompressible fluid flows. However, finite element numerical scemes often only impose global mass conservation weakly (e.g. Ω ( u q =, were u is te discrete velocity solution, and for eac q in te approximating pressure space. Depending on te element coice, tis can lead to mass conservation being very inaccurate [8]. Lack of pysical fidelity in suc a fundamental quantity callenges te pysical relevance of a numerical solution. One common element wic can lead to poor mass conservation is te Taylor-Hood (TH element. Te TH element uses continuous polynomials of degree k to approximate velocity and continuous polynomials of degree (k to approximate pressure. It was found in [4, 59, 6] tat adding te identically zero term γ ( u at te continuous level could improve mass conservation wen using TH elements. Te tecnique is referred to as grad-div stabilization and owes its success to a sarp energy bound. By adding te zero term at te continuous level, te usual [37] discrete model satisfies M u M + ν t n= u n+ M + γ t Tus, as γ increases, mass conservation will improve. n= u n+ C(data. Similar to te TH element, te Scott-Vogelius (SV element [68] uses continuous polynomials of degree k to approximate velocity. Unlike te TH element te SV element uses discontinuous polynomials of degree (k to approximate te pressure. Tis modest cange enforces pointwise mass conservation but sligtly increases te number of degrees of freedom in te system. It is known tat SV elements are LBB stable and admit optimal convergence properties under te mild restrictions tat k d, were d is te dimension of Ω, and solutions are computed on a barycenter refined mes [76] (e.g. see Figure.. A connection as been establised between te SV element solutions and te grad-div stabilized TH element solutions to te NSE [8]. On a fixed mes, as γ te TH solution tends to te SV solution; tat is, u T H u SV. Te autors ave sown an analytic convergence rate 3

12 Figure.: A barycenter refined triangle of γ. However, numerical experiments int at a faster convergence rate of γ. In Capter 5 we extend te connection between te SV element solutions and te grad-div stabilized TH solutions and sow an analytic convergence rate of γ for Stokes type problems. Te sacrifice of conserving pysical properties is in part motivated by reducing te number of degrees of freedom required to simulate flow problems. Turbulent flows especially devour computational resources. According to te Kolmogorov estimate, to solve NSE accurately, a mes must contain approximately Re 3 discretization points in d and Re 9 4 discretization points in 3d. Tus, direct numerical simulation of turbulent flows is expensive! Recent work on FEM for te α models [43, 44, 5, 6, 66, 5,,, 8] as sown teir effectiveness at producing accurate solutions to te NSE and te MHD on significantly coarser meses tan direct simulations require. One of te α models, called Navier-Stokes-ω (NS-ω appears particularly promising, and is given by u t + ( D N F u u + q ν u = f, (..7 u =. (..8 Here F denotes te Helmoltz filter: F := ( α + I were α > is te filtering radius, and D N is te N t order van Cittert approximate deconvolution operator, D N := N n= (I F n. Tis model is of particular interest because, in addition to energy conservation, te model conserves a model elicity. Moreover, it is an analog of NS-α, but can more efficiently be computed. Solving NS-ω wit SV elements enforces pointwise mass conservation in te sceme and gives more pysical significance to our solutions (Capter 4, [5]. Simulations for MHD flows require te same number of mes points as simulations for NSE flows to be successfully resolved. However, MHD flows contain an additional unknown, te magnetic field, making MHD flows more complex tan NSE flows. Tus, modeling is imperative for MHD flows [, 33, 35, 39, 75]. We extend te work done by Yu and Li on te MHD Leray-α model [75] 4

13 to te MHD Leray-deconvolution model wic is given by u t Re u + D N u u sd N B B + p = f, B t Re m B + D N u B D N B B = g, u =, B =. We perform an analytic study of te continuous model, and te numerical sceme presented for te model, and provide numerical experiments wic sow its effectiveness on bencmark problems (Capter 6. We remark tat te discretized MHD equations are overdetermined and ence require a Lagrange multiplier, wic leads to a second solenoidal constraint in te numerical sceme. Tus, te SV element is crucial in simulating MHD flows. Despite enforcing solenoidal constraints pointwise te SV element presents obstacles. Specifically, te SV element as a larger pressure space tan te TH element; in fact te SV pressure space can be comparable in size to te velocity space. Tus, efficiency is a concern wen using te SV element. Efficiency may be improved using Temam s penalty metod [69] as discussed in [65]. We study a finite element sceme for te 3D NSE tat employs te SV element and Temam s penalty metod tat conserves energy and elicity (Capter 7. Te body of tis tesis is comprised of seven capters. Capter presents matematical preliminaries and notations wic are used trougout te report, including te definitions of te TH and SV elements. Capter 3 extends te work done in [64] to te more general omogenous boundary conditions. We study te sceme wit two types of stabilization and sow tat te sceme is stable, conserves energy and under mild conditions conserves model elicity. Lastly, we give evidence of te advantages of te sceme by simulating a complex flow problem and comparing against te usual Crank-Nicolson sceme for NSE. Capter 4 presents a study of te NS-ω turbulence model wit SV elements. Cannel flow around a cylinder and over a step are bencmark flow problems wic we simulate to sow improved mass conservation using SV rater tan TH elements. Capter 5 extends te work done in [8] to sow analytically grad-div stabilized TH solutions to Stokes type problems converge to te SV solutions wit rate γ. Capter 6 presents te MHD Leray-deconvolution model and provides analysis of te continuous model, te numerical sceme presented for te model, and solutions to te bencmark problems of cannel flow over a step and te Orszag-Tang vortex problem. Capter 7 presents a numerical sceme for NSE and an analytic study of te sceme wic is based on Temam s penalty 5

14 metod and leads to increased efficiency in 3D computations of te bencmark problem of cannel flow over a step. 6

15 Capter Preliminaries For te analysis in tis tesis we consider eiter periodic or no slip boundary conditions. For periodic boundary conditions we assume te domain Ω is te L-periodic box (, L d, and for no slip boundary conditions we assume Ω denotes a bounded, polyedral domain in R d (d=, or 3. Te L (Ω norm and inner product will be denoted by and (,. Likewise, te L p (Ω norms and te Sobolev W k p (Ω norms are denoted by L p and W k p, respectively. For te semi-norm in W k p (Ω we use W k p. H k is used to represent te Sobolev space W k (Ω, and k denotes te norm in H k. For functions v(x, t defined on te entire time interval (, T, we define ( m < v,k := ess sup v(t, k, and v m,k := <t<t ( /m T v(t, m k dt. In te discrete case we use te analogous norms: v,k := max v n k, v /,k := max v n+/ k, n M n M ( /m ( M M /m v m,k := t v n m k, v / m,k := t v n+/ m k. n= n= For s R we recall te spaces H s and H s p H s (Ω := {w H s (Ω, w = on Ω}, and H s p(ω := {w H s loc (Rd, L-periodic, w =, Ω w = }. Te space X is H p(ω in te periodic boundary case and (H d for Diriclet boundary conditions. We use as te norm on X te H seminorm wic, because of te boundary condition, is equivalent to te H norm; for v X, v X := v. We denote te dual space of X by X, wit 7

16 te norm. We denote te pressure space by Q = L (Ω := {q L (Ω : q = } corresponding Ω to Diriclet boundary conditions and in te periodic case Q = L,p(Ω := {q L p : q = }. Te Ω space of weakly divergent free functions, V := {v X : ( v, q =, q Q}, and te dual space of V will be denoted by V. For TH elements, (X, Q is made of ((P k d, P k, k velocity/pressure elements. Tus we ave, for a given regular mes T, and omogenous Diriclet boundary conditions, X := { v : v e P k (e, e T, v [C (Ω] d, v Ω = }, Q := { q : q e P k (e, e T, q C (Ω, q L (Ω }. For periodic boundary conditions te spaces are defined by X := { v : v e P k (e, e T, v [C (Ω] d, v Ω = periodic }, Q := { q : q e P k (e, e T, q C (Ω, q L p(ω }. In addition to te TH approximation spaces we define te SV velocity and pressure approximation spaces. Let T denote te mes tat ensures te SV pair is LBB stable, for example: (A in d, k 4 and te mes as no singular vertices [63], (A in 3d, k 6 [77], (A3 wen k d and te mes is a barycenter refinement of a regular mes [76, 63], or (A4 on Powell-Sabin meses wen k = and d = or wen k = and d = 3 [78]. We note tat te SV velocity approximation space is te same as te TH velocity approximation space and define te SV pressure space to be Q SV := {q L (Ω : q T P k T T }, or Q SV := {q L p(ω : q T P k T T }. Since it is discontinuous, te dimension of te pressure space for SV elements is significantly larger tan for TH elements. Tis creates a greater total number of degrees of freedom needed for linear solves using SV elements. Altoug te velocity spaces of te TH and SV elements are te same, te spaces of discretely divergence free subspaces are different, and tus we denote te TH and SV 8

17 discretely divergence free subspaces respectively as V := {v X : ( v, q = q Q }, V SV := {v X SV : ( v, q = q Q SV }. Te SV elements are very attractive from te mass conservation point of view since teir discrete velocity space and discrete pressure space fulfill an important property, namely X SV Q SV. Tus, using SV elements, weak mass conservation via ( v, q =, q Q SV implies strong (pointwise mass conservation, since v = by coosing q = v. Suc a result, or coice of test function, is not possible wit many element coices, including TH. Capter 3 and Capter 4 make use te curl of te velocity, wic is known as vorticity. Tese capters prescribe omogenous Diriclet boundary conditions for te velocity, and so we do not define a discrete vorticity space in te context of periodic boundary conditions. However, we use a more general space for te discrete vorticity space wen velocity is prescribed no slip boundary conditions. Even toug te velocity satisfies omogeneous Diriclet boundary conditions, it is believed to be inappropriate to enforce omogeneous Diriclet boundary conditions for te vorticity. We coose te boundary condition to be a no-slip boundary condition along te boundary, and ence we define te space W := { v : v [C (Ω] 3, e T (v e P k (e, v n Ω = } X. We use t n := n t, and for bot continuous and discrete functions of time v n+ := v((n + t + v(n t. 9

18 For te convergence studies, we make use of te following approximation properties: inf u v C(β v V SV inf u v C k+ u k+, u (H k+ (Ω d, v X SV inf u v C k u k+, u (H k+ (Ω d, v X SV inf p r C s+ p s+, p H s+ (Ω r Q SV For te analysis in tis tesis we use tree trilinear operators wic are defined below along wit teir respective bounds. Te first trilinear operator is used wen te nonlinear term in te NSE is in rotational form and is given by Definition... Define b : X X X R, by b (u, v, w := (( u v, w. Te following bounds will be used often in our analysis Lemma... For u, v, w X, or L (Ω and u L (Ω, wen indicated, te trilinear term b(u, v, w satisfies b (u, v, w u v w, (.. b (u, v, w u v w, (.. b (u, v, w C (Ω u v w, (..3 b (u, v, w C (Ω v / v / u w, (..4 and if u, v, w V and w (H (Ω d, ten b (u, v, w C w v u (..5 Proof. Te first two estimates follow immediately from te definition of b. Te proof of te next two bounds are easily adapted from te usual bounds of te nonlinearity in non-rotational form. Te last bound takes more work. Begin wit a simple vector identity and tat te curl is self adjoint

19 wit u, v, w X: (( u v, w = (w v, u = ( (w v, u (..6 Continuing wit anoter vector identity for te curl of te cross product of two vectors, (w v = v w w v + ( vw ( wv, (..7 wic reduces, since v, w V, to (w v = v w w v. (..8 Combining tis wit (..6, we ave by Holder and Poincare s inequalities, (( u v, w (v w, u + (w v, u C w v u. (..9 by Te second trilinear operator is te skew-symmetric operator b : X X X R is defined b (u, v, w := (u v, w (u w, v. Te following bounds on b will be used. Lemma... Tere exists a constant C s dependent only on te size of Ω suc tat u, v, w X, b (u, v, w C s u v w b (u, v, w C s u v w / w / Proof. Tis well known lemma is proven, e.g., in [37]. We define te final trilinear operator,b 3 (,,, wic will be used in Capter 7. Definition... Define te trilinear operator b 3 : L (Ω W (Ω L (Ω R as b 3 (u, v, w = (u v, w,

20 wenever te integral exists finitely. Lemma..3. Let n denote te dimension of Ω. Ten te form b 3 (,, is trilinear continuous on H m (Ω H m+ (Ω H m3 (Ω were m i i and m + m + m 3 n if m i n, or m + m + m 3 > n if m i = n for some i. Proof. Tis operator as been studied extensively and is well known in te teory of te NSE. Te interested reader is referred to [7]. Corollary... Te following inequalities will be utilized trougout Capter 7. For every u, v, w H p(ω b 3 (u, v, w C u v w. (... For all u, v, w H p(ω b 3 (u, v, w = b 3 (u, w, v, (.. wic implies tat b 3 (u, v, v =. (.. 3. For all u H p(ω, v H p(ω and w H p(ω, b 3 (u, v, w u v w. (..3 Proof. Tese are well known identities follow immediately from Lemma..3. Te following two lemmas are also employed in te studies of te finite element analysis. Lemma..4. Assume u C (t n, t n+ ; L (Ω. If u is twice differentiable in time and u tt L ((t n, t n+ Ω ten u n+/ u(t n+/ tn+ 48 ( t3 u tt dt. (..4 t n

21 If u t C (t n, t n+ ; L (Ω and u ttt L ((t n, t n+ Ω ten u n+ u n t u t (t n+/ tn+ 8 ( t3 u ttt dt and (..5 t n if u C (t n, t n+ ; L (Ω and u tt L ((t n, t n+ Ω ten (u n+/ u(t n+/ ( t3 48 tn+ t n u tt dt. (..6 Te proof of Lemma..4 is based on te Taylor expansion wit remainder. It is more of tecnical nature and terefore omitted erein. Te analysis in tis tesis uses two discrete Gronwall inequalities, recalled from [37], for example, and a continuous Gronwall inequality. Tese are given below. Lemma..5 (Discrete Gronwall Lemma (version. Let t, H, and a n, b n, c n, d n (for integers n be finite nonnegative numbers suc tat Ten for t > l l l a l + t b n t d n a n + t c n + H for l. (..7 n= n= n= ( l l l a l + t b n exp t d n ( t c n + H n= n= n= for l. (..8 Lemma..6 (Discrete Gronwall Lemma (version. Let t, H, and a n, b n, c n, d n (for integers n be finite nonnegative numbers suc tat l l l a l + t b n t d n a n + t c n + H for l. (..9 n= n= n= Suppose tat td n < n. Ten, for t > ( ( l l d n a l + t b n exp t t td n n= n= l c n + H n= for l. (.. Lemma..7. (Continuous Gronwall inequality Let f(x and B(x be functions wic are piecewise continuous on te interval [a, b] and let K be a nonnegative scalar. Furter, assume tat f(x 3

22 and B(x satisfy t [a, b] t g(sds + f(t K + t a a B(sf(sds. (.. Ten, t [a, b] we ave te following upper bound s a g(sds + f(t Ke t a B(sds. (.. Lemma..8. (Aubin-Lions Lemma. Let X, X, and X be Banac spaces suc tat X X X. Suppose tat X is compactly embedded in X and tat X is continuously embedded in X. Additionally, assume tat X and X are reflexive spaces. For < p, q <, let W := {u L p ([, T ]; X t u L q ([, T ]; X }. Ten te embedding of W into L p ([, T ]; X is also compact. Define R to be te ortogonal complement of V SV product. Tat is in V wit respect to te (, inner V =: V SV R, Lemma..9. Tere exists a constant M < satisfying r R, r M r. Proof. Define M := max v R, v = v Observe M < since v R, v >, and te max is taken over a compact set of R n. For any r R, tere is an e R satisfying e = and r = r e. Taking divergence of bot sides, ten L norms gives r = r e. Rearranging and using te definition of M finises te proof. 4

23 Since we study discretizations of a fluid model, we must deal wit discrete differential filters. Continuous differential filters were introduced into turbulence modeling by Germano [4] and used for various models and regularizations [3, 5, 8]. Tey can arise, for example, as approximations to Gaussian filters of ig qualitative and quantitative accuracy []. Definition..3 (Continuous Helmoltz-filter. For v (L (Ω d and α > fixed, denote te filtering operation on v by v, were v is te unique solution in X to α v + v = v. (..3 We denote by F := ( α + I, so F v = v. We define next te discrete differential filter following Manica and Kaya-Merdan [53], but also enforcing incompressibility: Definition..4 (Discrete Helmoltz filter. Given v (L (Ω d, for a given filtering radius α >, v = F v is te unique solution in X SV of: Find (v, λ ( X SV, Q SV satisfying α ( v, χ + (v, χ (λ, χ + ( v, r = (v, χ (χ, r ( X SV, Q SV. (..4 Remark... Te definition of te discrete Helmoltz filter is defined using SV elements. Te definition of te discrete filter using TH elements is a straigt forward extension and so we omit it. by data. We next introduce te following lemma wic bounds te solution to te filtered problem Lemma... For v X, we ave te following bounds v v, v v and v v. (..5 Proof. Te proof can be found in [44]. We now define te van Cittert approximate deconvolution operators. Definition..5. Te continuous and discrete van Cittert deconvolution operators D N and D N are N N D N v := (I F n v, DNv := (Π F n v. (..6 n= n= were Π denotes te L projection Π : (L (Ω d X. 5

24 For order of deconvolution N =,,, 3 and v X we ave D v = v, D v = v v, D v = 3v 3v + v, D3 v = 4v 6v + 4v v. D N was sown to be an O(α N+ approximate inverse to te filter operator F in Lemma. of [6]. Te proof is an algebraic identity and olds in te discrete case as well, giving te following. Lemma... D N and D N are bounded, self-adjoint positive operators. For v (L (Ω d, v = D N v + ( (N+ α N+ N+ F (N+ v and v = D Nv + ( (N+ α N+ N+ Lemma... For v X, we ave te following bounds F (N+ v D N v C(N v. (..7 Proof. Te proof follows from an inductive argument based on te definition of te deconvolution operator D N and Lemma... Lemma..3. For smoot φ te discrete approximate deconvolution operator satisfies N+ v DNv Cα N+ N+ F N+ v + C(α k + k+ ( F n v k+. (..8 Tis is proven in [43]. Te dependence of te F n (v k+ terms in (..8 upon te filter radius α, for a general smoot function φ, is not fully understood for deconvolution order N (i.e. for n [7, 36]. In te case of v periodic F n (v k+ is independent of α. Also, for v satisfying omogeneous Diriclet boundary conditions, wit te additional property tat j v = on Ω for j [ ] k+, te F n (v k+ are independent of α. Our analysis of te metod is for general N, and tus for N, n= we make tis assumption of independence. However, our computations are for N =, and our experience as sown tere is typically little or no gain for larger N wit polynomials approximating 6

25 velocities wit degree tree or less. For elements wit iger order polynomials, we would expect a difference. Te Leray-deconvolution model first filters te velocity and ten approximately unfilters it. Hence we are interested in properties of te operator (D N F, wic is denoted by H N by Layton and Lewandowski [4], and we continue tis notation. Definition..6. Te operator H N : H p(ω H p(ω is defined by H N w := (D N F w. analysis. Next we list some properties of te operator H N proved in [4], wic will be used in our Lemma..4. Let s R be nonnegative. Ten. For s H N maps H s p into itself compactly.. If w H s p(ω ten H N (w H s+ p (Ω. Tat is H N w s+ C(α, N w s. 3. H N commutes wit te gradient operator, i.e. H N ( = H N (. 4. If w H s p(ω ten H N (w w strongly in H s p(ω wen α is fixed and N. Remark... Te constant, C(α, N, in can go to infinity as α or as N. 7

26 Capter 3 Stable Computing wit an Enanced Pysics Based Sceme for te 3D Navier-Stokes Equations In tis capter we extend te enanced-pysics based elicity sceme of [64] to omogeneous Diriclet boundary conditions for te velocity wit appropriate stabilizations. We propose tree numerical scemes tat are extensions of te elicity conserving sceme studied by Rebolz. Te first sceme we propose is a direct extension of te enanced pysics based sceme to omogenous Diriclet boundary conditions (i.e. we do not stabilize te sceme. Te oter two scemes are stabilized. Te first sceme is stabilized using te usual grad-div stabilization [6], and te second is stabilized using a modified grad-div stabilization term. Te usual grad-div stabilization term in our numerical sceme is derived by adding te identically term γ ( u to te continuous NSE. In te numerical sceme te term penalizes for lack of discrete mass conservation, and nullifies te effect of te pressure error on te velocity error [4, 5, 6]. Te model we study uses te rotational form of te NSE nonlinearity and so te pressure used is te more complex Bernoulli pressure, wic can adversely effect te velocity error. Tte modified grad-div stabilization term is derived similarly by adding te identically term γ ( u t to te continuous NSE. Te modified grad-div stabilization also penalizes for lack of discrete mass conservation, and nullifies te effect of te pressure error on te velocity error. Te use of stabilization is not witout potential drawbacks. Adding te stabilization terms to te numerical sceme canges te energy balance, wic is not ideal but often advantageous 8

27 in practice. We sow tat te modified grad-div stabilization terms alters te energy balance less tan te typical grad-div stabilization term, and provide a numerical experiments tat sows te improvement wen te stabilization terms are used. 3. Algoritms Algoritm 3.. (Enanced-pysics based elicity scemes for omogeneous Diriclet boundary conditions. Given a time step t >, finite end time T := M t, and initial velocity u V, find w W and λ Q satisfying (χ, r (W, Q (w, χ + (λ, χ = ( u, χ, (3.. ( w, r =. (3.. Ten for n =,,..., M, find (u n+, w n+, p n+, λ n+ (X, W, Q, Q satisfying (v, χ, q, r (X, W, Q, Q ( un+ u n t, v + STAB (p n+, v +(w n+ u n+, v + ν( u n+, v = (f(t n+, v (3..3 ( u n+, q = (3..4 (w n+, χ + (λ n+, χ = ( u n+, χ (3..5 ( w n+, r =. (3..6 were STAB = elicity sceme γ( u n+, v elicity sceme γ t ( (un+ u n, v elicity sceme 3 Remark 3... We ave found it computationally advantageous to decouple te 4 equation system (3..3-(3..6 into a velocity-pressure system (3..3-(3..4 and a projection system (3..5-(3..6, ten solve (3..3-(3..6 by iterating between te two sub-systems. Tis typically requires more iterations and linear solves to converge tan solving te fully-coupled system using a Newton metod. However te linear solves are muc easier in te decoupled system. Note also tat for te decoupled system te work required is only sligtly more tan a usual implicit Crank-Nicolson metod (i.e. witout vorticity projection since te extra work is (relatively inexpensive projection solves. Moreover, 9

28 for nonomogeneous boundary conditions, tis decoupling leads to a simplified boundary condition for te vorticity: w = I ( u on te boundary, were I is an appropriate interpolation operator. 3. Stability, conservation laws, and existence of solutions In tis section we prove fundamental matematical and pysical properties of te 3 elicity scemes: unconditional stability, solution existence and conservation laws. We begin wit stability. 3.. Stability Lemma 3... Solutions to Algoritm 3.. are nonlinearly stable. Tat is, tey satisfy: elicity sceme : elicity sceme : u M + ν t M n= u n+ t ν M n= M u M ( + t γ u n+ + ν u n+ t ν n= elicity sceme 3: elicity scemes,,3: u M + γ u M + ν t t M t ν w n+ n= M n= M n= t u n+ f + u = C(data. (3.. M n= f + u = C(data. (3.. f + u + γ u = C(data. (3..3 M n= u n+ = C(data. (3..4 elicity scemes,,3: t M n= ( p n + λ n C(data. (3..5 C(data is a constant dependent on T, ν, γ, f, u and Ω. Proof. To prove te bounds on velocity for eac of te elicity scemes, coose v = u n+ in (3..3. Te nonlinear and pressure terms are ten zero. Te triangle inequality and summing over time steps ten completes te proofs of (3..,(3..,(3..3.

29 To prove (3..4 coose χ = w n+ in (3..5 and r = λ n+ in (3..6. After combining te equations we obtain w n+ = ( u n+, w n+ u n+ u n+ w n+ + w n+ u n+ + w n+. Rearranging, and summing over time steps we obtain (3..4. To obtain te stated bound for λ n, we begin wit te inf-sup condition satisfied by X ( W and Q and use (3..5 to obtain λ n β sup (λ n, χ χ X χ X β sup ( u n + w n β β n ( u χ X ( u n, χ (w n, χ χ X. + w n Using te bounds for u n+ (see (3..-(3..3 and w n+ λ n. Te bound for te pressure is establised in an analogous manner. (see (3..4 we obtain te bound for 3.. Existence Lemma 3... Solutions exist to eac of te tree elicity scemes presented in Algoritm 3... Proof. For eac of te elicity scemes, tis is a straigt-forward extension of te existence proof given for te periodic case in [64]. Te result is a consequence of te Leray-Scauder fixed point teorem and te stability bounds of Lemma 3... We now study te conservation laws for energy and elicity in te elicity scemes. It is sown in [64] tat, wen restricted to te periodic case, te non-stabilized elicity sceme of Algoritm 3.. (elicity sceme conserves energy and elicity. In te case of omogeneous boundary conditions for velocity, tis pysically important feature for energy is still preserved. However, as one migt expect, te stabilization term in elicity scemes and 3 alters te energy balance. Lemma 3..3 sows tese energy balances. Te energy balance of elicity sceme, te unstabilized elicity sceme, is analogous to tat for te continuous NSE. However, for elicity sceme, we see te effect of te stabilization on te energy balance in te term γ t M n= u n+ on te left and side of (3..7. For most coices of elements, one may ave tat eac term in tis sum is small, but over a long time interval tis sum can grow to significantly (and non-pysically alter te balance. Te energy balance for

30 elicity sceme 3 differs from elicity sceme s energy balance in te addition of only two small terms, instead of a sum. Hence tis indicates tat te modified grad-div stabilization, for problems over a long time interval, offers a more pysically relevant energy balance tan te usual grad-div stabilization (elicity sceme Conservation Laws Lemma Te elicity scemes of Algoritm 3.. admit te following energy conservation laws: elicity sceme : u M + ν t M n= u n+ = t M n= (f(t n+, u n+ + u. (3..6 elicity sceme : M u M + ν t u n+ + γ t n= M n= u n+ = t M n= (f(t n+, u n+ + u. (3..7 elicity sceme 3: M ( u M + γ u M + ν t u n+ = t n= M n= (f(t n+, u n+ + ( u + γ u. (3..8 Proof. Te proofs of tese results follow from coosing v = u n+ in Algoritm 3.. for eac of te elicity scemes. Te key point is tat te nonlinear term vanises wit tis coice of test function, and tus does not contribute to te energy balance equations. We now consider te discrete elicity conservation in Algoritm 3... We begin wit te case of imposing Diriclet boundary conditions on te projected vorticity, i.e. W = X. Altoug tis case is nonpysical, analysis of it is te first step in understanding more complex boundary conditions. In tis case, te elicity scemes discrete nonlinearity preserves elicity, owever te sta-

31 bilization terms do not. We state te precise results in te next lemma. Denote te discrete elicity at time level n by H n := (un, un. Note tat from (3..4,(3..5, Hn := (un, wn. Lemma If W := X, te elicity scemes of Algoritm 3.. admit te following elicity conservation laws. elicity sceme : H M M + ν t ( u n+ n=, w n+ M = ν t n= (f(t n+, w n+ + H. (3..9 elicity sceme : H M M + ν t ( u n+ n=, w n+ M + γ t n= ( u n+, w n+ M = t (f(t n+ n+, w + H. (3.. n= elicity sceme 3: H M M + ν t ( u n+ n=, w n+ M + γ n= ( (u n+ u n, w n+ M = t (f(t n+ n+, w + H. (3.. n= Proof. Coosing v = w n+ elimates te nonlinear term and te pressure term from (3..3 for eac of te 3 elicity scemes, and reduces te time difference term to = t u n, w n+ = t (un+ u n, u n+ ( (u n+, u n+ + (u n+, u n (u n, u n+ (u n, u n t (un+ = t ( H n+ H n, (3.. as, for v, w H (Ω, (v, w = (w, v. Using (3.. elicity sceme becomes ( H n+ H n n+ + ν( u, w n+ = (f(t n+ n+, w (3..3 t 3

32 Multiplying by t and summing over time steps completes te proof of (3..9. Te proofs of (3.. and (3.. follow te same way, except tey will contain teir respective stabilization terms. Lemma 3..4 sows tat if we impose Diriclet boundary conditions on te vorticity, ten te nonlinearity is able to preserve elicity. Hence for elicity sceme, we see a elicity balance analogous to tat of te true pysics. However, te stabilization terms do not preserve elicity, and tus appear in te elicity balances for elicity scemes and 3. Interestingly, if te term γ( w n+, χ is added to te left and side of te vorticity projection equation (3..5, one can sow tat elicity sceme 3 conserves bot elicity and energy. Tis results from te cancellation of te stabilization term in elicity sceme 3 s momentum equation wen v is cosen to be w n+ and χ is cosen as u n+ and u n respectively. However, computations using tis additional term wit elicity sceme 3 were inferior to tose of elicity sceme 3 defined above. Similar conservation laws for elicity, even for elicity sceme, do not appear to old for te nonomogeneous boundary condition for vorticity, i.e. X W. Due to te definitions of tese spaces, extra terms arise in te balance tat correspond to te difference between te projection of te curl into discretely divergence-free subspaces of W and X. Tese extra terms will be small except at strips along te boundary, but noneteless global elicity conservation will fail to old. However, more typical elicity scemes, e.g. usual trapezoidal convective form or rotational form [37], introduce nonpysical elicity over te entire domain and tus te elicity scemes of Algoritm. still provide a better treatment of elicity tan suc elicity scemes. 3.3 Convergence Tree numerical elicity scemes are described in Algoritm.. We prove in detail convergence of solutions of elicity sceme 3 to an NSE solution. Convergence results for elicity scemes and can be establised in an analogous manner. Let P V : L V denote te projection of L onto V, i.e. P V (w := s were (s, v = (w, v, v V. For simplicity in stating te a priori teorem we summarize ere te regularity assumptions for te solution u(x, t to te NSE. 4

33 u L (, T ; H k+ (Ω L (, T ; H (Ω, (3.3. u(, t H (Ω, (3.3. u t L (, T ; H k+ (Ω L (, T ; H k+ (Ω, (3.3.3 u tt L (, T ; H k+ (Ω, (3.3.4 u ttt L (, T ; L (Ω (3.3.5 (u ( u tt L (, T ; L (Ω. (3.3.6 Teorem For u, p solutions of te NSE wit p L (, T ; H k (Ω, u satisfying (3.3.- (3.3.6, f L (, T ; X (Ω, and u V, (u n, wn given by elicity sceme 3 of Algoritm. for n =,..., M and t sufficiently small, we ave tat u(t u M + (u(t u M ( + ν t M n= / (u n+ n+ u C(γ, T, ν 3, u ( k u(t k+ + k u,k+ + k p,k + k u t,k+ + k u t,k+ + k u t, u,k+ + ( t / k u tt,k+ + ( t u ttt, + ( t u tt, + ( t (u ( u tt, + k+ u, u,k+. (3.3.7 Proof of Teorem. Since (u, p solves te NSE, we ave v X tat (u t (t n+, v (u(t n+ ( u(t n+, v (p(t n+, v + ν( u(t n+, v = (f(t n+, v. (3.3.8 Adding ( un+ u n t, v and ν( u n+, v to bot sides of (3.3.8 we obtain t (un+ u n, v + (( u(t n+ u(t n+, v (p(t n+, v + ν( u n+, v ( = (f(t n+ u n+ u n, v + u t (t n+, v + ν( u n+ u(t n+, v. (3.3.9 t Next, subtracting (3..3 from (3.3.9, label e n := u n u n, and adding te identically zero term 5

34 γ t ( ( un+ u n t, v to te LHS gives t (en+ e n, v + ν( e n+, v + γ t ( (en+ e n, v ( ( = ( u(t n+ u(t n+, v + w n+ u n+, v + p(t n+ p n+, v ( u n+ u n + u t (t n+, v + ν ( u n+ u(t n+, v. (3.3. t We split te error into two pieces Φ and η: e n = u n u n = (un U n + (U n u n := ηn + Φ n, were U n denotes te interpolant of u n in V, yielding t (Φn+, v + γ ( (Φn+ t Φ n, v + ν( Φ n+ Φ n, v = t (ηn+ η n, v ν( η n+, v γ t ( (ηn+ η n, v (( u(t n+ u(t n+, v ( + (w n+ u n+, v + (p(t n+ p n+ u n+ u n, v + u t (t n+, v t + ν( u n+ u(t n+, v. (3.3. Coosing v = Φ n+ yields ( Φ n+ Φ n t + ν Φ n+ + γ ( Φ n+ Φ n t = t (ηn+ η n, Φ n+ ν( η n+ n+, Φ γ ( (η n+ η n, Φ n+ t ( u(t n+ u(t n+ n+, Φ + (w n+ u n+, Φ n+ + (p(t n+ p n+, Φ n+ ( u n+ u n + u t (t n+ n+, Φ + ν( u n+ u(t n+ n+, Φ. (3.3. t We ave te following bounds for te terms on te RHS (see [8]. ν( η n+ n+, Φ ν Φ n+ + 3ν η n+ (

35 t (ηn+ η n, Φ n+ η n+ η n t + Φ n+ = ( t n+ η t dt dω + Φ n+ Ω t t n ( t n+ η t (t n+ + η tt dt dω + Similarly, Ω t n = η t (t n+ + t n+ Φ n+ t n η tt dt + Φ n+. (3.3.4 γ ( (η n+ η n, Φ n+ t γ ηt (t n+ t n+ + γ η tt dt + γ Φ n+. t n (3.3.5 ( u n+ u n t u t (t n+, Φ n+ u n+ u n u t (t n+ + t t n+ = ( t3 u ttt dt + 56 t n ν( u n+ u(t n+ n+, Φ 3ν u n+ ν u(t n+ ν + = ν( t3 6 t n+ t n u tt dt + ν Φ n+ Φ n+ (3.3.6 Φ n+ Φ n+ (3.3.7 (3.3.8 For te pressure term, since Φ n+ V, for any q Q, (p(t n+ p n+, Φ n+ = (p(t n+ q, Φ n+, (3.3.9 wic implies (p(t n+ p n+, Φ n+ γ p(t inf n+ γ q + q Q Φ n+. (3.3. 7

36 Utilizing (3.3.3-(3.3. we now ave t ( Φ n+ Φ n + 3ν η n+ + γ t ( t n+ + C( + ν t 3 For te nonlinear terms we ave γ t ( Φ n+ Φ n + 5ν 6 ηt (t n+ + t n+ u ttt dt + t n t n ( + (w n+ u n+, Φ n+ t n+ Φ n+ γ η tt dt + p(t t t γ inf n+ q n q Q u tt dt + ν + Φ n+ + γ Φ n+ ( u(t n+ u(t n+, Φ n+ η t (t n+ + t n+ t n η tt dt. (3.3. (w n+ u n+, Φ n+ = ( ( u(t n+ u(t n+ n+, Φ ( ( (w n+ u n+ u n+ n+, Φ ( = ( u n+ u n+, Φ n+ + ( (w n+ u n+ u n+ n+, Φ ( ( + ( u n+ u n+ n+, Φ ( + w n+ (u n+ u n+ n+, Φ ( u n+ u n+ ( u(t n+ u(t n+, Φ n+ + ( w n+ η n+ n+, Φ ( u n+ u n+ ( u(t n+ u(t n+, Φ n+ (3.3. We bound te second to last and last terms in (3.3. by (w n+ η n+ n+, Φ C w n+ η n+ Φ n+ ν Φ n+ + 3ν w n+ η n+ (3.3.3 (u(t n+ ( u(t n+ u n+ ( u n+ n+, Φ ν Φ n+ + 3ν u(t n+ ( u(t n+ u n+ ( u n+ ν Φ n+ + 3 t n+ 48 ν ( t 3 (u ( u tt dt. (3.3.4 t n For te first term in (3.3., we need a bound on u n+ w n+. Tis is obtained by restricting χ to V in (3..5 and ten subtracting ( u n+, χ from bot sides of (3..5, wic 8

37 gives us ( u n+ w n+, χ = ( (u n+ u n+, χ = ( η n+, χ + ( Φ n+, χ. By te definition of P V, (P V ( u n+ w n+, χ = ( u n+ w n+, χ = ( (u n+ u n+, χ = ( η n+, χ + ( Φ n+, χ Coosing χ = P V ( u n+ w n+ we obtain ( η P V ( u n+ n+ w n+ + Φ n+. (3.3.5 Now using (3.3.5 and, from Poincare s inequality, Φ n+ ( (P V ( u n+ n+ w u n+ n+, Φ C u n+ P V ( u n+ n+ w ( η C u n+ n+ Φ n+ + C Φ n+ we obtain Φ n+ Φ n+ n+ Φ Φ n+ 3 ν Φ n+ + Cν u n+ η n+ + ν Φ n+ + Cν 3 u n+ 4 Φ n+. (3.3.6 Also, we ave tat ( ( u n+ PV ( u n+ u n+ n+, Φ C u n+ PV ( u n+ u n+ Φ n+ ν Φ n+ + C u n+ u n+ PV ( u n+ (3.3.7 ( Combining (3.3.7 and (3.3.6 we obtain te required bound for Noting tat Φ n+ / ( Φ n+ (w n+ u n+ u n+, Φ n+ + Φ n, substituting te bounds derived in. 9

38 (3.3.3, (3.3.4, (3.3.6, and (3.3.7 into (3.3. yields ( Φ n+ Φ n t + γ ( Φ n+ Φ n t + ν Φ n+ ( ν Cν 3 u n+ 4 Φ n+ + γ ( Φ n+ + Φ n + γ inf p(t n+ q η + Cν n+ + η t (t n+ + γ η t (t n+ q Q + Cν w n+ η n+ + ν u n+ η n+ t n+ + η tt dt t ( n t n+ t n+ t n+ + γ η tt dt + C t 3 u ttt dt + u tt dt t n t n t n+ + Cν ( t 3 t n (u ( u tt dt + C u n+ u n+ PV ( u n+ t n (3.3.8 Next multiply by t, sum over time steps, and use te Gronwall inequality Lemma?? to yield M Φ M + γ Φ M + ν t Φ n+ C exp n= ( t M n= n= γ + ν + 4 M M M + t ν η n + t η t (t n + t n= n= + Cν 3 u n+ ( t 4 M n= M + t ν u n+ η n+ + t n= γ M γ η t (t n + t M n= n= t n+ t n inf p(t n q q Q ν w n+ η tt dt η n+ M t n+ + t γ η tt dt + ( t 4 u ttt, + ( t4 u tt, + ( t4 (u ( u tt, t n n= + t M n= u n+ u n+ PV ( u n+ Recall te approximation properties of U n V, q Q, and P V [37]: (3.3.9 η(t n s C k+ s u(t n k+, s =,, and inf q Q p(t n q C k p(t n k w n P V (w n C k+ w n k+. 3

39 Estimate (3.3.9 ten becomes Φ M + γ Φ M + ν t C exp ( t M n= M n= Φ n+ γ + ν + 4 ( + Cν 3 u n+ 4 γ k p,k + ν k u,k+ + k+ u t,k+ + γ k u t,k+ + ν k u t, u,k+ + t γ k u tt,k+ + t k+ u tt,k+ + ( t 4 u ttt, + ( t4 u tt, + ( t4 (u ( u tt, ( M + ν t ν k u t,k+ + k+ u, u,k+. (3.3.3 w n+ n= Finally, using te stability estimate for ν t M n= of te triangle inequality, we obtain ( w n+ from (3..4, and an application Remark As expected, if (X, Q is cosen to be te inf-sup stable pair (P k, P k, k, ten wit te smootness assumptions (3.3.-(3.3.6 and p L (, T ; H k (Ω te H convergence for te velocity is u u, C( t + k (3.3.3 Remark Te significant computational improvement of elicity scemes and 3 over elicity sceme is somewat masked in te statement of te a priori error bound for te velocity (for elicity sceme 3 given in ( For elicity sceme te pressure contribution to te bound is C ν p q, wereas for elicity scemes and 3 te pressure contribution is given by C p q, see (3.3.. Te presence of ν in te denominator for elicity sceme suggests a superior numerical performance of elicity scemes and 3 if a large pressure error is present. 3.4 Numerical Experiments Tis section presents two numerical experiments, te first to confirm convergence rates and te second to compare te elicity scemes accuracies over a longer time interval, against eac oter and a commonly used elicity sceme. For bot experiments, we will compute approximations to te Etier-Steinman exact Navier-Stokes solution on [, ] 3 [9], altoug we coose different parameters and viscosities for te two tests. We find in te first numerical experiment, computed 3

40 Figure 3.: Te velocity solution to te Etier-Steinman problem wit a =.5, d = at t = on te (, 3 domain. Te complex flow structure is seen in te stream ribbons in te box and te velocity streamlines and speed contours on te sides. convergence rates from successive mes and time-step refinements indeed matc te predicted rates from section 4. For te second experiment, te advantage of using te stabilized enanced pysics based elicity sceme is demonstrated. For cosen parameters a, d and viscosity ν, te exact Etier-Steinman NSE solution is given by u = a (e ax sin(ay + dz + e az cos(ax + dy e νd t u = a (e ay sin(az + dx + e ax cos(ay + dz e νd t u 3 = a (e az sin(ax + dy + e ay cos(az + dx e νd t (3.4. (3.4. (3.4.3 p = a (eax + e ay + e az + sin(ax + dy cos(az + dxe a(y+z + sin(ay + dz cos(ax + dye a(z+x + sin(az + dx cos(ay + dze a(x+y e νd t (3.4.4 We give te pressure in its usual form, altoug our elicity sceme approximates instead te Bernoulli pressure P = p + u. Tis problem was developed as a 3d analogue to te Taylor vortex problem, for te purpose of bencmarking. Altoug unlikely to be pysically realized, it is a good test problem because it is not only an exact NSE solution, but also it as non-trivial elicity wic implies te existence of complex structure [54] in te velocity field. Te t = solution for a =.5 and d = is illustrated in Figure 3.. For bot experiments below, we use 3

41 u = (u (, u (, u 3 ( T as te initial condition and enforce Diriclet boundary conditions for velocity to be te interpolant of u(t on te boundary, wile a do-noting boundary condition is used for te vorticity projection. All computations wit elicity scemes and 3 use stabilization parameter γ = Numerical Test : Convergence rate verification t u u S, rate u u S, rate u u S3, rate Table 3.: Te u NSE u, errors and convergence rates for eac of te tree elicity sceme of algoritm 3... To verify convergence rates predicted in section 4, we compute approximations to (3.4.- (3.4.4 wit parameters a = d = π/4, viscosity ν =, and end-time T =.. Since (P, P elements are being used, we expect O( + t convergence of u NSE u, for eac of te tree elicity scemes of Algoritm 3... Errors and rates in tis norm are sown in table 3., and we find tey matc tose predicted by te teory. Note te finest mes provides,454 total degrees of freedom Numerical Test : Comparison of te elicity scemes For our second test, we compute approximations to (3.4.-(3.4.4 wit a =.5, d =, kinematic viscosity ν =., end time T =.5, using all 3 elicity scemes from Algoritm 3... We use 3,7 tetraedral elements, wic provides 4,47 velocity degrees of freedom, and 46,875 degrees of freedom for te projected vorticity since ere tere are degrees of freedom on te boundary. It is important to note tat due to te splitting of te projection equations from te NSE system in te solver and since te projection equation is well-conditioned, te time spent for assembling and solving te projection equations is negligible. In addition to te 3 elicity scemes of Algoritm 3.., for comparison, we also compute approximations using te well-known convective form Crank-Nicolson (CCN FEM for te Navier- Stokes equations [37, 5, 9]. We run te simulations wit time-step t =.5. Results of te simulations are sown in Figures 3. and 3.3, were te L (Ω error and te elicity error are plotted against time. It is clear from te pictures tat te enanced pysics based elicity sceme is 33

42 . L Error vs Time u NSE (t n u n Usual Crank Nicolson Sceme (No Stab Sceme (Grad div stab Sceme 3 (altered GD stab t Figure 3.: Te plot above sows L error of te velocity vs time for te four elicity scemes of Test. We see in te plot tat te stabilizations add accuracy to te enanced-pysics elicity sceme, and tat te altered grad-div stabilization gives sligtly better results tan te usual graddiv stabilization. It can also be seen tat te enanced-pysics elicity sceme is far more accurate in tis metric tan te usual Crank-Nicolson elicity sceme. superior to te usual Crank-Nicolson elicity sceme, and its advantage becomes more pronounced wit larger time. Also it is seen tat te stabilizations of te enanced-pysics elicity sceme improve accuracy. 34

43 Helicity error vs Time.4.3. Usual Crank Nicolson Sceme (No Stab Sceme (Grad div Stab Sceme 3 (Altered GD Stab H(t n H n t Figure 3.3: Te plot above sows elicity error vs time for te four elicity scemes of Test. We see in te plot tat elicity is far more accurate in te enanced-pysics elicity sceme, and even better wit stabilizations, tan te usual Crank-Nicolson elicity sceme. 35

44 Capter 4 Large scale NSE computations witout a pressure space We study a finite element sceme for te 3d NSE tat globally conserves energy and elicity and, troug te use of SV elements, enforces pointwise te solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also sow te metod can be efficiently computed by exploiting a connection between tis metod, its associated penalty metod, and te metod arising from use of grad-div stabilized TH elements. Finally, we give numerical examples wic verify te teory and demonstrate te effectiveness of te sceme. 4. Te Algoritm Algoritm 4... Given a time step t >, finite end time T := M t, and initial velocity u V, find w W and λ Q satisfying (χ, r (W, Q (w, χ + (λ, χ = ( u, χ, (4.. ( w, r =. (4.. 36

45 Ten for n =,,,..., M, find (u n+, w n+, p n+, λ n+ (X, W, Q, Q satisfying (v, χ, q, r (X, W, Q, Q ( un+ u n t, v (P n+, v +(w n+ u n+, v + ν( u n+, v = (f(t n+, v (4..3 ( u n+, q = (4..4 (w n+, χ + (λ n+, χ = ( u n+, χ (4..5 ( w n+, r =. ( Numerical analysis of te sceme In tis section we provide a complete numerical analysis of te sceme. We prove unconditional stability, solution existence, conservation laws, and optimal convergence. 4.. Stability and solution existence Lemma 4... Solutions to Algoritm 4.. are unconditionally stable. Tat is, tey satisfy: M u M + ν t u n+ t ν n= t M w n+ n= t M n= M n= C t f(t n+ + u = C(data. (4.. M n= u n+ = C(data. (4.. ( P n + λ n C(data. (4..3 C(data is a constant dependent on T, ν, γ, f, u and Ω, but independent of and t. Proof. To prove te bound on te velocity coose v = u n+ in (4..3. Te nonlinear and pressure terms are ten zero. Te triangle inequality, and summing over time steps ten completes te proof of (4... To prove (4.. coose χ = w n+ in (4..5 and r = λ n+ in (4..6. After combining 37

46 te equations we obtain w n+ = ( u n+, w n+ u n+ u n+ u n+ w n+ + + w n+ w n+. Rearranging, and summing over time steps we obtain (4... To obtain te stated bound for λ n, we begin wit te inf-sup condition satisfied by X ( W and Q and use (4..5 to obtain λ n β sup (λ n, χ χ X χ X β sup ( u n + w n β β n ( u χ X ( u n, χ (w n, χ χ X. + w n Using te bounds for u n+ in (4.. and w n+ for te pressure is establised in an analogous manner. in (4.. we obtain te bound for λ n. Te bound We sow existence for te equivalent nonlinear problem: Given, ν, t >, f n+ V, and u n V, find (u, w V V satisfying t (u, v + (u w, v + ν ( u, v + ν (w, v = (f n+, v v V, (4..4 (w u, χ = χ V. (4..5 Restricting te test functions to V ensures equations (4..4-(4..5 are equivalent to (??- (??. We now formulate (4..4-(4..5 as a fixed point problem, y = F (y, and use te Leray- Scauder fixed point teorem. We first prove several preliminary lemmas, followed by a teorem wic proves tat a solution to (4..4-(4..5 exists. Lemma 4... For ν, t >, tere exists a unique solution (u, w to te following: Given g V, find (u, w V V satisfying t (u, v + ν ( u, v + ν (w, v = (g, v v V, (4..6 (w u, χ = χ V. (

47 Proof. We will prove uniqueness of solutions to (4..6-(4..7 by sowing only te trivial solutions solves te omogeneous problem, wic will also imply te existence of solutions to te finitedimensional problem. Since te space V includes only zero-mean functions, functions and operators are uniquely solvable, and tus we need not consider te adjoint problem. Coose v = u in (4..6, χ = w in (4..7, and substitute (4..7 into (4..6. Tis gives t u + ν u + ν w =, wic implies u = w =. Tis lemma allows us to define a solution operator to (4..6- (4..7. Definition 4... We define te solution operator T : V (V V to be te solution operator of (4..6- (4..7: if g V, ten T (g = (u, w solves (4..6- (4..7. linear. We ave tat T is well defined by te previous lemma, and we now prove it is bounded and Lemma Te solution operator T is linear, bounded, and continuous. Proof. Te linearity of T follows from te fact tat T is a solution operator to a linear problem. To see tat T is bounded (and tus continuous since it is linear, we let v = u, χ = w in (4..6- (4..7, multiply (4..7 by ν, and add te equations. Tis gives u t + ν 4 u + ν w ν g V. (4..8 Ten since u, w are finite-dimensional, u, w V V C g V. Hence, T (g u, w V V T = sup = sup C. (4..9 g V g V g V g V We next define te operator N. Te function F tat will be used in te formulation of te fixed point problem will be a composition of T and N. Definition 4... We define te operator N on (V V by N(u, w := f n+ + t un + u w. (4.. 39

48 We now prove properties for N necessary for use in Leray-Scauder. Lemma For te nonlinear operator N, we ave tat N : V V V, N is bounded, and N is continuous. Proof. To sow N maps as stated, we let (u, w V V and write N(u, w V = sup v V (N(u, w, v v. From te definition of N, we ave tat (f n+,v +(( t u n,v v f V + C u n C, and tat (u w, v v u w C 3 (4.. since u and w are given to be in V, and all norms are equivalent in finite dimension. Hence N(u, w V < C, and so N maps as stated. Note we ave also proven N is bounded. Te equivalence of norms in finite dimension is also key in sowing tat N is continuous, as N(u, w N(u k, w k V u (w w k V + (u u k w k V u w w k + w k u u k (4.. and tus as (u, w (u k, w k. We now define te operator F : (V V (V V to be te composition of T and N: F (y = T (N(Y. Lemma F is well defined and compact, and a solution to y = F (y solves (4..4-(4..5. Proof. F is well defined because N and T are. Te fact tat F is compact follows from te fact tat bot N and T are continuous and bounded. It can easily be seen tat a fixed point of F solves (4..4-(4..5 by expanding F. We are now ready to prove existence to (4..4-(4..5. Teorem 4... Let y λ = (u λ, w λ V V and consider te family of fixed point problems y λ = λf (y λ, λ. A solution y λ to any of tese fixed point problems satisfies y λ < K, independent of λ. Since F is compact, and fixed points of F solve (4..4-(4..5, by te Leray- Scauder teorem tere exist solutions to (4..4-(

49 Proof. All we ave to sow to prove tis teorem is tat solutions to y λ = λf (y λ are bounded independent of λ. Using te definition of F and te linearity of T we ave tat y λ = λf (y λ = λt (N(y λ = T (λn(y λ = T (λ(f n+ + t un + u λ w λ, (4..3 wic implies tat t (u λ, v λ(u λ w λ, v + ν ( u λ, v + ν (w λ, v = (λf n+, v + λ t (un, v v V,(4..4 (w λ u λ, χ = χ V. (4..5 Multiply (4..5 by ν, let χ = w λ in (4..5, v = u λ in (4..4, and add te equations. Similarly to te stability estimate, tis gives t u λ + ν 4 u λ + ν w λ λ ( ν f n+ + t un ( ν f n+ + t un C, (4..6 wic is a bound independent of λ. Tus te teorem is proven. 4.. Conservation laws for discrete solutions In tis section we study te discrete conservation laws of Algoritm 4... Specifically, we sow te incompressibility constraints are satisfied pointwise and tat te sceme admits an energy and elicity balance wic is analogous to tese balances for te continuous NSE. Lemma Assuming (X, Q = (P k, Pk disc wit k d and periodic boundary conditions, we ave te discrete velocity and vorticity are divergence free at every timestep, tat is u n+ =, (4..7 w n+ =. (4..8 Proof. Note tat te SV pair satisfies X Q and terefore we can coose q = u n+ (4..5. Tis gives in u n+ =. (4..9 4

50 Identity (4..7 follows from taking te square root of bot sides of (4..9. To derive (4..8 note tat te use of periodic boundary conditions guarantees tat W X and tus W Q. Now coosing r = w n+ and proceeding as before gives (4..8. We now study te energy and elicity conservation of te sceme. Denote te discrete energy and elicity at time level n by E n := un and H n := (un, un respectively. Notice tat from (4..4 and (4..5, H n := (un, wn. Lemma Algoritm 4.. admits te following energy and elicity conservation laws. H M M EM + ν t n= u n+ M = t n= M + tν ( u n+, w n+ = H + tν n= (f(t n+, u n+ + M n= E. (4.. (f(t n+, u n+ (4.. Proof. To prove (4.. and (4.. we coose v = u n+ and v = w n+ in (4..3. Te key point is tat te nonlinear term vanises wit tese coices of test functions, and tus does not contribute to te energy and elicity balance equations Convergence We now prove optimal convergence of solutions to Algoritm 4.. to an NSE solution. Let P V : L V denote te projection of L onto V, i.e. P V (w := s were (s, v = (w, v, v V. For simplicity in stating te a priori teorem we summarize ere te regularity assumptions for te solution u(x, t to te NSE. u L (, T ; H k+ (Ω L (, T ; H (Ω, (4.. u(, t H (Ω, u L (, T ; H k+ (Ω, (4..3 u t L (, T ; H k+ (Ω L (, T ; H k+ (Ω, (4..4 u tt L (, T ; H k+ (Ω, (4..5 u ttt L (, T ; L (Ω (4..6 (u ( u tt L (, T ; L (Ω. (4..7 4

51 Teorem 4... For u, p solutions of te NSE wit p L (, T ; H k (Ω, u satisfying (4..- (4..7, f L (, T ; X (Ω, and u V, (u n, wn given by Algoritm 4.. for n =,..., M and t sufficiently small, we ave tat ( u(t u M + ν t M n= (u n+ n+ u / + u n+/ w n+/ C(T, ν 3, u ( k u(t k+ + k u,k+ + k u t,k+ + k u t,k+ + k u t, u,k+ + ( t / k u tt,k+ + ( t u ttt, + ( t u tt, + ( t (u ( u tt, + k+ u, u,k+. (4..8 Remark 4... Tis proof is similar to te convergence proof in Capter 3. Te fundamental difference in te proofs is tat now we are assuming te use of te SV element, wic wile maintaing optimal approximation properties removes te adverse (often large effect of te Bernoulli pressure occurring wen computing a sceme in rotational form of te NSE. Proof of Teorem. Since (u, p solves te NSE, we ave v X tat (u t (t n+, v + (u(t n+ ( u(t n+, v (p(t n+, v + ν( u(t n+, v = (f(t n+, v. (4..9 Adding ( un+ u n t, v and ν( u n+, v to bot sides of (4..9 we obtain t (un+ u n, v + (( u(t n+ u(t n+, v (p(t n+, v ( + ν( u n+, v = (f(t n+ u n+ u n, v + u t (t n+, v t + ν( u n+ u(t n+, v. (4..3 Next, subtracting (4..3 from (4..3 and labelling e n := u n u n gives ( (( u(t n+ u(t n+, v + w n+ t (en+ e n, v + ν( e n+, v = ( ( + p(t n+ p n+ u n+ u n, v + u t (t n+, v + ν t u n+, v ( u n+ u(t n+, v. We split te error into two pieces Φ and η: e n = u n u n = (un U n + (U n u n := ηn + Φ n, 43

52 were U n denotes te interpolant of u n in V, yielding t (Φn+ Φ n, v + ν( Φ n+, v = t (ηn+ η n, v ν( η n+, v (( u(t n+ u(t n+, v + (w n+ u n+, v + (p(t n+ p n+, v ( u n+ u n + u t (t n+, v + ν( u n+ u(t n+, v. (4..3 t Coosing v = Φ n+ vanises te pressure term and yields ( Φ n+ Φ n t + ν Φ n+ = t (ηn+ η n, Φ n+ ν( η n+ n+, Φ ( u(t n+ u(t n+ n+, Φ + (w n+ u n+, Φ n+ ( u n+ u n + u t (t n+ n+, Φ + ν( u n+ u(t n+ n+, Φ. (4..3 t We ave te following bounds for te terms on te RHS (see [8]. ν( η n+ n+, Φ ν Φ n+ + 3ν η n+ (4..33 t (ηn+ η n, Φ n+ η n+ η n t + Φ n+ = ( t n+ η t dt dω + Φ n+ Ω t t n ( t n+ η t (t n+ + η tt dt dω + Similarly, ( u n+ u n t Ω u t (t n+, Φ n+ t n = ηt (t n+ + t n+ Φ n+ t n η tt dt + Φ n+ u n+ u n u t (t n+ + t t n+ = ( t3 u ttt dt + 56 t n ν( u n+ u(t n+ n+, Φ 3ν u n+ ν u(t n+ ν + = ν( t3 6 t n+ t n u tt dt + ν Φ n+ Φ n+. (4..34, (4..35 Φ n+ Φ n+. (

53 Utilizing (4..33-(4..36 we now ave t ( Φ n+ Φ n + 5ν 6 ( t n+ + C( + ν t 3 Φ n+ 3ν η n+ + t n+ u ttt dt + t n t n + ηt (t n+ ( + (w n+ u n+, Φ n+ t n+ t n u tt dt η tt dt + ν + Φ n+ ( u(t n+ u(t n+, Φ n+. (4..37 Treatment of te nonlinear terms is te same as in [9] wic gives. ( Φ n+ Φ n t + ν ( Φ n+ ν Cν 3 u n+ 4 Φ n+ + Cν η n+ + ηt (t n+ + Cν w n+ η n+ + ν u n+ η n+ ( t n+ t n+ t n+ + η tt dt + C t 3 u ttt dt + u tt dt t n t n t n+ + Cν ( t 3 t n (u ( u tt dt + C u n+ u n+ PV ( u n+ t n (4..38 Next multiplying by t, summing over time steps, and using Lemma?? yields Φ M + ν t Cexp ( M n= t Φ n+ M n= ν Cν 3 u n+ 4 M M ( t ν η n + t η t (t n n= n= M + t ν w n+ η n+ + t n= M t n+ n= M n= ν u n+ η n+ + t η tt dt + ( t 4 u ttt, + ( t4 u tt, n= t n M + ( t 4 (u ( u tt, + t u n+ u n+ PV ( u n+ (4..39 Recall te approximation properties of U n V, q Q, and P V [37] η(t n s C k+ s u(t n k+, s =,, and w n P V (w n C k+ w n k+. 45

54 Estimate (4..39 ten becomes Φ M + ν t M n= Φ n+ C exp ( t M n= ν Cν 3 u n+ 4 (ν k u,k+ + k+ u t,k+ + ν k u t, u,k+ + t k+ u tt,k+ + ( t 4 u ttt, + ( t4 u tt, + ( t4 (u ( u tt, ( M + ν t ν k u t,k+ + k+ u, u,k+. (4..4 w n+ n= Finally, from te boundness estimate for ν t M n= from (4.., and an application of te triangle inequality we obtain (4..8. w n+ 4.3 Improved efficiency troug decoupling Algoritm 4.. can be twice decoupled to allow for more efficient solves of te nonlinear system. First, te nonlinear system sould be decoupled into velocity-pressure and vorticity projection pieces. Once tis is done, one is left to solve several saddle point systems at eac timestep. Due to te use of SV elements, we are able to use te classical penalty metod to obtain optimally accurate solutions wile eliminating te saddle point structure of te linear systems. Te key point ere is tat te pointwise div-free subspace of te velocity space is guaranteed to ave optimal approximation properties in te setting were SV elements are LBB stable [65]. We prove now a connection between te (P k, P disc k SV solution and (P k, Q element solutions, were Q can be cosen from {P k, P k,..., P, P, {}}, te last of wic leads directly to te penalty metod for bot te reduced velocity and vorticity systems. Teorem Let Q be specified so tat te (P k, Q pair is LBB stable. Ten on a fixed mes, te (P k, Q velocity solutions to (4..-(4..6 converge to te SV solution wit convergence order γ in te energy norm, as γ ; tat is, if u is te TH velocity solution and u is te SV velocity solution, ten u u l (,T :H (Ω C γ (

55 Remark Te pair (P k, Q is LBB stable wen Q is cosen to be P r (for r < k. Of particular interest is te case wen Q = {} wic is equivalent to a dual-penalty metod, similar to tat of Temam s in [69]. Proof. Let (u n+, w n+, p n+, l n+ and (u,n+, w,n+, p,n+, λ,n+ denote te SV and (P k, Q solutions to (4..3-(4..6 respectively. Additionally, denote te velocity and vorticity differences of te solutions as r n+ and s n+ so tat r n+ := u n+ u,n+, (4.3. s n+ := w n+ w,n+. (4.3.3 Ortogonally decompose r n+ = r n+ +r n+ and s n+ = s n+ +s n+ so tat r n+, s n+ V and r n+, s n+ q Q. R. Note it follows from (4..4 and te above decomposition tat ( r n+, q = Pick v = r n+ in (4..3, ten te (P k, Q and SV solutions satisfy respectively ( un+ ( u,n+ u n t, r n+ n+ + γ( u, r n+ n+ + (w u n+, r n+ u,n, r n+,n+ + (p, r n+,n+ + (w u,n+, r n+ t +ν( u n+, r n+ = (f(t n+, r n+ ( ν( u,n+, r n+ = (f(t n+, r n+ (4.3.5 Subtracting (4.3.5 from (4.3.4, rearranging and reducing wit te following identities u n+ = r n+ = r n+ (4.3.6 ( r n+, r n+ = ν r n+ (4.3.7 gives t (rn+, r n+ t (rn, r n+ + ν r n+ + γ r n+ = (w n+ u n+, r n+ (w,n+ u,n+, r n+ (p,n+, r n+. (4.3.8 Standard inequalities, bounds on solutions and Lemma.. yields t (rn+, r n+ t (rn, r n+ + ν r n+ + γ r n+ C r n+. (

56 Similarly, if χ = s n+ in (4..5, te (P k, Q and SV solutions respectively satisfy (w n+, s n+ n+ + γ( w, s n+ n+ + (λ, s n+ n+ = ( u, s n+, (4.3. (w,n+, s n+ + (l,n+, s n+ = ( u,n+, s n+. (4.3. Subtracting (4.3. from (4.3. and rearranging gives s n+ + γ( w n+, s n+ = (s n+, s n+ n+ (λ, s n+ n+ + ( u, s n+ +(l,n+, s n+ ( u,n+, s n+. (4.3. Reducing wit standard inequalities, bounds on solutions, lemma and te following identity w n+ = s n+ = s n+ (4.3.3 gives s n+ C γ. (4.3.4 Next for χ = s n+ in (4..5 te (P k, Q and SV solutions ten satisfy te following equalities (w n+, s n+ = ( u n+, s n+, (4.3.5 (w,n+, s n+ = ( u,n+, s n+. (4.3.6 Subtracting (4.3.6 from (4.3.5,rearranging and using standard inequalities gives s n+ s n+ s n+ + r n+ s n+. (4.3.7 Reducing gives s n+ C( s n+ + r n+. (4.3.8 Coosing v = r n+ in (4..3 admits te following equalities for te (P k, Q and SV 48

57 solution, respectively: t (un+ u n, r n+ n+ + (w u n+, r n+ n+ + ν( u, r n+ = (f(t n+, r n+, (4.3.9 t (u,n+ u,n, rn+,n+ + (w u,n+, r n+,n+ + ν( u, r n+ = (f(t n+, r n+ (4.3.. Subtracting (4.3. from (4.3.9 and rearranging gives t (rn+, r n+ t (rn, r n+ + ν r n+ (4.3. = (w n+ u n+, r n+ (w,n+ u,n+, r n+. (4.3. We majorize te left and side by rewriting te nonlinear terms using a standard identity wic yields t (rn+, r n+ t (rn, r n+ + ν r n+ (4.3.3 (w n+ r n+, r n+ + (s n+ u,n+, r n+. (4.3.4 Te first nonlinear term reduces using ortogonality and bounds on solutions. We bound te second nonlinear term by splitting s n+ into its ortogonal components wic gives t (rn+, r n+ t (rn, r n+ + ν r n+ C r n+ ( C s n+ r n+ + C s n+ r n+. (4.3.6 To te rigt and side we add C s n+ r n+ and C s n+ r n+. Ten reducing wit (4.3.4, (4.3.8 and standard inequalities gives t (rn+, r n+ t (rn, r n+ + ν r n+ C γ + C r n+. (4.3.7 Adding (4.3.9 and (4.3.7, dropping te stiffness term and reducing gives t rn+ t rn + γ r n+ C γ + C r n+ + C r n+. (4.3.8 Using standard inequalities, multiplying by t and summing over time steps yields M r M + γ t n= M r n+ r + C t n= r n+ + CT γ. (

58 Te discrete Gronwall inequality finises te proof Penalty Metod Formulation We now sow tat Algoritm 4.. using te SV elements is equivalent to a dual penalty metod wit te TH elements (P k, {}. Te penalty metod is Algoritm Given a time step t >, finite end time T := M t, and initial velocity u V, find w W and λ Q satisfying (χ, r (W, Q (w, χ + (λ, χ = ( u, χ, (4.3.3 ( w, r =. (4.3.3 Ten for n =,,,..., M, find (u n+, w n+, P n+, λ n+ (X, W, Q, Q satisfying (v, χ, q, r (X, W, Q, Q ( un+ u n t, v (P n+, v +(w n+ u n+, v + ν( u n+, v = (f(t n+, v (4.3.3 ( u n+, q + ε(p n+, q = ( (w n+, χ (λ n+, χ = ( u n+, χ ( ( w n+, r + ε(λ n+, r =. ( Now coosing q = v and r = χ in ( and ( respectively ten rearranging gives ε ( u n+, v = (P n+, v ( ε ( w n+, χ = (λ n+, χ ( Substituting ( and ( into (4..3 and (4..5 respectively gives ( un+ u n t, v + ε ( u n+, v +(w n+ u n+, v + ν( u n+, v = (f(t n+, v ( (w n+, χ + ε ( w n+, χ = ( u n+, χ ; ( (

59 wic is identical to te sceme for te stabilized (P k, {} element pair, wit te identification of γ = ɛ. 4.4 Numerical Experiments In tis section we present two numerical experiments. Tis first is a verification of predicted convergence rates, and te second is a simulation of cannel flow over a 3d forward-backward step. All computations were performed in MATLAB. Linear solves were performed using backslas, wic is very efficient wen using te penalty metod formulation of te discrete problem Numerical Test : Convergence rate verification To verify convergence rates predicted in Section 3, we compute approximations to to te model problem wit solution u = cos(πz( +.t (4.4. u = sin(πz( +.t (4.4. u 3 = sin(πz( +.t (4.4.3 p = sin(π(x + y + z. (4.4.4 from t = to T =, wit ν =. We use te penalty metod wit grad-div parameter γ =,, and compute wit X = P 3, on barycenter refinements of uniform meses. Errors and rates are sown in Tables 4. and 4., for successively finer meses and reduced timesteps. Optimal rates are observed for velocity and vorticity in te indicated norms, verifying te results of Section 3. We note tat on te finest mes, (decoupled linear solves averaged 66 seconds on a x.66 GHz Quad-Core Intel Xeon processor wit GB 66 MZ DDR3 memory, and seconds on te second finest mes, and 6 seconds on te finest one. Eac nonlinear solve required between 3 and 4 iterations, and tus between 6 and 8 (reduced linear solves for eac of velocity and vorticity dimensional cannel flow over a forward-backward step We now present results for time dependent 3d cannel flow over a forward-backward facing step wit Re =. A diagram of te flow domain is given in Figure 4.. Te flow we study is an altered version of te flow studied in [3], but wit a different treatment of boundary conditions. 5

60 t dim(x Total dof u u L (L rate u u L (H rate / T 3,89,8 4.34e- 9.5e- /4 T/3 3,87 78,46.76e e- 3. /6 T/6 78,987 6, e e- 3. /8 T/9 85,5 65,99.794e e- 3. / T/8 359,373,98, e e Table 4.: Te errors and rates for te velocity solution in numerical experiment. Rates appear optimal. Te Total dof column sows te total degrees of freedom required if SV elements were used. t w w L (L rate u L (L w L (L / T 5.445e-.863e e-6 /4 T/3 6.86e e-5.687e-7 /6 T/6.675e- 3..e-5.797e-8 /8 T/9 7.85e e e-9 / T/ e e e-9 Table 4.: Te errors and rates for te vorticity in numerical experiment, as well as te errors in te velocity and vorticity divergences. First, we coose no-slip boundaries for te cannel walls. For te inflow conditions, [3] uses te constant inflow profile u in =<,, >, wic is bot nonpysical and not appropriate for a velocityvorticity metod since vorticity at te inflow edges will approac infinity as te meswidt decreases. Tus, instead, we treat te problem as an infinite cannel and enforce u in = u out, and taking te initial condition to be te steady Re = 5 solution. We use X = P 3, and compute on a barycenter refined tetraedral mes, wic provides,8,9 total degrees of freedom for te full SV discretization, 398, of wic form te velocity space. Te system is solved wit te penalty metod described above wit γ =,. A timestep of t =.5 is used, and we compute to T =. Visualizations of te solution are sown in Figures 4. and 4.3, and te correct pysical beavior is realized - an eddy detaces from te step and moves down te cannel, and a new eddy forms. 5

61 Figure 4.: Sown above is domain for te 3d cannel flow over a step problem. Figure 4.: Sown above is te x = 5 sliceplane of speed contours and velocity streamlines at T =. An eddy can be observed to ave moved down te cannel, and a second eddy as formed beind te step. 53

62 Figure 4.3: Sown above is a plot of te same velocity field as in Figure 4., but zoomed in near te step. Streamribbons are used to visualize te flow, and more clearly sow te eddies. 54

63 Capter 5 Convergence Rate of Grad-div Stabilized TH solutions to SV solutions 5. Order of convergence for NSE approximations We consider te rate of convergence of finite element approximations of te NSE using grad-div stabilized TH formulations to te solution of SV elements, as te grad-div stabilization parameter γ tends to zero. We sow first for te steady case, ten for te time-dependent case, tat te rate is O(γ. 5.. Te steady NSE case Consider te discrete steady convective NSE formulation: Find (u, p (X, P suc tat (v, q (X, P, were P = Q (TH or Q (SV, b (u, u, v (p, v + ν( u, v + γ( u, v = (f, v (5.. ( u, q =. (5.. We note tat for te case of SV elements, te grad-div term trivially vanises. Define α := C s ν f. Te formulation (5..-(5.. is known to be well-posed under te small data condition α > [37], for eiter element coice, due to assumptions on te 55

64 mes and polynomial degree. Lemma 5... Solutions to (5..-(5.. exist and satisfy ν u + γ u ν f (5..3 If P = Q : p γ( u f ( + Cs ν f + ν (5..4 If P = Q : p f ( + Cs ν f + ν (5..5 If α >, ten solutions are unique. Proof. Taking v = u in (5.. and using Caucy-Scwarz and Young s inequalities gives (5..3. Te pressure bounds follow directly from te discrete LBB condition and te bound (5..3. Te SV pressure bound does not include te term wit γ since te grad-div term is trivially zero in tis case. Remark 5... We consider limiting beavior as γ, and tus te bound (5..4 seems insufficient to guarantee stability of te pressure in te limit. However, te following teorem implies tat u C γ, and te TH pressure solution is indeed bounded by a data-dependent constant, independent of γ. Teorem 5... On a fixed mes and wit data satisfying α >, te TH velocity solutions to (5..-(5.. converge to te SV velocity solution wit convergence order γ in te energy norm, as γ ; tat is, if u is te TH solution and u is te SV solution, ten (u u C γ. Proof. Let (u, p (V, Q denote te solution of (5..-(5.. using SV elements, (u, p (V, Q for te TH solution, and te difference between tem to be r V, so tat u = u + r. For te TH solution u, setting v = w V and s R, respectively, in (5.. gives te equations b (u, u, w + ν( u, w = (f, w, (5..6 b (u, u, s + ν( u, s + γ( u, s = (f, s. (

65 Similarly, te SV solution u V satisfies b (u, u, w + ν( u, w = (f, w, (5..8 b (u, u, s (p, s = (f, s. (5..9 From (5..7 and (5..9, we ave b (u, u, s + ν( u, s + γ( u, s = b (u, u, s (p, s, (5.. and since ( u, s = ( r, s and u = r, ν( r, s + γ( r, s = b (u, u, s b (u, u, s (p, s = b (r, u, s b (u, r, s (p, s. (5.. Ortogonally decompose r =: r + r, were r V and r r. Now setting v = r in (5.. gives, after reducing wit ortogonality properties and using Lemma..9, ν r + γ r = b (r, u, r b (u, r, r (p, r C ( M r u + M r u + p r (5.. Since u, u are uniformly bounded by te data by (5..3, independent of γ, r is also. Using tis and (5..5 provides ν r + γ r C r. (5..3 Dropping te first term on te left and dividing by r gives r C γ, (5..4 wic implies from Lemma..9 tat r C γ. (

66 It remains to bound r. From (5..6, (5..8, and taking w = r, we get b (u, u, r + ν( u, r = b (u, u, r + ν( u, r, (5..6 wic reduces to ν( r, r = b (u, u, r b (u, u, r, = b (u, r, r b (r, u, r. (5..7 Skew symmetry properties and decomposing r gives ν r = b (u, r, r b (r, u, r b (r, u, r. (5..8 Standard inequalities and (5..3 now provides ν r C r r + C s ν f r. (5..9 Using te small data condition, ten dividing troug by r gives r C r C γ. (5.. Te triangle inequality completes te proof, as (u u = r r + r C γ. (5.. Lemma 5... If p is te TH pressure and p is te SV pressure ten p (p γ u C γ. Proof. Te TH and SV solutions to (5..-(5.. satisfy respectively b (u, u, v (p, v + ν( u, v + γ( u, v = (f, v, (5.. b (u, u, v (p, v + ν( u, v = (f, v. (

67 Subtracting (5..3 from (5.. and rearranging gives (p (p γ u, v = b (u, u u, v + b (u u, u, v + ν( (u u, v. (5..4 From Lemma.., Teorem 5.. and bounds on solutions it follows tat (p (p γ u, v C γ v. (5..5 Dividing (5..5 by v and te LBB condition (of te SV element finises te proof. 5.. Te time-dependent case for te NSE For te time-dependent case, we find an analogous result to te steady case. We consider te semi-discrete formulation, and extension to te usual temporal discretizations suc as backward Euler and Crank-Nicolson is straigt-forward, altoug tecnical. Tus we proceed to study te following problems: Given u ( V, find (u (t, p (t (X, P (, T ] suc tat (v, q (X, P, were P = Q (TH or Q (SV, ((u t, v + b (u, u, v (p, v + ν( u, v +γ( u, v = (f, v (5..6 ( u, q =. (5..7 It is straigt-forward to sow (e.g. [37] tat tis formulation admits unique solutions satisfying for t T, u (t + ν t u (s ds + γ t u (s ds C(data, (5..8 If P = Q : p ( + γ C(data, (5..9 If P = Q : p C(data. (5..3 Remark 5... For te fully discrete case, tere is a restriction tat te time-step be small enoug to get uniqueness; oterwise an analogous result olds. Remark Wit te following teorem, te bound (5..9 can be improved to be independent of γ. 59

68 Teorem 5... On a fixed mes, te TH velocity solutions to (5..6-(5..7 converge to te SV solution wit convergence order γ in te energy norm, as γ. Tat is, if u is te TH solution and u is te SV solution, ten u u L (,T ;H (Ω C γ. Remark Te stability estimate (5..8 suggests te rate may be only γ / since te SV solution is pointwise divergence-free, but te teorem proves it is indeed faster. Proof. Our strategy for tis proof is similar to tat of te steady case. Let (u, p (V, Q [, T ] denote te solution of (5..6-(5..7 using SV elements, (u, p (V, Q [, T ] for te TH solution, and te difference between tem to be r V [, T ], so tat u (t = u (t + r (t. Again we ortogonally decompose r (t = r (t + r (t, were r (t r and r (t V ; recall V = V R in te X inner product. Consider (5..6 wit an arbitrary test function s R V. Te TH and SV solutions satisfy, respectively, ((u t, s + b (u, u, s + ν( u, s + γ( u, s = (f, s, (5..3 ((u t, s + b (u, u, s (p, s + ν( u, s = (f, s. (5..3 Subtracting and utilizing te following identities u = r = r (5..33 ( r, s = ( r, s. (5..34 provides te equation ((r t, s + ν( r, s + γ( r, s = b (r, u, s b (u, r, s (p, s. 6

69 Taking s = r, ten reducing wit Lemmas.. and..9, and (5..8 and (5..3 yields ((r t, r + ν r + γ r = b (r, u, r b (u, r, r (p, r C s r u r + C s u r r + p r C s r u M r + C s u r M r + p r ( C s M r u + C s M u r + p r C r. (5..35 We now bound r. Consider (5..6 wit an arbitrary test function w V. Te TH and SV solutions satisfy, respectively, ((u t, w + b (u, u, w + ν( u, w = (f, w, (5..36 ((u t, w + b (u, u, w + ν( u, w = (f, w. (5..37 Subtracting gives ((r t, w + ν( r, w = b (u, u, w + b (u, u, w, (5..38 wic reduces to ((r t, w + ν( r, w = b (r, u, w b (u, r, w. (5..39 Taking w = r gives ((r t, r + ν r = b (r, u, r b (u, r, r (5..4 = b (r, u, r b (r, u, r b (u, r, r (5..4 Using lemmas.. and..9, and te uniform bound on solutions yields ((r t, r + ν r C s r 3/ u r / + C s r u r + C s r u r C s r 3/ u r / + C r. (5..4 6

70 Adding (5..35 to (5..4 gives ((r t, r + ((r t, r + ν r + ν r + γ r C s r 3/ u r / + (C + p r, (5..43 wic reduces wit ortogonality properties, te uniform bounds on solutions, ten standard inequalities to d dt r + ν r + γ r C s r 3/ u r / + C r C r + ν r + C γ + γ r. (5..44 Tis leaves d dt r + γ r C r + C γ. (5..45 Te Gronwall inequality, u ( = u (, and reducing gives us t r dt C γ, (5..46 wic proves te teorem. 5. Extension to turbulence models Recent work on finite element metods for te α models of fluid flow as proven teir effectiveness at finding accurate solutions to flow problems on coarser spatial and temporal discretizations tan are necessary for successful simulations of te NSE [43, 44, 5, 6, 66, 5,,, 8]. We prove te convergence result for grad-div stabilized TH solutions to SV solutions of te Leray-α model; analogous results / proofs for te oter α models follow similarly. Since a goal of te α-models is to find solutions on coarser meses tan would be used for te NSE, mass conservation of solutions can be very poor and tus large grad-div stabilization tat preserves overall accuracy and improves te mass conservation will elp to provide more pysically relevant solutions. Te continuous Leray-α model formulation is: find (u, p, w, λ (X, P, X, P suc 6

71 tat (v, q, χ, ψ (X, P, X, P, were P = Q (TH or Q SV (SV, ((u t, v + b (w, u, v (p, v + ν( u, v +γ( u, v = (f, v, (5.. ( u, q =, (5.. (w, χ + α ( w, χ + (λ, χ + γ( w, χ = (u, χ, (5..3 ( w, ψ =. (5..4 Te equations (5..3-(5..4 are te discretization of te α-filter, wit discrete incompressiblity enforced. Advantages of using tis discretization for te filter instead of te usual one are discussed in [6]. Te following lemma will be useful for te analysis in tis section. Lemma 5... If (u, p, w, λ solves (5..-(5..4 ten w u. Proof. Te lemma can be verified quickly by coosing χ = w in (5..3 and using te Caucy- Scwarz inequality. Teorem 5... On a fixed mes te grad-div stabilized TH velocity solutions to (5..-(5..4 converge to te SV velocity solution wit convergence order γ in te energy norm, as γ. Tat is, if we denote te SV velocity solutions as u and grad-div stabilized TH solution as u ten u u L (,T ;H (Ω C γ. Proof. Let (u, w, p, λ (X, X, Q, Q [, T ] denote te solution of (5..-(5..4 using SV elements, (u, w, p, λ (X, X, Q, Q [, T ] for te TH solution. Let te difference between u and u be denoted by r u and te difference between w and w be denoted by r w so tat u (t = u (t + r u (t, and w (t = w (t + r w (t. Ortogonally decompose r u (t = r u(t + r u(t, were r u(t R and r u(t V SV. Similarly, ortogonally decompose r w (t = r w(t + r w(t so tat r w(t R and r w(t V SV. Consider (5.. and (5..3 wit an arbitrary test function s R V. Te TH and SV 63

72 solutions satisfy, respectively, ((u t, s + b (w, u, s + ν( u, s + γ( u, s = (f, s (5..5 ((u t, s + b (w, u, s (p, s + ν( u, s = (f, s. (5..6 Subtracting using previous identities gives ((r u t, s + ν( r u, s + γ( r u, s = b (w, u, s b (w, u, s (p, s. Taking s = r u, and reducing wit Lemmas..,..9 and 5.., and uniqueness of solutions yields ((r u t, r u + ν r u + γ r u = b (w, u, r u b (w, u, r u (p, r u C s ( w u r u + w u r u + p r u C r u. (5..7 We now derive a similar bound for r w. Consider tat te TH and SV solutions satisfy te following equations from (5..3: (w, χ + α ( w, χ + (λ, χ + γ( w, χ = (u, χ, (5..8 (w, χ + α ( w, χ + (λ, χ = (u, χ. (5..9 Subtracting, coosing χ = r w, and rearranging gives r w + α r w + γ r w = (r u, r w (λ, r w. (5.. Te Caucy-Scwarz inequality and Lemma. yields r w C γ. (5.. Next we derive a bound for r w. To do tis we subtract (5..9 from (5..8 and coose χ = r w wic gives (r w, r w + α r w = (r u, r w. (5.. 64

73 From ere we rearrange using Caucy-Scwarz and equivalence of norms over finite dimensional Hilbert spaces wic gives r w C ( r u + r w. (5..3 We proceed similar to te time-dependent NSE case and bound r u. Consider (5.. wit an arbitrary test function v SV V SV. Te TH and SV solutions satisfy ((u t, v SV ((u t, v SV + b (w, u, v SV + ν( u, v SV = (f, v SV, ( b (w, u, v SV + ν( u, v SV = (f, V SV. (5..5 Subtracting (5..5 from (5..4 rearranging and coosing v = r u gives ((r u t, r u + ν r u b (w, r u, r u + b (r w, u, r u. (5..6 To majorize te first trilinear term in (5..6 use Lemmas. and 4., bounds on solutions and note tat for ortogonal decompositions te triangle inequality is an equality. Lastly, using equivalence of norms gives b (w, r u, r u C r u r u C r u r u + C r u r u C r u C r u. (5..7 We bound te second trilinear using Lemma. and uniform bound on solutions. Ten we split te r w term using te triangle inequality and use (5..3, wic yields b (r w, u, r u C r w r u C r w r u + C r w r u. (5..8 Adding C r w r u and C r w r u to te rigt and side of (5..8 and using ortogonality gives b (r w, u, r u C r w r u + C r w r u. (

74 We majorize te first rigt and side term using Lemma., bounds on solutions and (5... Additionally, we majorize te second rigt and side term using (5..3. After we combine like terms we are left wit b (r w, u, r u C γ + C r u. (5.. From equivalence of norms we ave tat r u C r u. Terefore, ((r u t, r u + ν r u C γ + C r u. (5.. Adding (5.. and (5..7 gives d dt r u + γ r u C r u + C γ. (5.. Analogous to te time-dependent NSE proof using te Gronwall inequality, u ( = u ( and reducing finises te proof. 5.. Numerical Verification for te Leray-α model To numerically verify te velocity convergence rate sown above we consider te bencmark d problem of cannel flow over a forward-backward step. Te domain Ω is a 4 rectangle wit a step 5 units into te cannel at te bottom. Te top and bottom of te cannel as well as te step are prescribed wit no-slip boundary conditions, and te sides are given te parabolic profile (y( y/5, T. We use te initial condition u = (y( y/5, T inside Ω, coose te viscosity ν = /6 and run te test to T=. Te correct pysical beavior is for an eddy to form beind te step (at larger T, te eddy will move down te cannel and a new eddy will form. A barycenter-refinement of a Delauney triangulation of Ω is used, wic yields a total of 4,467 degrees of freedom for te (P, P disc SV computations and 9,47 for (P, P TH. A Crank- Nicolson time discretization is used wit a timestep of t =.. For te TH computations, we use grad-div stabilization parameters γ = {,,,,,,, }. Plots of te SV and TH solutions are sown in Figure 5., and te correct pysical beavior is observed in bot; in fact, tese solutions are nearly indistinguisable. Plots of te TH solutions wit γ > are also nearly identical and so are omitted. Differences between te TH solutions wit varying γ, and te SV solution are computed in te H norm, and are sown (wit rates in Table 5.; first order convergence is observed, in accordance wit our teory. Te divergence errors of te 66

75 TH solutions are given in Table 5., wic also display first order convergence. Also of particular interest is tat te TH solution wit γ = as very poor mass conservation, even toug its plot appears correct. γ u γ T H u SV H rate u γ T H E E E E E E-7 Table 5.: Convergence of te grad-div stabilized TH Leray-α solutions toward te SV Leray-α solution, first order as γ. SV solution TH solution wit γ = Figure 5.: SV and TH solutions of te Leray-α model at t =. 67

76 5.3 Extension to magnetoydrodynamic flows To understand a fluid flow wic is influenced by a magnetic field one must understand te mutual interaction of a magnetic field and a velocity field. Te system of differential equations wic describe te flow of an electrically conductive and nonmagnetic incompressible fluid (e.g. liquid sodium are called magnetoydrodynamics (MHD. Tese equations are commonly used in metallurgical industries to eat, pump, stir and levitate liquid metals [5]. We consider te steady MHD in te form studied in, e.g., [6, 7], wic is te NSE coupled to te pre-maxwell equations. For simplicity of te analysis, we restrict to omogeneous Diriclet boundary conditions (or periodic for bot velocity and te magnetic field and consider a convex domain. Te Galerkin finite element metod tat explicitly enforces incompressibility of bot te velocity and magnetic fields and wit grad-div stabilization of bot velocity and magnetic fields is, (v, χ, q, ψ (X, X, Q, Q, b (u, u, v + ν( u, v sb (B, B, v (P, v + γ( u, v = (f, v (5.3. ( u, q = (5.3. ν m ( B, χ b (B, u, χ + b (u, B, χ +(λ, χ + γ( B, χ = ( G, χ (5.3.3 ( B, ψ =. (5.3.4 Te Lagrange multiplier is added in (5.3.3 so tat te divergence of te magnetic field can be explicitly enforced via (5.3.4 witout overdetermining te discrete system. For te coice of (X, Q to be Taylor Hood elements, bot u = and B = are enforced weakly in (5.3.-(5.3.4, but if instead SV elements are cosen ten pointwise enforcement is recovered (coose q = u and ψ = B. Similar to te NSE case, tere is a middle ground of improved mass conservation wile using TH elements, if γ is cosen large. Note we consider te stabilization parameters to be equal only for simplicity since we consider teir limiting beavior; in practice it may be necessary to coose tem differently for optimal accuracy. Lemma Solutions to ( (5.3.4 exist and satisfy u ν f + s ν B ν ν m ν m G (=: M, (5.3.5 s f + ν m G (=: M. (

77 If ν C s M sc s M >, and (5.3.7 ν m C s M C s M > (5.3.8 ten solutions are unique. Proof. Existence of solutions is a straigt forward application of te Leray-Scauder Teorem. To derive (5.3.5 and (5.3.6 we multiply (5.3.3 by s and add it to (5.3.. Next we coose v = u and χ = B. Noting tat b (B, B, u = b (B, u, B leaves ν u + sν m B (f, u + s( G, B. (5.3.9 Te bounds can be derived from (5.3.9 by using Young s inequality. To derive sufficient conditions for uniqueness assume (to get a contradiction tat tere are two solutions to (5.3.-(5.3.4, (u, B, p, λ and (u, B, p, λ. Now let D u := u u and D B := B B. Substituting u, u into (5.3., coosing v = D u, subtracting and rearranging gives ν D u + γ D u = b (u, u, D u b (u, u, D u + sb (B, B, D u sb (B, B, D u. (5.3. Using standard inequalities and noting tat b (v, u, u = we can rewrite (5.3. as ν D u + γ D u = sb (B, D B, D u + sb (D B, B, D u b (D u, u, D u. (5.3. Scaling (5.3.3 by s and similar treatment gives sν m D B + sγ D B = sb (B, D u, D B + sb (D B, u, D B sb (D u, B, D B. (

78 Adding (5.3. and (5.3. and noting tat b (B, D u, D B = b (B, D B, D u, yields ν D u + sν m D B sb (D B, u, D B sb (D u, B, D B + sb (D B, B, D u b (D u, u, D u. (5.3.3 Utilizing Lemma. and Young s inequality we can now rewrite tis as D u (ν C s M sc s M + D B s(ν m C s M C s M. (5.3.4 Uniqueness ten follows from (5.3.7 and ( Convergence of velocity and magnetic field TH solutions to te SV solution for steady MHD We now extend te results above to te case of steady MHD, formulated by (5.3.-( Here tere are two grad-div stabilization terms tat arise in te analysis, but te main ideas of te proofs for te NSE carry troug to tis problem as well, altoug more tecnical details arise. An extension to time dependent MHD can be performed analogously to ow te NSE was extended in Section 3. Teorem On a fixed mes te grad-div stabilized TH velocity and magnetic field solutions to (5.3.-(5.3.4 converge to te SV velocity and magnetic field solutions wit convergence order γ in te energy norm, as γ ; if (u, B is te TH solution and (u, B is te SV solution, ten (u u + (B B C γ. Proof. Let (u, p, B, λ SV (V, Q, V SV, Q denote te solution of (5.3.-(5.3.4 using SV elements, (u, p, B, λ (V, Q, V, Q for te TH solution. Additionally, denote te difference between te velocity solutions and te magnetic field solutions by r u V and r B V, so tat u = u + r u, B = B + r B. 7

79 Plugging in te TH and SV solutions into (5.3. gives te following equations: v V, b (u, u, v + ν( u, v sb (B, B, v + γ( u, v = (f, v, (5.3.5 b (u, u, v + ν( u, v sb (B, B, v (p, v = (f, v. (5.3.6 Subtracting (5.3.6 from (5.3.5 gives ν( r u, v + γ( u, v = b (u, r u, v b (r u, u, v +sb (B, r B, v + sb (r b, B, v (p, v. (5.3.7 Similarly, plugging in te TH and SV solutions into (5.3.3 gives te following two equations: χ V, ν m ( B, χ b (B, u, χ + b (u, B, χ +γ( B, χ = ( G, χ, (5.3.8 ν m ( B, χ b (B, u, χ + b (u, B, χ +(λ, χ = ( G, χ. (5.3.9 Subtracting (5.3.9 from (5.3.8 results in te following equality, ν m ( r B, χ + γ( B, χ = b (B, r u, χ + b (r B, u, χ b (u, r B, χ b (r u, B, χ + (λ, χ. (5.3. Ortogonally decompose r u =: r u + r u and r B =: r B + r B were r u, r B V SV and r u, r B R. Coosing v = r u in (5.3.7, χ = r B in (5.3. and adding te two resulting equations yields ν r u + γ r u + ν m r B + γ r B = b (u, ru, r u b (r u, u, r u + sb (B, r B, r u + sb (r b, B, r u (p, r u +b (B, r u, r B + b (r B, u, r B b (u, rb, r B b (r u, B, r B + (λ, r B (5.3. 7

80 From (5.3.5, (5.3.6 and Lemmas. and..9, we can transform (5.3. to γ r u + r B r u + r B C sm M r u + C s M M r u +sc s M M r B + sc s M M r B + p + C s M M r u +C s M M r B + C s M M r B + C s M M r B + M λ. (5.3. Since u, u, B and B are all bounded by data, r u, r u, r B and r B are as well. Terefore, r u + r B C γ. (5.3.3 It remains to bound r u and r B. We will majorize te terms individually and ten combine te results. First, setting v = r u in (5.3.5 and (5.3.6, and rearranging gives te following ν( u, r u = b (u, u, r u + sb (B, B, r u + (f, r u, (5.3.4 ν( u, r u = b (u, u, r u + sb (B, B, r u + (f, r u. (5.3.5 Subtracting (5.3.5 from (5.3.4, rewriting te nonlinear terms wit standard identities and reducing wit ortogonality properties gives ν r u b (u, r u, r u + b (r u, u, r u + sb (B, r B, r u + sb (r B, B, r u. (5.3.6 Coosing χ = r B in (5.3.8 and (5.3.9, and rearranging gives te following equalities ν m ( B, r B = b (B, u, r B b (u, B, r B + ( G, r B, (5.3.7 ν m ( B, r B = b (B, u, r B b (u, B, r B + ( G, r B. (5.3.8 Subtracting (5.3.8 from (5.3.7, rewriting te nonlinear terms and reducing wit ortogonality properties gives ν m r B b (B, r u, r B + b (r B, u, r B + b (u, r B, r B + b (r u, B, r B. (

81 Adding (5.3.6 and (5.3.9 gives te following upper bound ν r u + ν m r B b (u, r u, r u + b (r u, u, r u + sb (B, r B, r u + sb (r B, B, r u + b (B, r u, r B + b (r B, u, r B + b (u, r B, r B + b (r u, B, r B. (5.3.3 Now using Lemma., (5.3.5, (5.3.6 and te triangle inequality yields ν ru + ν m rb C s (M ru + M rb + sm r B r u + M r u r u + M r u r B + +M r B r B + sm rb r u + M ru r B. (5.3.3 Te first terms may be subtracted from bot sides of (5.3.3 immediately. Te subsequent terms may be andled using Young s inequality to yield ( ν C sm sc s M C s M r u + ( ν m C sm sc s M C s M r B 6ν s C s M r B + 6ν C M r u + 6ν m C s M r u + 6ν m C s M r B. Provided tat ν C s M sc s M C s M >, and ν m C s M sc s M C s M >, (5.3.3 it follows from te triangle inequality tat (u u + (B B C γ. (

82 5.3. Numerical verification for steady MHD To numerically verify te MHD convergence teory, we select te test problem wit solution u = cos(y sin(x, B = x y, P = sin(x + y, ( on te unit square wit ν = ν m =, s = and f and g calculated from tis information. Te mes used was a barycenter-refined uniform triangulation of Ω, wic provided a total of 4, 34 degrees of freedom for te (P, P TH computations and 6, 6 for (P, P disc SV. Te results are sown in Table 5., and first order convergence in te H norm is observed for bot velocity and te magnetic field. γ u γ T H u SV H rate u γ T H Bγ T H B SV H rate B γ T H 7.5E E E E E-4-3.9E-4.93E E-7.79E E-5.38E E-7.688E-5.8.6E-5.83E E E-6.97.E-6.936E E E-7..3E-7.947E-9. 5.E- Table 5.: Convergence of te grad-div stabilized TH steady MHD solutions toward te SV steady MHD solution, first order as γ. 5.4 Extrapolating to approximate te γ = solution Te previous sections verified tat, provided te SV element is stable, te grad-div stabilized TH solutions to Stokes type problems converge to te SV solution as γ. However, in practice tere are limitations on ow large γ may be cosen, because as γ increases te resulting linear system becomes ill-conditioned. In tis section we consider linearly and quadratically extrapolating from grad-div stabilized TH velocity solutions found wit smaller γ to approximate te SV solution in an effort to improve mass conservation. Let te true solutions to (5.. (5.. be given by u = (x4 x 3 + x (4y 3 6y + y (y 4 y 3 + y (4x 3 6x + x, (5.4. P = x + y + (cos(y + sin(x, (

83 on te unit square wit ν =. Let γ k (k =, or 3 denote a distinct stabilization parameter and let (u γ k, pγ k denote Taylor-Hood solutions of (5..-(5.. wit stabilization parameters γ k. Additionally, let (u Ex, p Ex denote te extrapolated solution and (u, p denote te Scott-Vogelius solution to (5..-(5... Computations were done on a barycenter-refined uniform triangulation of Ω, wic provided 6 degrees of freedom for te (P, P TH elements and 33 degrees of freedom for te (P, P disc SV element. Te results in Table 5.3 are for linear extrapolated solutions, and and Table 5.4 summarizes te results for quadratic extrapolated solutions. Little improvement is seen in linear extrapolation, but a dramatic improvement is observed for quadratic. γ γ u γ uγ u Ex u Ex u H.946e-4.585e e e-6.964e-4.653e-6.38e-7.98e e e e e-6.585e-5.653e-6.6e e e-6.653e-6.47e e-6 Table 5.3: Improved mass conservation using linear extrapolation. γ γ γ 3 u Ex H u EX u H 7.383e-.73e e e- Table 5.4: Improved mass conservation using quadratic extrapolation 75

84 Capter 6 NS-Omega Tis capter studies a finite element metod for an α-model known as te NS-ω model. Successful DNS of fluid flows described by te incompressible NSE, if possible, is expensive for complex flows. Recently it as been found tat α-models are able to more accurately predict fluid flows on coarser spatial and temporal discretizations tan DNS [43, 44, 5, 6, 66, 5,,, 8]. Te NS-ω model is particularly attractive because te model is well-posed, conserves energy, conserves a model elicity [46, 38], and it can be computed efficiently wit unconditionally stable algoritms [44]. Te continuous model is given by u t + ( D N F u u + q ν u = f, (6.. u =, (6.. were u and f represent te same entities as tey do for te NSE. Te operators F and D N are te Helmoltz filter and van Cittert approximate deconvolution operator respectively. Te use of van Cittert approximate deconvolution increases te spatial accuracy of te model. We note tat te model uses te rotational form of te NSE nonlinearity and so its Bernoulli pressure (q = p + u is more complex tan te usual pressure if te convective form of te nonlinearity is used for te NSE. Specifically, Bernoulli pressure contains a velocity component wic can lead to large pressure approximation errors in finite element discretizations, and tis can in turn adversely effect te velocity error. Using SV elements decouples te velocity and pressure error, and provides pointwise divergence free velocity approximations. In addition to using SV elements te finite element discretization we use for te model 76

85 (6..-(6.. also linearizes te regularized terms via Baker s metod [3]. Tis decouples te momentum-mass system from te filtering and deconvolution, wic makes te cost of filtering and iger orders of deconvolution negligible in comparison to te momentum-mass solve. 6. A numerical sceme for NS-ω We now are ready to present te NS-ω algoritm we study erein. Te sceme uses a trapezoidal temporal discretization, and uses a Baker-type [3] extrapolation to linearize and maintain unconditional stability. Algoritm 6... Given kinematic viscosity ν >, end-time T >, te time step is cosen t < T = M t, f L (, T ; (H (Ω d, te initial condition u V, te filtering radius α >, deconvolution order N, first find u XSV satisfying (u, v (λ, v = (u, v, v X SV (6.. ( u, r = r Q SV, (6.. ten set u := u, and find (un+, q n+ ( X SV, Q SV for n =,,..., M satisfying t (un+ u n, v + (( DNF ( 3 un un u n+, v (q n+, v + ν( u n+, v = (f n+, v v X SV, (6..3 ( u n+, r = r Q SV. ( Unconditional stability and well-posedness Lemma 6... Consider te NS-ω algoritm 6... A solution u l, l =,... M, exists at eac time-step and is unique. Te algoritm is also unconditionally stable: te solutions satisfy te á priori bound: M u M + ν t u n+/ u + t M f n+/ ν. (6..5 n= n= Proof. Te existence of a solution u n to te sceme in Algoritm 6.. follows from te Leray- Scauder Principle [37]. Te main step is deriving an á priori estimate, wic can be obtained by setting v = u n+/ in (6..3. Te nonlinear term in te sceme vanises wit tis coice. Tus, 77

86 for every n t ( un+ u n + ν u n+/ ν f n+/ + ν un+/, i.e., t ( un+ u n + ν u n+/ ν f n+/. Summing from n =... M gives te desired result. Eac time step in te sceme of Algoritm 6.. only requires te solution of a linear system. Tus, te above stability estimate also implies tat solutions at eac time level exist uniquely. Remark 6... Since te kinetic energy KE(u n := un and energy dissipation ε(u n := ν u n of NS-ω, take te usual form, Lemma 6.. implies KE(u M + t M n= ε(u n+/ KE(u + t M ν n= f n+/. (6..6 Tus, if ν = and f =, KE(u M = KE(u. Hence Algoritm 6.. is energy conserving. 6.. Convergence Analysis given next. Our main convergence result for te discrete NS-ω model described in Algoritm 6.. is Teorem 6.. (Convergence for discrete NS-ω. Consider te discrete NS-ω model. Let (w(t, p(t be a smoot, strong solution of te NSE suc tat te norms on te rigt and side of (6..7-(6..8 are finite. Suppose (u, p SV are te V and Q SV interpolants of (w(, p(, respectively. Suppose (u, q satisfies te sceme (6..3-(6..4. Ten tere is a constant C = C(w, p suc tat ( ν t M n= w u, F ( t,, α + C k+ w,k+, (6..7 (w n+/ (u n+ + u n / / F ( t,, α + Cν / ( t w tt, + Cν / k w,k+, (

87 were F ( t,, α := C {(ν + ν / k w,k+ +ν / k ( w 4,k+ + w / ( 4, + C(N( t w ttt, + f tt, + (ν + ν / w tt, + ν / ( t + α N+ + α k + k+ w /, }. (6..9 Remark 6... Tere are two important points to note from teorem. First, te velocity error does not depend at all on te pressure error. Second, optimal accuracy can be acieved if α O(, and N + k, wic provides a guide for parameter selection. Proof of Teorem 6... Let b ω (u n+/, v n+/, χ n+/ := (( DN F ( 3 un un v n+/, χ n+/, and, ten by adding and subtracting terms, we can write b ω (u n+/, v n+/, χ n+/ = b(u n+/, v n+/, χ n+/ F E(u n+/, v n+/, χ n+/, were te linear extrapolated deconvolved filtering error FE is given by F E(u n+/, v n+/, χ n+/ := (( u n+/ D NF ( 3 un un v n+/, χ n+/. At time t n+/, te solution of te NSE (w, p satisfies ( w n+ w n, v t + b ω (w n+/, w n+/, v + ν( w n+/, v = (f n+/, v + Intp(w n, v, v V SV (6.. were te pressure term disappears since V SV is now pointwise div-free, as stated in Capter. Te term Intp(w n, v collects te interpolation error, te above linear extrapolated deconvolved 79

88 filtering error and te consistency error. It is given by Intp(w n, v = ( w n+ w n w t (t n+/, v + ν( w n+/ w(t n+/, v t +b ω (w n+/, w n+/, v b ω (w(t n+/, w(t n+/, v F E(w(t n+/, w(t n+/, v +(f(t n+/ f n+/, v. (6.. Subtracting (6.. from (6..3 and letting e n = w n u n we ave t (en+ e n, v + b ω (w n+/, w n+/, v b ω (u n+/, u n+/, v + ν( e n+/, v = Intp(w n, v, v V SV, (6.. were te pressure term of NS-ω disappears since V SV is now pointwise div-free. Decompose te error as e n = (w n U n (u n U n := η n φ n were φn V SV, and U is te L projection of w in V SV. Setting v = φ n+/ in (6.. we obtain (φ n+ φ n, φ n+/ + ν t φ n+/ t b ω (u n+/, e n+/, φ n+/ t b ω (e n+/, w n+/, φ n+/ = (η n+ η n, φ n+/ + tν( η n+/, φ n+/ + t Intp(w n, φ, i.e. ( φn+ φ n + ν t φ n+/ = (η n+ η n, φ n+/ + tν( η n+/, φ n+/ + t b ω (η n+/, w n+/, φ n+/ + t b ω (u n+/ t b ω (φ n+/, w n+/, φ n+/, η n+/, φ n+/ + t Intp(w n, φ n+/. (6..3 We now bound te terms in te RHS of (6..3 individually. According to te coice of U, (η n+ η n, φ n+/ =. Te Caucy-Scwarz and Young s inequalities give ν t( η n+/, φ n+/ ν t η n+/ φ n+/ ν t φn+/ + Cν t η n+/. (6..4 8

89 Lemmas..,.. and.. and standard inequalities give t b ω (η n+/, w n+/, φ n+/ = t( DNF ( 3 ηn ηn w n+/, φ n+/ C t DNF ( 3 ηn ηn w n+/ φ n+/ C(N t ( 3 ηn ηn w n+/ φ n+/ C(N t( η n + η n w n+/ φ n+/ ν t φn+/ + C(N t ν ( η n + η n w n+/, (6..5 Using (..5, we get t b ω (φ n+/, w n+/, φ n+/ = t ( DN F ( 3 φn φn w n+/, φ n+/ C t w n+/ φ n+/ D NF ( 3 φn φn C(N t w n+/ φ n+/ ( φ n + φ n ν t φn+/ + C(N t ν ( φ n + φ n w n+/. (6..6 Te final trilinear term requires a bit more effort. Begin by splitting te first entry of tis term by adding and subtracting w n+/, followed by rewriting te resulting error term as pieces inside and outside of te finite element space. tb ω (u n+/, η n+/, φ n+/ = tb ω (η n+/, η n+/, φ n+/ + tb ω (φ n+/, η n+/, φ n+/ + tb ω (w n+/, η n+/, φ n+/. (6..7 We bound eac of te terms on te rigt and side of (6..7 using te same inequalities and lemmas as above: tb ω (η n+/, η n+/, φ n+/ t DNF ( 3 ηn ηn η n+/ φ n+/ C(N t ( 3 ηn ηn η n+/ φ n+/ ν t 4 φn+/ + C(N tν ( η n + η n η n+/, (6..8 8

90 tb ω (φ n+/, η n+/, φ n+/ = t( DNF ( 3 φn φn η n+/, φ n+/ t (φ n+/ t (φ n+/ C(N t η n+/, D NF ( 3 φn φn η n+/, D NF ( 3 φn φn η n+/ C(N / t ν t φn+/ + t (η n+/ φ n+/, DNF ( 3 φn φn φ n+/ 3 φn / φn ( 3 φn φn φ n+/ 3 φn φn η n+/ + C(N t ν ( φ n + φ n η n+/, (6..9 / tb ω (w n+/, η n+/, φ n+/ t DN F ( 3 wn wn η n+/ φ n+/ C(N t ( 3 wn wn η n+/ φ n+/ ν t 4 φn+/ + C(N tν ( w n + w n η n+/ ν t 4 φn+/ + C(N tν η n+/. (6.. Combining (6..4-(6.. and summing from n = to M (assuming tat φ = reduces (6..3 to M φ M + ν t φ n+/ n= M C t{ Cν ( w n+/ + η n+/ ( φ n + φ n + M n= n= ( (ν + ν η n+/ + ν ( η n + η n w n+/ +ν ( η n + η n η n+/ M + Intp(w n, φ n+/ } n= M C t{ Cν ( w n+/ + η n+/ φ n n= M M +(ν + ν η n+/ + ν η n w n+/ +ν M n= n= n= M η n 4 + Intp(w n, φ n+/ }. (6.. n= 8

91 Now, we continue to bound te terms on te RHS of (6... We ave tat M C t(ν + ν η n+/ C t(ν + ν n= C t(ν + ν M η n n= M k w n k+ n= C(ν + ν k w,k+, (6.. and similarly, C tν M M η n 4 tcν 4k w n 4 k+ n= n= Cν 4k w 4 4,k+. (6..3 For te term M C tν n= M η n w n+/ C tν k w n k+ w n+/ n= Cν k ( w 4 4,k+ + w / 4 4,. (6..4 We now bound te terms in Intp(w n, φ n+/. Using Caucy-Scwarz and Young s inequalities, and Lemmas..4,.. and.. and te regularity assumptions on w, ( w n+ w n t w t (t n+/, φ n+/ φn+/ + w n wn+ w t (t n+/ t φn+ + φn + ( t 3 tn+ w ttt dt, ( t n (f(t n+/ f n+/, φ n+/ φn+/ + f(t n+/ f n+/ φn+ + φn + ( t3 48 tn+ t n f tt dt, (

92 ν( w n+/ w(t n+/, φ n+/ ε ν φ n+/ + C ν w n+/ w(t n+/ ε ν φ n+/ + C ν ( t3 48 tn+/ t n w tt dt, (6..7 b ω (w n+/, w n+/, φ n+/ b ω (w(t n+/, w(t n+/, φ n+/ = b ω (w n+/ w(t n+/, w n+/, φ n+/ + b ω (w(t n+/, w n+/ w(t n+/, φ n+/ C DN F ( 3 (wn w(t n (wn w(t n φ n+/ w n+/ +C DNF ( 3 w(t n w(t n φ n+/ w n+/ w(t n+/ + C(N ( 3 w(t n w(t n φ n+/ w n+/ w(t n+/ tn+ C(N ν ( w(t n + w(t n ( t 3 w tt dt + ε 3 ν φ n+/ 48 t n tn+ ε 3 ν φ n+/ ( t3 + C(N ν w tt dt. ( t n Next we will bound te linear extrapolated deconvolved filtering error FE using w(t n+/ ( 3 w(t n w(t n = O( t (6..9 and Lemma..3 as well. Tus, F E ( w(t n+/ DN F ( 3 w(t n w(t n w(t n+/, φ n+/ C (w(t n+/ DNF ( 3 w(t n w(t n w(t n+/ φ n+/ ε 4 ν φ n+/ + Cν w(t n+/ (w(t n+/ D NF ( 3 w(t n w(t n ε 4 ν φ n+/ + Cν w(t n+/ ( (I D NF w(t n+/ + D N F (w(t n+/ ( 3 w(t n w(t n ε 4 ν φ n+/ + C(Nν ( t 4 + α 4N+4 + α k + k+ w(t n+/. (

93 Combine (6..5-(6..3 to obtain t M n= M t C n= Intp(w n, φ n+/ M φ n+ + (ε + ε + ε 3 + ε 4 t ν n= φ n+/ + C(N( t 4 ( w ttt, + f tt, + (ν + ν w tt, + ν ( t 4 + α 4N+4 + α k + k+ w /,. (6..3 Let ε = ε = ε 3 = ε 4 = / and wit (6..-(6..4, (6..3, from (6.. we obtain M φ M + ν t φ n+/ M C t n= n= M Cν ( w n+/ + k φ n + C t φ n +C(ν + ν k w,k+ + Cν 4k w 4 4,k+ + Cν k ( w 4 4,k+ + w / 4 4, + C(N( t 4 ( w ttt, + f tt, + (ν + ν w tt, n= + ν ( t 4 + α 4N+4 + α k + k+ w /,. (6..3 Hence, wit k, from Gronwall s Lemma (see Lemma..5, we ave M φ M + ν t φ n+/ n= C {(ν + ν k w,k+ + ν 4k w 4 4,k+ + ν k ( w 4 4,k+ + w / 4 4, + C(N( t 4 ( w ttt, + f tt, + (ν + ν w tt, + ν ( t 4 + α 4N+4 + α k + k+ w /,}, (6..33 were C = C exp(cν T. Estimate (6..7 ten follows from te triangle inequality and ( To obtain (6..8, we use (6..33 and ( w(t n+/ (u n+ + u n/ (w(t n+/ w n+/ + η n+/ + φ n+/ ( t3 48 tn+ t n w tt dt + C k w n+ k+ + C k w n k+ + φ n+/. 85

94 6..3 An alternative coice of α It is common in α-models for te coice of filtering radius parameter to be cosen on te order of te meswidt, α = O(. From te preceding error analysis, it can be seen tat suc a coice of α is te largest it can be witout creating suboptimal asymptotic accuracy. Altoug tis provides some guidance on te coice of α, finding an optimal α on a particular fixed mes still may require some tuning. We describe now a connection between NS-ω and te velocity-vorticity-elicity (VVH formulation of te NSE [6], tat suggests an alternative coice of α tat may aid in tis process. NS-ω can be considered as a rotational form NSE formulation were te vorticity term is andled by oter equations, wic for NS-ω is te regularization equations. Suc a formulation is quite similar to a velocity-vorticity metod, were te vorticity comes directly from solving te vorticity equation. In particular, consider te numerical metod devised in [6] for te VVH NSE formulation: Algoritm 6... Step. Given u n, u n, w n and u = 3 un un, find w n+ and el n+/ from w n+ w n ν w n+/ + D(w n+/ u el n+/ = f n+/ t (6..34 w n+ = (6..35 w n+ = (u n u n on Ω (6..36 Step. Given u n, w n and w n+, find u n+ and P n+/ u n+ u n ν u n+/ + w n+/ u n+/ P n+/ = f n+/ t (6..37 u n+ = (6..38 u n+ = φ on Ω (6..39 were φ is te Diriclet boundary condition function and D( denotes te deformation tensor, i.e. D(v = ( v+( vt, u, w denote velocity and vorticity, P is te Bernoulli pressure and el is elical density. If f is irrotational and we remove te nonlinear term from te vorticity equation, tis system is analogous te NS-ω sceme erein if we identify elical density wit te Lagrange multiplier λ corresponding to te incompressibility of te filtered velocity, and ν t wit α, i.e. α = ν t. 86

95 Coosing an optimal filtering radius α is certainly problem dependent, and by no means are we suggesting tis coice is always optimal. However, our numerical experiments sow it can be a good starting point for coosing α wen using NS-ω. 6. Numerical experiments In tis section we present several numerical experiments tat demonstrate te effectiveness of te numerical metod studied erein. Te first two experiments are for bencmark tests of cannel flow over a step and around a cylinder, respectively, and bot sow excellent results. Te tird and fourt tests are done wit SV elements and TH elements, and compare solutions for a problem wit known analytical solution and te cylinder problem. 6.. Experiment : Cannel flow over a forward-backward facing step Our first numerical experiment is for te bencmark d problem of cannel flow over a forward-backward facing step. Te domain Ω is a 4x rectangle wit a x step 5 units into te cannel at te bottom. Te top and bottom of te cannel as well as te step are prescribed wit no-slip boundary conditions, and te sides are given te parabolic profile (y( y/5, T. We use te initial condition of u = (y( y/5, T inside Ω, and run te test to T = 4. For a cosen viscosity ν = /6, it is known tat te correct beavior is for an eddy to form beind te step, grow, detac from te step to move down te cannel, and a new eddy forms. For a more detailed description of te problem, see [5, 3]. Te eddy formation and separation present in tis test problem is part of a complex flow structure, and its capture is critical for an effective fluid flow model. Moreover, a useful fluid model will correctly predict tis beavior on a coarser mes tan can a direct numerical simulation of te NSE. For te following tests, we computed Algoritm 6.. wit (P, P disc SV elements on a barycenter-refined mes, yielding 4,467 total degrees of freedom, wit deconvolution order N =, and varying α. For comparison, we also directly compute te (linearized NSE (α = N =. We compute first wit timestep t =.5, and te solutions at T = 4 are sown in Figure 6.. Several interesting observations can be made. First, we note tat te optimal coice of α appears to be near α = ν t, as tis is te only solution to predict a smoot flow field and eddies forming and detacing beind te step. For te te NSE (α =, a smoot flow field is predicted; owever, te eddies beind te step appear to be stretcing instead of detacing. Larger values of α, including te common coice of te average element widt α =, give increasingly worse solutions. Tis is 87

96 somewat counterintuitive, as α is a filtering radius tat is supposed to regularize and tus smoot oscillations. A closer examination reveals tat te oscillations are arising from an inability of te more regularized models to resolve te flow at te top left corner of te step, were te flow near te bottom of te cannel is forced up to intersect wit te free stream. To test te scaling of optimal α wit t, we compute wit te same data, but wit timesteps t =. and.5, wit parameter α = ν t. Te results at T = 4 are sown in Figure 6., and results are good in bot cases. However, for te smaller timestep, we see te eddies stretcing instead of detacing. Tis is not surprising, as one sould expect some -dependence on te coice of α. 6.. Experiment : Cannel flow around a cylinder Te bencmark problem of d cannel flow around a cylinder as been studied in numerous works, e.g. [67, 3, 34, 4], and is well documented in [67]. Te domain is te rectangle [,.] [,.4] representing te cannel wit flow in te positive x direction, wit a circle radius.5 centered at (.,. representing te cylinder. No slip boundary conditions are prescribed on te top and bottom of te cannel as well as on te cylinder, and te time dependent inflow and outflow velocity profiles are given by u(, y, t = u(., y, t = [ ] T 6 sin(πt/8y(.4 y,, y.4..4 Te forcing function is set to zero, f =, and te viscosity at ν =., providing a time dependent Reynolds number, Re(t. Te initial condition is u =, and we compute to final time T = 8 wit time-step t =.5. An accurate approximation of tis flow s velocity field will sow a vortex street forming beind te cylinder by t = 4, and a fully formed vortex street by t = 7. We test te algoritm wit α = and α = ν t, on a barycenter refined mes tat provides 6,656 degrees of freedom for (P, P disc SV elements, and again find tat α = ν t provides a better solution tan α =. Tese results are sown for t = 7 in Figure 6.3. Te α = ν t solution agrees wit documented DNS results [8, 4], but te α = solution at t = 7 is observed to be incorrect, as it does not fully resolve te wake, and its speed contours sow it gives a muc different (and tus incorrect solution beind te cylinder. 88

97 nse dt=.5 NSE, t = NS-ω, t =.5, α = omega alpa = sqrt(nudt ν t = omega alpa =.3 NS-ω, t =.5, α = NS-ω, t omega = alpa.5, =.6 α = = Figure 6.: Sown above are te T = 4 SV solutions as velocity streamlines over speed contours for te step problem from Experiment. Sown are te NSE (top wic is somewat underresolved on tis mes as te eddies are not fully detacing, NS-ω wit α = ν t (second from top wic agrees wit te known true solution, NS-ω wit α =.3 (tird from top wic as oscillations present in te speed contours, and NS-ω wit α = =.6 (bottom wic is a poor approximation. All of te solutions are pointwise divergence-free. 89

98 NS-ω, t =.5, α = ν t NS-ω, t =., α = ν t Figure 6.: Sown above are te T = 4 SV solutions as velocity streamlines over speed contours for te step problem from Experiment, wit parameter cosen as α = ν t, for varying timesteps SV elements, α = = SV elements, α = ν t = Figure 6.3: Te above pictures sow te velocity fields and speed contours at t = 7 using SV. elements wit. α = (top and α = ν t (bottom. Te α = solution is under-resolved, as it loses resolution of te vortex street, and its speed contours are inaccurate. Te α = ν t solution captures te entire wake, and its speed contours agree well wit te known solution. 9

99 6... Comparison to TH element solution Using te same problem data as Experiment above, we also compute using (P, P TH elements, wit α = ν t (wic gave about te same answer as for α =. Since tis element pair is widely used and is closely related to SV elements (tey differ only in te pressure space being continuous or not, a comparison is of interest. Since TH uses a continuous pressure space, wit te same mes te total degrees of freedom is 7,36. All of problem data is kept te same, and results are sown in Figures 6.4 and 6.5. In Figure 6.4, we observe tat te TH solution is muc worse tan te SV solution sown in Figure 6.3; te TH solution fails to resolve te important beavior beind te cylinder. Figure 6.5 sows mass conservation versus time for te TH and SV solutions. As expected, te SV solution is divergence-free up to macine precision. Te mass conservation offered by te TH solution is very poor. It is not surprising tat te TH solution is muc worse tan te SV solution. It was sown in [4] tat for te rotational form NSE, te Bernoulli pressure error can be large enoug to dramatically increase velocity error for tis problem. Since NS-ω is also rotational form, tis same effect can be expected. However, for te SV solution, as sown erein, te velocity error is independent of te pressure error. Tus even toug pressure error may be large, it as no adverse effect on te velocity error, leaving te good solution seen in Figure TH elements, α = ν t Figure 6.4: Te above picture sow te t = 7 solution using TH elements, as a velocity vector field and speed contours. Tis solution is incorrect, as it fails to capture any wake beind te cylinder Experiment 3: Effect of pressure error on velocity error In tis experiment, we investigate more closely te effect of te pressure error on te velocity error, wic caused a dramatic difference between SV and TH solutions in te above experiment of flow around a cylinder. Te error analysis in Section 6.. sowed tat in Algoritm 6.., wic uses SV elements, te velocity error is not affected by te pressure error. If TH elements are used, owever, 9

100 5 u (t n 5 Taylor Hood Scott Vogelius t n Figure 6.5: Sown above are te plots of te L norms of te divergence of te velocity solutions versus time, for te SV and TH solutions, bot wit α = ν t =.. As expected, te SV solution is incompressible to near macine precision. Te TH solution, owever, gives very poor mass conservation. ten te energy error of te velocity can be sown to depend on C(ν t M n= inf r Q T H q r, e.g. [44], altoug te scaling by C(ν of tis term can be reduced by using grad-div stabilization [6, 4, 5]. To better demonstrate tis effect, we compute Algoritm 6.. wit bot SV and TH elements, for a series of simple test problems wit increasing pressure complexity and te same velocity solution. On te domain, Ω = (, and t. = T, we coose u = ( +.t cos(y sin(x, p = x + y + sin(n(x + y, wic will solve te NSE wit an appropriate function f. Solutions are approximated to tis problem on a quasi-uniform barycenter-refined mes tat provides,64 degrees of freedom wit (P, P disc SV elements (7,58 for velocity and 5,364 for pressure and 8,8 degrees of freedom wit (P, P TH (7,58 for velocity and 94 for pressure, kinematic viscosity is set to be ν =., timestep t =.5, α = ν t =.58, N =, and te parameter for pressure complexity n =,,, 3. Te results are sown in Table 6., and as expected te error in te SV velocity solution is unaffected by te increase in pressure complexity. However, te TH velocity solution significanty loses accuracy. Also included in te table is te size of te velocity divergence, measured in L (, T ; L (Ω. As expected, for te SV solution, near macine epsilon is found for eac n, but for TH, te quantity is non-negligible and gets significantly worse wit increasing pressure complexity. 9

101 n u NSE u SV, u SV, u NSE u T H, u T H, 7.33E-5.6E E-3.775E E-5.67E E E E-5.E-4.76E-.533E E E E- 3.35E- Table 6.: Errors in velocity and divergence for Experiment for SV and TH elements used wit Algoritm

102 Capter 7 Leray-deconvolution for MHD In tis capter we study te Leray-deconvolution model for te MHD. We prove conservation laws for te continuous model, well-posedness, and study limiting beavior as α and N. Te model is studied in te context of periodic boundary conditions, wic are not pysically recognized, but restricting to periodic boundary conditions is an important first step in model development. Extensions to omogenous Diriclet boundary conditions for te velocity (and te regularized velocity would work in te same way, but suc a boundary condition for te regularized velocity is likely not appropriate. To date te correct treatment of oter types of boundary conditions for te velocity and te regularized velocity remains an open problem [45] Error, Existence, and Uniqueness of te Continuous Model Te analytical study of te MHD Leray-deconvolution models begins by establising existence and uniqueness of solutions and an energy balance. We are also interested in te consistency error of te model. First, we make precise te definition of te MHD Leray-deconvolution model. Let T >, f, g L ((, T ; Hp and u, B Hp be given. Ten for α > and N <, te problem is: find (u, B, p satisfying u, B L ([, T ], H p L ([, T ]; H p, (7.. u t, B t L ([, T ]; H p, (7.. p L ([, T ]; L p,, (7..3 (

103 and te following relation in a distributional sense u t + H N(u u Re u + P sh N (B B = f, (7..5 B t + H N(u B Re m B H N (B u = g, (7..6 u(x, = u, (7..7 B(x, = B. (7..8 Before we study te well-posedness of te model we study te accuracy and sow tat as α te asymptotic consistency error is O(α N+. Tis is done by rearranging te MHD so tat te Leray-α deconvolution model appears on te left, and te residual of te true solution of te MHD in te model on te rigt. Rearranging gives u t + H N (u u Re u + P sh N (B B f = [H N (uu uu] +s [BB H N (BB], (7..9 B t + H N (u B Re m B H N (B u g = [H N (ub ub] + [Bu H N (Bu]. (7.. Tus te error tensors in (7..9 and (7.., τ, and τ respectively are given by τ := H N (uu uu + sbb H N (BB, (7.. τ := H N (ub ub + Bu H N (Bu. (7.. Adding (7..9 and (7.. we see tat te MHD Leray-α deconvolution model s consistency error tensor, τ, is τ := τ + τ Analysis of modeling error for various models utilizing deconvolution [5, 6, 4, 4], as sown tat te error is actually driven by τ rater tan τ. Teorem 7... Te consistency error of te MHD Leray-deconvolution wit order N is O(α N+. Tat is Ω ( τ dx α N+ N+ ( α + (N+ u u + s N+ ( α + (N+ B B ( +α N+ N+ ( α + (N+ u B + N+ ( α + (N+ B u. 95

104 Proof. We rewrite (7.. and (7.. to get τ = (H N (u uu + s(b H N (BB, (7..3 τ = (H N (u ub + (B H N (Bu. (7..4 Integrating τ over Ω, using Caucy-Scwarz, and Lemma.. gives τ dx α N+ ( N+ F N+ u u + s N+ F N+ B B Ω α N+ ( N+ F N+ u B + N+ F N+ B u. (7..5 Substituting in for te definition of te filter finises te proof. Teorem 7... Let α and let N. Assume u, B H p and f, g L ([, T ]; H p. Ten te MHD Leray-deconvolution model admits a unique solution (u, B, p, wit u, B L ([, T ]; H p L ([, T ]; H p. Te solution satisfies te integral relation v, χ V T T ( d u(τ, vdτ Re dτ ( d B(τ, χ Re dτ T T ( u(τ, vdτ + s ( B(τ, χdτ + T T T T (H N u(τ u(τ, vdτ (H N B(τ B(τ, vdτ = (H N u(τ B(τ, χdτ (H N B(τ u(τ, χdτ = T T (f(τ, vdτ,(7..6 ( g(τ, χdτ.(7..7 Furter, te solution satisfies te regularity u, B L ([, T ]; H p L ([, T ]; H p, (7..8 conserves energy u(t + s B(t + Re = u + s B + t t u dxdτ + sre m t (f, udxdτ + s Ω Ω Ω t Ω B dxdτ ( g, Bdxdτ, (7..9 and conserves cross elicity, (u(t, B(T + T (Re + Re m ( u, B = (u, B + T (f, B + T ( g, u. (7.. 96

105 Proof. We prove existence following te Galerkin metod, wic makes use of te spaces D(Ω := {u C (Ω : u is periodic on Ω}, V := {u D(Ω : u = }, H := te closure of V in L, and V := te closure of V in H. Te space V is separable and we coose te basis for V to be te eigenfunctions of te Stokes operator, A, wic we will denote by {w i } i=. We denote te ortogonal projection operator from H onto te space spanned by w, w,..., w m by P m. Te Galerkin system of order m corresponding to (7..5-(7..8 is tus d dt u m + Re Au m + P m H N u m u m sp m H N B m B m = P m f m, (7.. d dt B m + Re AB m + P m H N u m B m P m H N B m u m = P m g m, (7.. u m ( = P m u, (7..3 B m ( = P m B. (7..4 Classical results ensure te existence of solutions to te Galerkin system for any finite m on some interval [, t m ], and tat P m u u as m (see for example [7, 7, 7]. Our goal is to derive a priori estimates for te Galerkin system tat will ensure t m = T, and allow us to pass limits as m using te Aubion-Lions Lemma. We begin deriving te estimates by multiplying (7.. by u m, and multiplying (7.. by B m to see ( d dt u m, u m + Re A um sp m (H N B m B m, u m = (P m f m, u m, (7..5 ( d dt B m, B m + Re m A Bm P m (H N B m u m, B m = (P m g m, B m. (7..6 Multiplying (7..6 by s, adding te result to (7..5, and reducing gives ( d dt u m, u m + s( d dt B m, B m + Re A um + sre m A Bm = (P m f m, u m + s(p m g m, B m. (

106 Integrating (7..7 from to s, and standard inequalities gives u m (s + B m (s P m u + P m B + Re u + B + Re s s f(τ V dτ + Re m f(τ V dτ + Re m s s g(τ V dτ g(τ V dτ. (7..8 It now follows tat sup u m (s u + B + Re s [,T ] sup B m (s u + B + Re s [,T ] s s f(τ V dτ + Re m f(τ V dτ + Re m s s g(τ V dτ, and (7..9 g(τ V dτ. (7..3 Tus, te sequences u m and B m remain in a bounded set of L ([, T ]; H. We now integrate (7..7 from to T to get Re T T A um (τ dτ + sre m A Bm (τ dτ u + s B + Re T f(τ V dτ + sre m T g(τ V dτ. (7..3 Tis estimate sows te sequences u m and B m remain in a bounded set of L ([, T ]; V. To pass te limit as m we use te Aubion-Lions Lemma, wic requires us to derive bounds for d dt u m and d dt B m. Rearranging (7.. and (7.. gives d dt u m = Re Au m P m H N u m u m + sp m H N B m B m + P m f m, (7..3 d dt B m = Re AB m P m H N u m B m + P m H N B m u m P m g m. (7..33 We argue tat d dt u m and d dt B m remain in a bounded set in L 4 3 ([, T ]; V. It is clear tat P m f m, P m g m, Au m, and AB m are in L 4 3 ([, T ]; V. So, we argue te nonlinear terms satisfy te desired regularity. We begin by majorizing te first nonlinear term in (7..3 as follows s P m H N B m B m P m H N B m Bm 3 C(N B m Bm 3 = s P m H N B m B m 4 3 C(N B m B m (7..34 It now follows from te boundedness of B m in L ([, T ]; H, and integrating (7..34 from to T 98

107 tat T T s P m H N B m (τ B m (τ 4 3 dτ C(N B m (τ dτ. (7..35 Te remaining nonlinear terms can be sown to satisfy te desired regularity similarly. Tus, we may conclude tat d dt u m and d dt B m remain in a bounded set of L 4 3 ([, T ]; H. At tis point we ave argued sufficient regularity to use te Aubion-Lions Lemma and sow solutions exist. However, te regularity we ave establised so far will not allow us to sow tat solutions are unique. Nor will it allow us to sow tat te solutions conserve energy and crosselicity. Tese properties require more regularity of solutions and so we now establis tat u m and B m are in L ([, T ]; V L ([, T ]; Hp. We begin deriving te estimates by multiplying (7.. by Au m, and multiplying (7.. by AB m to see d dt A um + Re Au m sp m (H N B m B m, Au m = (P m f m, Au m, (7..36 d dt A Bm + Re m A B m P m (H N B m u m, AB m = (P m g m, AB m.(7..37 Adding (7..36 to (7..37 and applying Young s inequality gives d dt A um + d dt A Bm + 5Re Au m + 5Re m AB m 6 6 CRe f + CRe m g (H N u m B m, AB m + (H N B m u m, AB m (H N u m u m, Au m + s(h N B m B m, Au m. (7..38 We now bound te first nonlinear term in (7..38 using standard bounds on te nonlinearity and Agmon s inequality as follows: (H N u m B m, AB m H N u m A Bm AB m H N u m H A Bm AB m. (7..39 Note tat te last inequality in (7..39 makes sense because of te regularity of u and Lemma..4. Applying Young s inequality gives (H N u m B m, AB m CRe H N u m H A Bm + Re AB m. (

108 Similar treatment of te remaining nonlinear terms of (7..38 and rearranging gives d dt A um + d dt A Bm + Re Au m + Re m AB m CRe f + CRe m g + CRe m H N u m H A Bm + CRe m H N B m H A um + CRe H N u m H A um + CRe H N B m H A Bm. (7..4 We now consider two cases, Case : A um + A Bm <, and Case : A um + A Bm. If A u m + A B m < ten d dt A um + d dt A Bm CRe f + CRe m g + CRe m H N u m H + CRe m H N B m H + CRe H N u m H + CRe H N B m H. (7..4 Integrating (7..4 from to t gives A um (t + A Bm (t t CRe f(τ dτ t t + CRe m H N u m (τ H dτ + CRe m + CRe t +CRe m g(τ dτ t H N B m (τ H dτ H N u m (τ H dτ + CRe t H N B m (τ Hdτ (7..43 Taking a supremum over t [, T ] ensures u, B m L ([, T ]; V. We now consider te case wen A u m + A B m. Let ν = max(re, Re m. Te assumption A u m + A B m implies d dt A um + d dt A Bm Cν max{( f + g, ( H N u m H + H N B m H }( A um + A Bm. (7..44

109 Integrating from to t and using Gronwalls inequality gives A um (t + A Bm (t ( A um ( + A Bm ( exp( t Cν max{( f + g, ( H N u m H + H NB m H }. (7..45 Tus, we ave sown tat u m, B m L ([, T ]; V. Integrating (7..4 from to T and using u m, B m L ([, T ]; V implies u m, B m L ([, T ]; H p V. Te Aubion-Lions Lemma implies te existence of u, B L ([, T ]; V L ([, T ]; H p V wit subsequence u mj and B mj so tat u mj u weakly in L ([, T ]; D(A, u mj u strongly in L ([, T ]; V, B mj B weakly in L ([, T ]; D(A, and B mj B strongly in L ([, T ]; V. (7..46 We now sow te solution is unique. Assume tere are two solutions (u, B and (u, B. Te solutions satisfy d dt u Re u = H N u u + sh N B B + f, (7..47 d m B = H N u B + H N B u + g, (7..48 dt B Re d dt u Re u = H N u u + sh N B B + f, (7..49 d dt B Re m B = H N u B + H N B u + g. (7..5 Let e u := u u and e b := B B. Subtracting (7..5 from (7..49, and (7..48 from (7..47, and rearranging te nonlinear terms gives d dt e u Re e u = H N u e u H N e u u d dt e B Re m e B = H N u e B H N e u B +sh N B e B + sh N e B B, (7..5 +H N B e u + H N e B u. (7..5

110 Multiplying (7..5 by e u, and (7..5 by e B, and reducing gives d dt e u + Re u = (H N e u u, e u + s(h N B e B, e u +s(h N e B B, e u (7..53 d dt e B + Re m B = (H N e u B, e B + (H N B e u, e B +(H N e B u, e B. (7..54 Scaling (7..54 by s and adding te result to (7..53 and reducing gives d dt e u + d dt e B + Re u + Re m B = (H N e u u, e u + (H N e B B, e u s(h N e u B, e B + s(h N e B u, e B. (7..55 We majorize te first nonlinear term in (7..55 using standard inequalities and te regularity of u as follows (H N e u u, e u H N e u u e u C(N, u, Re e u + Re e u. ( Similar treatment of te remaining nonlinear terms and rearranging gives d dt e u + d dt e B C(N, u, u, B, B, Re, Re m, s( e u + e B. (7..57 Gronwalls inequality finises te proof of uniqueness. Remark 7... Te energy and cross-elicity balances of te model are a result of te strong convergence of u mj to u and B mj to B in L ([, T ]; V. Tis allows us to pass limits in te Galerkin system wic ave energy and cross-elicity balances. If we tried to construct a proof similar to te one above to sow energy and cross elicity conservation for te 3d MHD te proof would fail at ( Tus, for te MHD Galerkin system we can only sow strong convergence in L ([, T ]; H. Hence we would not be able to pass a limit in terms like u mj. Instead we can only argue an inequality using u lim inf j u mj, and oter suc inequalities.

111 7..5 Limiting beavior Teorem 7.. states tat solutions to te Leray-deconvolution model exist uniquely, provided α >, and N <, and tat te model conserves energy and cross elicity for ideal MHD. Considering te model as a way to understand te MHD equations, it is natural to study beavior of te model as α wit N fixed. If N is fixed to be, ten te model is te Leray-α MHD model and Yu and Li [75] proved tat as α tere is a subsequence of solutions to te Leray α model wic converges to a weak solution of te MHD. Computationally, te Leray-deconvolution model is O(α N+ + t + k accurate (see Teorem 7... Wen computing wit te model te filtering radius, α, is cosen to be on te order of, te mes widt. Tus, to improve accuracy one may eiter srink α or increase N. Te first approac requires using a finer mes, wic increases te total degrees of freedom and tus increases te runtime. However, increasing N only requires us to solve one more sifted Poisson problem per deconvolution step. Tis leads one to study te beavior of te model wen N wile α remains constant. Tis was studied by Layton and Lewandowski for te Leray-Deconvolution model of te NSE [4]. Teorem Let α > be fixed, and assume tat f, g L ((, T ]; H p(ω and u, B H p(ω. Let (u N, B N, p denote te solution to (7..5-(7..8 for N >. Ten tere exist subsequences {u Nj } j N of {u N } N N and {B Nj } j N of {B N } N N as N suc tat (u Nj, B Nj converge to (u, B, te weak solutions of te MHD equations. Te convergence is weak in L ((, T ]; H p(ω and strong in L ((, T ]; H p(ω. Proof. Tis proof requires te use of te Aubion-Lions Lemma and so we must argue regularity of te solutions. It follows immediately from Teorem 7.. tat u N and B N are bounded in L ([, T ]; V L ([, T ]; H, and tat t u j, t B j L 4 3 ([, T ]; Hp (Ω. Wit te establised regularity we may use te Aubion-Lions Lemma to extract subsequences, wic we continue to denote by {u j } j N and {B j } j N, wic converge to functions u and B respectively, and tat u, B L ([, T ]; H p(ω L ([, T ]; H p(ω [4, 4, 7]. Te convergence is strong in L ([, T ]; H p(ω and weak in L ([, T ]; H p(ω. 3

112 We next verify tat u, B satisfy te integral relation v, χ V T T ( d u(τ, vdτ Re dτ ( d B(τ, χ Re dτ T T ( u(τ, vdτ + s ( B(τ, χdτ + T T T T (u(τ u(τ, vdτ (B(τ B(τ, vdτ = (u(τ B(τ, χdτ (B(τ u(τ, χdτ = T T (f(τ, vdτ, (7..58 ( g(τ, χdτ. (7..59 We now sow te solutions satisfy v, χ V T T ( t u j, v = ( t B j, χ = T T T Re ( u j (τ, vdτ +s T Re m ( B j (τ, χdτ +s T T b(h Nj u j (τ, u j (τ, vdτ (H Nj B j (τ B j (τ, vdτ + b(h Nj u j (τ, B j (τ, χdτ (H Nj B j (τ u j (τ, χdτ + T T (f(τ, vdτ,(7..6 ( g(τ, χdτ.(7..6 It is obvious tat we may pass to limits in te linear terms, and so we restrict attention to te nonlinear terms. For te first nonlinear term in (7..6 we ave T T b(h Nj u j, u j, v b(u, u, v T b(h Nj u j, (u j u, v + T b((h Nj u j uu, v (7..6 We majorize (7..6 wit (.. and (..3 and ten apply Caucy-Scwarz wic gives T T b(h Nj u j, u j, v + T C v { b(u, u, v T H Nj u j (u j u + C v {( T H Nj u j T T (H Nj u j u u } ( u j u T T ( H Nj (u j u ( ( u + ( (H Nj u u T ( u }. (7..63 It follows from Lemma..4 and te strong convergence of u j to u in L ([, T ]; H p(ω tat te RHS of (7..63 goes to zero as j. 4

113 Te second nonlinear term in (7..6 is treated te same way as te first nonlinear term. Te beavior of te nonlinear terms in (7..6 is less clear because eac ave a magnetic component and a velocity component, so, we focus on te first nonlinear term in (7..6: T b(h Nj u j, B j, χ T (u, B, χ b(h Nj u j, (B j B, χ + T C χ { T T H Nj u j (B j B + b((h Nj u j u, B, χ T (H Nj u j u B } (7..64 Te nonlinearity is now bounded te same way as before wit (.., (..3 and Caucy-Scwarz. Teorem Let N be fixed, and assume tat f, g L ([, T ]; Hp (Ω and u, B Hp(Ω. Let (u α, B α, p denote te solution to (7..5-(7..8 for α >. Ten tere exist subsequences {u αj } j N of {u α } α R and {B αj } j N of {B α } α R as α + suc tat (u αj, B αj converge to (u, B, te weak solutions of te MHD equations. Te convergence is weak in L ((, T ]; H p(ω and strong in L ((, T ]; H p(ω. Proof. Te proof of Teorem 7..4 follows immediately from te limiting beavior proof for te Leray-α model for te MHD equations done in [75] and Lemma A Numerical Sceme for te Leray-deconvolution model for MHD In tis section we present a numerical sceme for te Leray-deconvolution model of te MHD equations. Te numerical sceme is derived wit a Galerkin finite element discretization in space and a Crank-Nicolson discretization in time. We sow tat te metod is well-posed, conserves energy and cross elicity, is unconditionally stable wit respect to timestep, and is optimally convergent. Additionally, we study te numerical sceme wit te Scott-Vogelius (SV element pair wic enforces strong mass and electrical carge conservation. 7.. Te numerical sceme We are now ready to introduce te numerical sceme for te model (7..5-(7..8. Following [3,, 43] we ave found it advantageous to linearize te sceme in a manner tat maintains stability, 5

114 asymptotic accuracy, and conservation laws. Tis is done by approximating te first term in te nonlinearities wit a second order accurate in time extrapolation. Let Ũ n B n := 3 Bn Bn. := 3 un un and Algoritm 7... Assume te initial conditions are divergence free, i.e. u, B V. Ten te linearized sceme at eac time step is given by, (v, χ, q, r (X, X, Q, Q t (un+ u n, v + b(dn Ũ n n+, u, v + Re ( u n+, v sb(dn B n n+, B, v (P n+, v = (f(t n+, v, (7.. t (Bn+ B n, χ + Re m ( B n+ +b(d NŨ n, χ b(d N B n ( u n+, q =, (7.., u n+, χ, B n+, χ + (λ n+, χ = ( g(t n+, χ, (7..3 ( B n+, χ =. (7..4 It is sufficient to maintain all te teoretical results to define u := u and B := B. Here P n+ := p n+ + s Bn+ is a modified pressure derived from te vector identity B = ( B + ( B, and λ n+ := Re m B n+ is a Lagrange multiplier. Remark 7... Extrapolating te nonlinearities in te sceme is computationally attractive as it makes te sceme linear, and decouples te filtering and deconvolution. Tus, te velocity and magnetic fields may be filtered independently of eac oter. Tese are fast solves wen compared to te full MHD system. Tus, te extra solves required to implement te Leray-α deconvolution model wit N = do not significantly increase te run time over DNS. Teorem 7... Assuming f, g L ((, T, V (Ω, solutions to (7..-(7..4 exist at eac timestep. Furter, te sceme is unconditionally stable and satisfies te a priori bound: ( u M + s B M M + t n= (Re u n+ + sre m B n+ ( u + s B M + t (Re f(t n+ V + sre m g(t n+ (7..5 n= Remark 7... Existence of solutions to (7..-(7..4 follows directly from te stability estimate 6

115 and is a straigtforward extension of te work done in []. Te a priori bound can be derived from te following energy conservation lemma by applying standard inequalities. Lemma 7... Solutions to (7..-(7..4 admit te following conservation laws Mass conservation u n = (pointwise Incompressibility of te magnetic field B n = (pointwise Global energy conservation = ( um + s M BM + t ( u + s B + t M n= n= (Re u n+ + sre m B n+ ( (f(t n+, u n+ + s( g(t n+, B n+ (7..6 Global cross-elicity conservation (u M, B M + t = (u, B + t M n= M n= ( (Re + Re m ( u n+, B n+ ( ( g(t n+, u n+ + (f(t n+, B n+ (7..7 Proof. To prove pointwise mass conservation and incompressibility of te magnetic field, we note tat for Scott-Vogelius elements X Q. Tus, we may coose q in (7.. to be u n+ we may coose r in (7..4 to be B n+. Te results now follow. To prove energy conservation we cose v = u n+ in (7.. and χ = B n+ in (7..3. Tis anniilates te first nonlinear term in (7.. as well as te pressure term. In (7..3, tis anniilates te second nonlinear terms and te Lagrange multiplier term. Tis leaves and t ( un+ u n + Re u n+ t ( Bn+ B n + Re m B n+ sb(d N B n b(d N B n, B n+, u n+ = (f(t n+, u n+, (7..8, u n+, B n+ = ( g(t n+, B n+. (7..9 7

116 Next, scale (7..9 by s, add (7..8 to (7..9, and rewrite te nonlinear term in (7..8 using (... Tis gives t ( un+ u n + s t ( Bn+ B n + Re u n+ +sre m B n+ = (f(t n+, u n+ + s( g(t n+, B n+. To finis te te proof of (7..6 we multiply by t, and sum over timesteps. To prove cross elicity is conserved, we begin by coosing v = B n+ anniilates te fourt and fift terms. Next we coose χ = u n+ tird and fift terms. Wat remains is in (7.. wic in (7..3 wic anniilates te t (un+ u n, B n+ t (Bn+ B n, u n+ + b(d N Ũ n, u n+, B n+ + Re ( u n+, B n+ + Re m ( B n+, u n+ + b(d N Ũ n = (f(t n+, B n+, (7.., B n+, u n+ = ( g(t n+, u n+. (7.. Adding (7.. to (7.. and using (.. gives t (un+ u n, B n+ = t + t + t (Bn+ B n, u n+ ( (u n+, B n+ (u n, B n + t ( (B n+, u n+ (B n, u n t ( (u n+, B n (u n, B n+ ( (B n+, u n (B n, u n+, wic reduces to ( (u n+, B n+ (u n t, B n + (Re + Re m ( u n+, B n+ = ( g(t n+, u n+ + (f(t n+, B n+. (7.. Te proof of (7..7 is completed by multiplying (7.. by t, and summing over timesteps. Remark If f = g = Re = Re m conserved. = ten energy and cross elicity are exactly Teorem 7... Suppose (u, B solve te 3d MHD and tat u, B, f, g satisfy te following 8

117 regularity u, B L ([, T ]; H k+, (7..3 u, B L ([, T ]; H N+, (7..4 u t, B t L ([, T ]; H, (7..5 u tt, B tt L 4 ([, T ]; H, (7..6 u ttt, B ttt L ([, T ]; H, and (7..7 f tt, g tt L ([, T ]; H. (7..8 Ten te solution (u, B, p, λ to (7..-(7..4 converges to te true solution wit optimal rate, u u, + B B, = O( t + k + α N+. Proof. For te analysis we assume te use of SV elements. We begin by adding identically zero terms to te continuous MHD to get t (un+ u n, v + Re ( u n+, v = (f(t n+, v (u t (t n+, v + t (un+ u n, v + Re ( u n+, v Re ( u(t n+, v +(DN ( 3 un un u n+, v (u(t n+ u(t n+, v +s(b(t n+ B(t n+, v s(dn( 3 Bn Bn B n+, v (DN ( 3 un un u n+, v + s(dn( 3 Bn Bn B n+, v, (7..9 t (Bn+ B n, χ + Re m ( B n+, χ = ( g(t n+, χ (B t (t n+, χ + t (Bn+ B n, χ + Re m ( B n+, χ Re m ( B(t n+, χ +(B(t n+ u(t n+, χ (DN( 3 Bn B n u n+, χ (u(t n+ B(t n+, χ + (DN( 3 un un B n+, χ +(DN ( 3 Bn B n u n+, χ (DN( 3 un un B n+, χ. (7.. 9

118 Subtracting (7..9 from (7.. and labeling e n u := u n = un, and e n B := Bn Bn gives t (en+ u e n u, v + Re ( e n+ u, v = (DN ( 3 un un u n+, v (DN( 3 un un u n+, v + s(dn ( 3 Bn Bn B n+, v s(dn( 3 Bn Bn B n+, v + G (u, B, n, v, (7.. were G (u, B, n, v := (f n+ f(t n+, v (u t (t n+, v + t (un+ u n, v + Re ( u n+, v Re ( u(t n+, v + (DN ( 3 un un u n+, v (u(t n+ u(t n+, v + s(b(t n+ B(t n+, v s(dn ( 3 Bn Bn B n+, v. (7.. Next we decompose te velocity and magnetic errors respectively as e n u = (u(t n P V u(t n (P V u(t n u n =: ηn u φ n, and en B = (B(tn P V B(t n (P V B(t n B n =: ηn B ψn, were P V u(t n and P V B(t n denote te L projection of u(t n and B(t n respectively in V. Specifying v = φ n+ in (7.., rearranging, and noting tat (η n+ u η n u, φ n+ = gives t ( φn+ φ n + Re φ n+ = Re ( η n+ u, φ n+ + DN( 3 un un u n+ n+, φ (DN( 3 un un u n+, φ n+ + s(dn( 3 Bn Bn B n+, φ n+ s(dn( 3 Bn Bn B n+ n+, φ + G (u, B, n, φ n+. (7..3

119 Rewriting te explicitly written nonlinearities in (7..3 and reducing gives t ( φn+ φ n + Re φ n+ = Re ( η n+ u, φ n+ + DN( 3 un un η n+ s(dn ( 3 Bn Bn e n+ u, φ n+ B, φn+ (DN( 3 en u en u u n+, φ n+ s(dn( 3 en B en B B n+ n+, φ + G (u, B, n, φ n+. (7..4 Next subtracting (7.. from (7..3 gives t (en+ B en B, χ + Re m ( e n+ B, χ = (DN ( 3 Bn Bn u n+, χ (DN( 3 Bn Bn u n+, χ + (DN( 3 un un B n+, χ (DN( 3 un un B n+, χ + G (u, B, n, χ, (7..5 were G (u, B, n, χ := ( (g n+ g(t n+, χ (B t (t n+, χ + t (Bn+ B n, χ + Re m ( B n+, χ Re m ( B(t n+, χ + (DN( 3 Bn B n u n+, χ (B(t n+ u(t n+, χ + (u(t n+ B(t n+, χ (DN( 3 un un B n+, χ. (7..6 Coosing χ = ψ n+ in (7..5, reducing, and rearranging yields t ( ψn+ ψ n + Re m ψ n+ = Re m ( η n+ B, ψn+ + (DN ( 3 Bn Bn u n+, ψ n+ (DN( 3 Bn Bn u n+ n+, ψ + (DN( 3 un un B n+ n+, ψ (DN( 3 un un B n+, ψ n+ + G (u, B, n, ψ n+. (7..7

120 Rewriting te nonlinear terms in (7..7 and reducing gives t ( ψn+ ψ n + Re m ψ n+ (DN( 3 Bn Bn e n+ + (DN( 3 un un η n+ = Re m ( η n+ B, ψn+ u, ψ n+ B, ψn+ (DN ( 3 en B en B u n+ n+, ψ + (DN ( 3 en u en u B n+, ψ n+ + G (u, B, n, ψ n+. (7..8 Multiplying (7..8 by s, adding (7..8 and (7..4, and reducing wit te identity (u v, w = (u w, v yields t ( φn+ φ n + s t ( ψn+ ψ n + Re φ n+ = Re ( η n+ u, φ n+ + sre m ( η n+ B, ψn+ + DN ( 3 un un η n+ + (DN ( 3 φn φn u n+ s(dn ( 3 ηn B ηn B u, φ n+, φ n+ s(dn( 3 Bn Bn η n+ + sre m ψ n+ (DN ( 3 ηn u ηn u u n+, φ n+ s(dn( 3 Bn Bn η n+ B, φn+ B n+ n+, φ + s(dn( 3 φn φn B n+ n+, φ u, ψ n+ s(dn( 3 ηn B ηn B u n+ n+, ψ + s(dn( 3 ψn ψn u n+ n+, ψ + s(dn ( 3 un un η n+ B, ψn+ + s(dn( 3 ηn u ηn u B n+, ψ n+ s(dn ( 3 φn φn B n+, ψ n+ + G (u, B, n, φ n+ + G (u, B, n, ψ n+. (7..9 We now bound te RHS terms in (7..9 individually. For te first two terms we use Caucy- Scwarz and Young s inequalities Re ( η n+ u, φ n+ Re 4 φn+ + CRe η n+ u, and (7..3 sre m ( η n+ B, ψn+ sre m 4 ψ n+ + CRe m η n+ B. (7..3

121 We bound te first two nonlinear terms in (7..9 using Lemma..3, and standard inequalities DN( 3 un un η n+ u, φ n+ C DN( 3 un un D N ( 3 un un n+ η u φ n+ Re 4 φn+ + CRe 3 un un ( 3 un un η n+ u, (7..3 (D N( 3 ηn u ηn u Re u n+, φ n+ C DN( 3 ηn u ηn u u n+ φ n+ 4 φn+ + CRe u n+ ηu n + CRe u n+ ηu n. (7..33 Te tird and fourt nonlinear terms in (7..9 are bounded as follows: (DN( 3 φn φn u n+, φ n+ (DN 3 φn u n+, φ n+ + (DN φn u n+, φ n+ C φ n u n+ φ n+ + C φ n u n+ φ n+ Re 4 φn+ + CRe u n+ s (DN( 3 Bn Bn η n+ B, φn+ Cs (DN( 3 Bn Bn η n+ B φn+ Re ( φ n + φ n, ( φn+ + CRe η n+ B ( B n + B n. (7..35 Te fift and sixt nonlinear terms in (7..9 are bounded as follows s (DN( 3 ηn B ηn B B n+ n+, φ (DN( 3 ηn B ηn B B n+ n+ φ 4 φn+ + CRe B n+ ( ηb n + η n B, (7..36 s (DN( 3 φn φn B n+ n+, φ Re (DN 3 φn B n+, φ n+ + (DN φn B n+ n+, φ C φ n B n+ φ n+ + C φ n B n+ φ n+ Re 4 φn+ + CRe B n+ ( φ n + φ n. (

122 Te six remaining nonlinear terms are bounded similar to te first six nonlinear terms bounded above. Combining te upper bounds, using regularity of te continuous and discrete solutions, and grouping like terms gives t ( φn+ φ n + s t ( ψn+ ψ n + Re φ n+ + sre m ψ n+ C(Re + sre m ( φ n + φ n + scre m ( ψ n + ψ n + C(Re + Re m ( ηu n + ηu n + CRe( ηb n + η n B + C(Re + Re + sre m η n+ u + C(Re + Re m + sre m η n+ B + G (u, B, n, φ n+ + G (u, B, n, ψ n+ (7..38 It remains to bound te terms in G (u, B, n, φ n+ and G (u, B, n, ψ n+. Te bounds for G and G are derived similarly and so we only write out te details explicitly for G. Te linear terms are bounded wit Caucy-Scwarz and Young s inequality as follows: ( un+ u n t (f(t n+ f n+, φ n+ φn+ + f(tn+ f n+ t n+ φn + φn+ + ( t3 f tt dt, ( t n u t (t n+ n+, φ φn+ + u n un+ u t (t n+ t φn + φn+ + ( t 3 8 t n+ t n u ttt dt, (7..4 ( u n+ u(t n+ n+, φ εre φ n+ + CRe u n+ u(t n+ εre φ n+ + CRe ( t3 48 t n+ t n u tt dt.(7..4 To bound te first nonlinear term we add an identically zero term, and rearrange to get (D N ( 3 un un u n+, φ n+ (u(t n+ u(t n+, φ n+ = (D N( 3 un un u n+, φ n+ (D Nu(t n+ u(t n+, φ n+ + (D Nu(t n+ u(t n+, φ n+ (u(t n+ u(t n+, φ n+ = (DN( 3 un (u un n+ u(t n+ n+, φ +(DN( 3 un un u(t n+ u n+ n+, φ + ((D Nu(t n+ u(t n+ u(t n+, φ n+. (7..4 4

123 Next we bound te terms in (7..4 individually: (D N ( 3 un un (u n+ u(t n+, φ n+ DN ( 3 un (u un n+ u(t n+ n+ φ εre φ n+ εre φ n+ εre φ n+ t n+ + CRe( u n + u n t 3 t n+ + CRe t 3 t n t n u tt dt ( u n 4 + u n 4 + u tt 4 dt + CRe t 4 ( u n 4 + u n 4 + CRe t 3 t n+ t n u tt 4 dt. (7..43 (DN ( 3 un un u(t n+ u n+ n+, φ DN( 3 un un u(t n+ u n+ n+ φ εre φ n+ + CRe t D Nu tt (t n+ u n+ εre φ n+ + CRe t D N u tt (t n+ u n+ εre φ n+ εre φ n+ t n+ + CRe t 3 t n+ + CRe t 3 t n u tt 4 dt + CRe t 4 u n+ 4 t n u tt 4 dt + CRe t 4 ( u n+ 4 + u n 4 (7..44 ((D Nu(t n+ u(t n+ u(t n+, φ n+ C DNu(t n+ u(t n+ u(t n+ φ n+ εre φ n+ + CRe D Nu(t n+ u(t n+ εre φ n+ + CReα 4N+4 N+ A (N+ u N + CRe(α k + k+ ( A n u k+. (7..45 Te bound on te remaining nonlinear terms in G is derived similarly. Specifying ε = 8, substi- n= 5

124 tuting in te above bounds, and multiplying troug by t gives ( φ n+ φ n + s( ψ n+ ψ n + t Re φ n+ + s t Re m ψ n+ ( C(Re, Re m, s t φ n+ + φ n + φ n + ψ n+ + ψ n + ψ n t n+ + t 3 + t 3 t + η n u + η n u t n f tt dt + t 3 n+ t n u tt 4 dt + t 3 t n+ + ηb n + η n B + η n+ u + η n+ t n g tt dt + t 3 t n+ t n B ttt dt + t 3 t n+ t n u ttt dt + t 3 t n+ t n B tt dt + t 3 B t n+ t n t n+ t n + t 4 ( u n+ 4 + u n 4 + u n 4 + B n+ 4 + B n 4 + B n 4 N + α 4N+4 N+ A (N+ u + (α k + k+ ( A n u k+ n= u tt dt B tt 4 dt N + α 4N+4 N+ A (N+ B + (α k + k+ ( A n B k+. (7..46 Recall tat η n u = (u(t n P V u(t n and η n B = (B(tn P V B(t n implies n= η n α k+ u n k+ for α = u, B, (7..47 η n α k+ u n k+ for α = u, B. (7..48 Summing te terms in (7..46 and using te bounds above gives φ M + s ψ M + t M n= ( Re φ n+ + sre m ψ n+ C(Re, Re m, s t M n= ( φ n+ + ψ n+ ( + C(Re, Re m, s k u 4 4,k+ + k u 4,k+ + k B 4 4,k+ + k B 4,k+ + t 4 f, + t 4 g, + t 4 u ttt, + t 4 u tt, + t 4 u tt 4 4, + t 4 B ttt, + t 4 B tt, + t 4 B tt 4 4, + t 4 u 4 4, + t 4 B 4 4, + α 4N+4 N+ A (N+ u, + α 4N+4 N+ A (N+ B, N N + (α k + k+ ( A n u,k+ + (α k + k+ ( A n B,k+. (7..49 n= n= Te proof is finised wit an application of Gronwall s inequality and te triangle inequality. 6

125 Remark Te algoritm as two important features wic te error analysis reveals. Te first is tat te velocity and magnetic field errors (and convergence rates are independent of te pressure error. Tis follows because te SV element is pointwise divergence free. Te second important feature is as a result of linearizing te sceme tere is no restriction on te timestep, wic is analogous to te results in [34, 5]. Remark Te filtering parameter, α, is typically cosen on te order of. Tis simplifies te error estimate to O( t + l were l = min((n +, k. We note tat by coosing N = wen k = 3 we maintain tird order convergence. However, if we coose N = ten we only expect second order convergence. 7. Numerical Experiments In tis section we test te numerical sceme on some bencmark problems. All computations are done in d on a barycenter refinement of a regular mes (to ensure stability wit ((P 3, P disc SV elements, and α = O(. 7.. Convergence Rates converge as From Teorem 7.., wit k = 3 we expect te asymptotic error in Algoritm 7.. to u u, + B B, = O( t α N+. Tus, for te Leray-α model (i.e. N = we expect te error to be O( t + because α. However, if we use te Leray-deconvolution model wit N = we expect te error to be O( t + 3, wic is optimal using te SV element ((P 3, Pdisc, and a reduction in error over te case were N is cosen to be. To verify te convergence rates we compute solutions to a problem wit known solutions on a series of refined meses and timesteps. Te test problem is cosen to ave solution u = ( +.t cos(πy sin(πx, B = (.t sin(πy cos(πx, P = x + y. on te unit square wit Re = Re m =, s =, endtime T =., and f and g calculated from u, B, p and te MHD equations. Te meswidt and timestep are tied togeter so tat wen is alved, te timestep is divided by (approximately. 7

126 Table 7. sows tat we acieve tird order convergence wen N =, wile we only acieve second order convergence wen N =. Tis agrees wit our analysis. Furter, Table 7. sows tat on te tree finest discretizations te Leray-Deconvolution (N= model as an order of magnitude reduction in error compared to te Leray-α (N= model wit only a modest increase in runtime (approximately 8% on te finest mes. Tis confirms our teory, and demonstrates a clear advantage of deconvolution. t u u, + B B, rate time u u, + B B, rate time (N= (sec (N= (sec T T T T T e Table 7.: Convergence rates for te Leray-deconvolution (N = and te Leray -α (N = models. 7.. Cannel Flow over a Step For our second experiment, we consider a variation of te bencmark problem of flow troug a cannel over a step found in [4, 3, 57]. Te parameters are specified as follows Re = 5, Re m =, s =.5, and endtime T = 5. Te initial conditions are u = and B =. For te velocity we assume constant inflow and constant outflow on te left and rigt boundaries, and all oter boundaries are prescribed no-slip conditions. Te magnetic field is set to be in te y direction and in te x direction on all boundaries. Figure 7. sows te domain and boundary conditions. Figure 7.: Te domain and boundary conditions for cannel flow problem. 8

127 It is known from [4, 3, 57] tat te correct pysical beavior is for an eddy to form beind te step. To verify tis we compute a DNS of MHD flow on a mes wic gave,65 degrees of freedom and a timestep t =.. Figure 7. sows te correct beavior of te simulation Figure 7.: Te true solution. Te goal of a model is to provide more accurate (at least in an averaging sense simulations tan DNS on coarse spacial and temporal discritizations. Tus, to test our model we now compare te MHD Leray-deconvolution model and DNS of MHD on a mes wic gave 8,666 degrees of freedom and timestep t =.. Te model parameters were cosen to be N = and α =.8(. Te results are sown in Figure 7.3 as velocity streamlines over speed contours. Te coarse-mes DNS failed to provide a plausible solution due to numerical oscillations. Te Leray-deconvolution model does stretc te recirculation region but it as captured te recirculation beind te step and te solution still maintains smoot velocity contours. Tis demonstrates te effectiveness of te model at capturing (qualitative long term beavior of MHD flows on coarse discretizations Orszag-Tang Vortex Problem We conclude te section by repeating an experiment done by J.-G Liu and W. Wang in [49]. Consider te ideal d MHD equations wit Re = Re m =, f = g =, s =, and compute on te π periodic box wit initial conditions u = sin(y +. sin(x +.4, B = 3 sin(y sin(x +.3. Te computations were done on a mes wic gives a total of 9,4 degrees of freedom, and parameters were specified as follows: N =, α = 9, t =., and T =.7. Te configuration is 9

128 Figure 7.3: Top: MHD Leray-deconvolution, Bottom: DNS of MHD, at T = 5 on mes giving 8,666 dof.