Advancements In Finite Element Methods For Newtonian And Non-Newtonian Flows

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations 8-3 Advancements In Finite Element Metods For Newtonian And Non-Newtonian Flows Keit Galvin Clemson University, Follow tis and additional works at: ttps://tigerprints.clemson.edu/all_dissertations Part of te Applied Matematics Commons Recommended Citation Galvin, Keit, "Advancements In Finite Element Metods For Newtonian And Non-Newtonian Flows" (3). All Dissertations. 36. ttps://tigerprints.clemson.edu/all_dissertations/36 Tis Dissertation is brougt to you for free and open access by te Dissertations at TigerPrints. It as been accepted for inclusion in All Dissertations by an autorized administrator of TigerPrints. For more information, please contact

2 Advancements in finite element metods for Newtonian and non-newtonian flows A Dissertation Presented to te Graduate Scool of Clemson University In Partial Fulfillment of te Requirements for te Degree Doctor of Pilosopy Matematical Sciences by Keit J. Galvin August 3 Accepted by: Dr. Hyesuk Lee, Committee Cair Dr. Leo Rebolz, Co-Cair Dr. Cris Cox Dr. Vincent Ervin

3 Abstract Tis dissertation studies two important problems in te matematics of computational fluid dynamics. Te first problem concerns te accurate and efficient simulation of incompressible, viscous Newtonian flows, described by te Navier-Stokes equations. A direct numerical simulation of tese types of flows is, in most cases, not computationally feasible. Hence, te first alf of tis work studies two separate types of models designed to more accurately and efficient simulate tese flows. Te second alf focuses on te defective boundary problem for non-newtonian flows. Non-Newtonian flows are generally governed by more complex modeling equations, and te lack of standard Diriclet or Neumann boundary conditions furter complicates tese problems. We present two different numerical metods to solve tese defective boundary problems for non-newtonian flows, wit application to bot generalized-newtonian and viscoelastic flow models. Capter 3 studies a finite element metod for te 3D Navier-Stokes equations in velocityvorticity-elicity formulation, wic solves directly for velocity, vorticity, Bernoulli pressure and elical density. Te algoritm presented strongly enforces solenoidal constraints on bot te velocity (to enforce te pysical law for conservation of mass) and vorticity (to enforce te matematical law tat div(curl)= ). We prove unconditional stability of te velocity, and wit te use of a (consistent) penalty term on te difference between te computed vorticity and curl of te computed velocity, we are also able to prove unconditional stability of te vorticity in a weaker norm. Numerical experiments are given tat confirm expected convergence rates, and test te metod on a bencmark problem. Capter 4 focuses on one main issue from te metod presented in Capter 3, wic is te question of appropriate (and practical) vorticity boundary conditions. A new, natural vorticity boundary condition is derived directly from te Navier-Stokes equations. We propose a numerical sceme implementing tis new boundary condition to evaluate its effectiveness in a numerical ii

4 experiment. Capter 5 derives a new, reduced order, multiscale deconvolution model. Multiscale deconvolution models are a type of large eddy simulation models, wic filter out small energy scales and model teir effect on te large scales (wic significantly reduces te amount of degrees of freedom necessary for simulations). We present bot an efficient and stable numerical metod to approximate our new reduced order model, and evaluate its effectiveness on two 3d bencmark flow problems. In Capter 6 a numerical metod for a generalized-newtonian fluid wit flow rate boundary conditions is considered. Te defective boundary condition problem is formulated as a constrained optimal control problem, were a flow balance is forced on te inflow and outflow boundaries using a Neumann control. Te control problem is analyzed for an existence result and te Lagrange multiplier rule. A decoupling solution algoritm is presented and numerical experiments are provided to validate robustness of te algoritm. Finally, tis work concludes wit Capter 7, wic studies two numerical algoritms for viscoelastic fluid flows wit defective boundary conditions, were only flow rates or mean pressures are prescribed on parts of te boundary. As in Capter 6, te defective boundary condition problem is formulated as a minimization problem, were we seek boundary conditions of te flow equations wic yield an optimal functional value. Two different approaces are considered in developing computational algoritms for te constrained optimization problem, and results of numerical experiments are presented to compare performance of te algoritms. iii

5 Table of Contents Title Page i Abstract ii List of Tables vi List of Figures vii Introduction Preliminaries A Numerical Study for a Velocity-Vorticity-Helicity formulation of te 3D Time-Dependent NSE Discrete VVH Formulation Numerical Results Natural vorticity boundary conditions for coupled vorticity equations Derivation Numerical Results A New Reduced Order Multiscale Deconvolution Model Derivation Te Discrete Setting Error Analysis Numerical Results Analysis and approximation of te Cross model for quasi-newtonian flows wit defective boundary conditions Modeling Equations and Preliminaries Te Optimal Control Problem Te Optimality System Steepest descent approac Numerical Results Approximation of viscoelastic flows wit defective boundary conditions Model equations Te Optimality system Steepest descent approac Mean pressure boundary condition Nonlinear least squares approac Numerical Results iv

6 8 Conclusions Appendices A deal.ii code for 3d vorticity equation Bibliograpy v

7 List of Tables 3. Velocity and Vorticity errors and convergence rates using te nodal interpolant of te true vorticity for te vorticity boundary condition Velocity and Vorticity errors and convergence rates using te nodal interpolant of te L projection of te curl of te discrete velocity into V, for te vorticity boundary condition Velocity and Vorticity errors and convergence rates using nodal averages of te curl of te discrete velocity for te vorticity boundary condition Velocity errors and convergence rates for te first 3d numerical experiment Vorticity errors and convergence rates for te first 3d numerical experiment vi

8 List of Figures. Barycenter refined tetraedra and triangle Flow domain for te 3d step test problem Sown above are (top) speed contours and streamlines, (middle) vorticity magnitude, and (bottom) elical density, from te fine mes computation at time t = at te x = 5 mid-slice-plane for te 3d step problem wit nodal averaging vorticity boundary condition Fine mes used for te resolved NSE solution and te coarse mes used for te RMDM approximations Fine mes used for te resolved NSE solution and te coarse mes used for te RMDM approximations Diagram of te contraction domain, along wit te fine and coarse meses used in te computations for te contraction problem Speed contour plots of te resolved NSE solution as well as solutions of Algoritm at t = Domain for te flow problem. Red indicates an inflow boundary. Blue indicates an outflow boundary Streamlines and magnitude of te velocity approximation for r =.5 and g = [.,.] Inflow and outflow velocity profiles for r =.5 and g = [.,.] Streamlines and magnitude of te velocity approximation for r =.5 and g = [, ] Inflow and outflow velocity profiles for r =.5 and g = [, ] Sown above is te domain for te flow problem Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using Diriclet boundary conditions for te velocity and stress Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using te steepest descent algoritm for te flow rate matcing problem wit initial guess g = [.,...,.] Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using te Gauss- Newton algoritm for te flow rate matcing problem wit initial guess g = [.,...,.] Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using te steepest descent algoritm for te mean pressure matcing problem wit initial guess g = [.,...,.] Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using te steepest descent algoritm for te flow rate matcing problem wit initial guess g = [5,..., 5]. 97 vii

9 7.7 Plots of te magnitude of te velocity and streamlines, velocity and pressure profiles on S, S, and S 3, and stress contours of te solution generated using te Gauss- Newton algoritm for te flow rate matcing problem wit initial guess g = [5,..., 5]. 98 viii

10 Capter Introduction Te understanding of fluid flow as been a subject of scientific interest for undreds of years. More recently, te branc of fluid mecanics known as computational fluid dynamics (CFD) as been an area of intense interest for matematicians due to te multitude of scientific areas tat depend on it. Many industries (e.g. automotive, aerospace, environmental) rely on bot accurate and efficient simulations of various types of fluids. However, state of te art models and metods are far from being able to efficiently solve most problems of interest in CFD to a desired degree of precision. Moore s law states (rougly) tat te amount of computing power available doubles every two years, and as proven to be a fairly accurate estimate over te last 5 years. Despite te great advances made in computing power in tat time period, and even assuming Moore s law for computational speed increase continues, te accurate and timely simulation of most flows will not be acieved in te foreseeable future. Advances in matematics for CFD ave gained far more towards tis goal tan computing power, by developing robust and efficient algoritms built on solid matematical and pysical grounds. It is te goal of tis work to extend te state of te art in matematics of CFD for two important problems. Te first concerns te accurate and efficient simulation of incompressible, viscous Newtonian fluids. We will present and analyze a new numerical metod for approximating solutions to te velocity-vorticity-elicity formulation of te Navier-Stokes equations. Te driving force beind tis new metod is tat it offers increased pysical fidelity and numerical accuracy, along wit a step towards furter understanding te important but ill-understood pysical quantity elicity. Discussion of tis metod naturally raises te very difficult question of ow to accurately

11 impose boundary conditions on te vorticity, as well as ow to compute wit turbulent flows. For te former, we propose a new natural boundary condition for te vorticity equation wic increases bot te accuracy and pysical relevance of our discrete vorticity approximation. For te latter, we consider a new reduced-order multiscale model for simulating Newtonian fluids. Te second main problem we study in tis work concerns te robust simulation of non- Newtonian fluids in te absence of standard boundary conditions. Tis problem often arises wen modeling flow in an unbounded domain (e.g. modeling blood flow in a portion of a blood vessel). We consider two different approaces for developing accurate and efficient metods for tese defectiveboundary problems for non-newtonian flows. Te first, a gradient-descent metod, is presented and tested for bot generalized-newtonian and viscoelastic flow models, and analyzed in te case of te former. Te second, a nonlinear least squares metod, is presented and tested on a viscoelastic flow model. Te flow of time-dependent, incompressible, viscous Newtonian flows is modeled by te Navier-Stokes equations (NSE), wic may be derived from te continuity equation (describing conservation of mass) and te equation describing conservation of momentum. In dimensionless form, te NSE are formulated as u t ν u + u u + p = f, (.) u =, (.) were u and p denote te fluid velocity and pressure, respectively, f denotes an external body force, and ν > denotes te fluid s kinematic viscosity. Te Reynolds number Re = ν is a dimensionless parameter representing te ratio of inertial forces to viscous forces. In laminar flows, wic are caracterized by low Reynolds number, viscous forces dominate inertial forces, making te flow field smoot. Simulations of laminar flows can often be performed witout too muc complication. For flows wit moderate Reynolds numbers, inertial forces start to play a larger role, resulting in complex flow beaviors, making predictions muc more difficult. Turbulent flows, caracterized by ig Reynolds numbers, present very complex and caotic flow properties, often requiring special models and metods. In, a velocity-vorticity-elicity (VVH) formulation of te NSE was presented in [55]. Tis formulation was derived by taking te curl of mass and momentum equations (.)-(.), and

12 applying several vector identities to produce te vorticity-elical density equations w t ν w + D(w)u η = f, (.3) w =. (.4) were w := u is te fluid vorticity, η := u w is te elical density, and D(w) := ( w+( w)t ) is te symmetric part of te vorticity gradient. Te dimensionless VVH formulation of te NSE ten comes from coupling (.3)-(.4) to te NSE via te rotational form of te nonlinearity in te momentum equation w t u t ν u + w u + P = f, (.5) u =, (.6) ν w + D(w)u η = f, (.7) w =. (.8) Here P := u u + p denotes te Bernoulli pressure, wic is needed because of our use of te rotational form of te nonlinearity. Since its original derivation in, VVH as been studied in oter applications including numerical metods for solving steady incompresible flow [5], te Boussinesq equations [54], and as a selection criterion for te filtering radius in te NS-ω turbulence model [5], all wit excellent results. Te VVH formulation of te NSE as four important caracteristics tat make it attractive for use in simulations. First, numerical metods based on finding velocity and vorticity tend to be more accurate (usually for an added cost, but not necessarily wit VVH) [6, 63, 59, 6, 5], and especially in te boundary layer [7]. Second, it solves directly for te elical density η, wic may give insigt into te important but ill-understood quantity elicity, H = η dx, wic is believed Ω to play a fundamental role in turbulence [4, 53, 5, 9, 3,,, 9]. VVH is te first formulation to directly solve for tis elical quantity. Tird, te use of η in te vorticity equation enables η to act as a Lagrange multiplier corresponding to te divergence-free constraint for te vorticity, analogous to ow te pressure relates to te conservation of mass equation. VVH is te first velocity-vorticity metod to naturally enforce incompressibility of te vorticity, wic is important since (.4) is as 3

13 muc a matematical constraint as it is a pysical one, making its violation inconsistent on multiple levels. Finally, te structure of te VVH system allows for a natural splitting of te system into a two-step linearization, since lagging vorticity in te velocity equation linearizes te equation, and similarly lagging velocity in te vorticity equation linearizes tis equation as well. A numerical metod based on suc a splitting was proposed in [55], and wen coupled wit a finite element discretization, was sown to be accurate on some simple test problems. Capter 3 of tis work will precisely define and furter study tis discretization of te VVH formulation of te NSE by providing a rigorous stability analysis (for bot velocity and vorticity), and testing te metod on a bencmark problem. Amidst our study of tis discretization of te VVH formulation, an important, but difficult question is raised in regards to boundary conditions for te vorticity. Consider te basic vorticity equation, derived by taking te curl of te momentum equation (.), w t ν w + u w w u = f. (.9) Peraps te most natural and reasonable boundary condition for te vorticity is w = u on Ω. (.) Unfortunately, tis boundary condition presents some difficulty wen employed wit finite elements. In general, differentiating te piecewise-polynomial u can often lead to a decrease in convergence order [5]. Recently, various metods for avoiding tis loss in accuracy ave been proposed. In [6], a finite difference approximation of (.) using nodal values of te finite element functions is employed. Tis metod is fairly successful on uniform meses wen second-order accuracy is desired, owever, it s implementation on non-uniform meses and for iger-order elements can be quite complex. In general, in te presence of sarp boundary layers of te velocity (e.g. for flows wit moderate or ig Re), te use of te vorticity boundary condition (.) may require extreme mes refinement around te boundary to avoid inaccurate vorticity approximations. Oter metodologies for implementing vorticity boundary conditions ave also been tried, wit some success. In [55], te vorticity on te boundary was set to be te L projection of te discontinuous finite element function u into te continuous finite element space. Tis metod is one of tree implemented 4

14 in te numerical testing of our metod for te VVH system presented in Capter 3, providing fairly accurate results on a bencmark flow problem. Oter strategies include using te boundary element metod [4], or te lattice Boltzmann metod [8]. In Capter 4, we employ a different approac in deriving a new vorticity boundary condition, in opes of avoiding any unnecessary complication. Te proposed metod includes natural boundary conditions for a weak formulation of te vorticity equation. Te boundary conditions are derived directly from te pysical equations and te finite element metod, making tem simpler to understand tan some of te aforementioned strategies. A full derivation of tese vorticity boundary conditions will be presented in Capter 4, along wit a numerical sceme to evaluate teir effectiveness in a numerical experiment. Anoter clear need in te development of te VVH algoritm is for some kind of stabilization/subgrid model to allow us to andle iger Re flows. In Capter 5 we consider a new reduced order, multiscale, approximate deconvolution model for Newtonian flows. Approximate deconvolution models (ADM) are a form of large eddy simulation (LES) models introduced in [, 3] for te purpose of simulating large-scale flow strctures at a reduced computational cost compared wit direct numerical simulation (DNS). We know from Kolmogorov s 94 teory tat eddies below a critical size (O(Re 3/4 ) for 3d flow) are dominated by viscous forces and disappear very quickly, wile tose above tis critical size are deterministic in nature. Hence, a DNS requires O(Re 9/4 ) mes points in space per time step to accurately simulate eddies in 3d. Even for moderate Re flows, tis requirement makes DNS computationally infeasible. ADM models (and LES models in general) aim to avoid tis problem by filtering out small scales, wile modeling teir effect on te large scales. Because only large scales are being solved for, tese models require a significantly smaller amount of mes points tan DNS. Recently, a promising new multiscale deconvolution model (MDM) [] as been proposed wic avoids some of te drawbacks of general ADM models, and is given by v t + G γ v G γ v + q ν v = f (.) v =. (.) Tis formulation makes use of two different Helmoltz filters (associated wit two different filtering radii α and γ) and a deconvolution operator G γ wic connects te two filter scales. Tis formulation and tese filters and operators will all be defined in detail in Capter 5, were we derive (in detail) a new, reduced order MDM, along wit an efficient and stable algoritm to approximate it. 5

15 Te second alf of tis work is concerned wit te accurate and efficient simulation of te defective boundary problem for two types of non-newtonian fluids. Te modeling of flow in an unbounded domain requires te introduction of artificial boundaries. Often, te flow is assumed to satisfy some Diriclet or Neumann boundary condition on a portion of tese artificial boundaries (e.g. inflow or outflow boundaries). However, te amount of boundary data available for a given flow is often very limited, making tese types of boundary conditions very ard to impose. In many practical applications te only flow data available are quantitative (e.g. average flow rates, mean pressure values, etc.). In situations like tese, it is often more realistic to model te flow using defective boundary conditions. Typically, governing equations are cosen depending on te flow being modeled, and instead of completing tese equations wit standard Diriclet or Neumann type boundary conditions, te defective boundary problem consists of only considering information suc as flow rates (or mean pressure values) on te inflow or outflow boundaries S i, i.e. u n ds = Q i for i =,..., m. (.3) S i We note tat tese boundary conditions are known as defective because tey are insufficient to close te differential model (i.e. our flow problem is ill-posed) [6]. Te goal of te second alf of tis work is to study tis problem in te context of two different types of flows (and ence two different types of modeling equations). Before we proceed we note tat flow problems wit defective boundary conditions ave been studied in various applications in te past. In [4], te defective boundary problem for te NSE was studied were flow rates are specified on inflow and outflow boundaries. In tis work a do-noting approac is presented were te flow rate conditions are implicitly incorporated into te variational formulation troug te coice of appropriate boundary conditions and function spaces, resulting in a well-posed variational problem. An alternative approac to te defective boundary problem for te NSE subject to flow rate conditions was presented in [6]. In tis study, te flow rate conditions are enforced weakly via te Lagrange multiplier metod. In [4] te defective boundary problem for quasi-newtonian flows subject to flow rate conditions was investigated using te Lagrange multiplier metod. Bot te continuous and discrete variational formulations of a generalized set of modeling equations were proven to be well-posed, and error analysis of te numerical approximation was also presented. In [7], a new approac to te defective boundary problem for Stokes flow was proposed. 6

16 Tis approac formulates te defective boundary problem as an optimal control problem troug te coice of a suitable functional to minimize. Tis approac proved to be versatile, as te functional to minimize can be altered to matc various kinds of defective boundaries (flow rates, mean pressure, etc). In te optimal control formulation, te control was cosen to be a constant normal stress on eac of te inflow and outflow boundaries, and appears in te modeling equations troug te addition of a boundary integral (often referred to as a boundary control [35]). Te study of optimal control problems for Newtonian and non-newtonian fluids as been istelf an active researc area in te recent past, e.g. [35, 36, 37]. One approac to solve tese types of optimization problems is based off of solving sensitivity equations, wic are derived troug te Frecet derivative of te constraint operator wit respect to te control variables [35,, 38]. An alternative approac studied in [35, 49] is an adjoint-based optimization metod, in wic te metod of Lagrange multipliers is used to derive an optimality system consisting of constraint equations, adjoint equations, and a necessary condition. In [] an optimal control problem for te Ladyzenskaya model for generalized-newtonian flows was studied. Additionally, a sape optimization problem for blood flow modeled by te Cross model was presented in []. In [48] a defective boundary problem for generalized-newtonian flows was studied. In tat work te model problem considered was te tree-field power law model subject to flow rate or mean pressure conditions on portions of te boundary. Te defective boundary problem was formulated as an optimal control problem wic was ten transformed into an unconstrained optimization problem via te Lagrange multiplier metod. However, analysis of te adjoint problem and te metod of Lagrange multipliers was limited, in part due to te coice of modeling equations. In Capter 6, we begin by considering te defective boundary problem for generalized- Newtonian fluids governed by te Cross modeling equations [6] (wic will be explicitly defined later in tis work). Newtonian fluids are caracterized by aving a sear stress, denoted by σ, tat is directly proportional to its sear rate (given by D(u)), i.e. σ = νd(u), (.4) were te fluid viscosity ν is constant. On te oter and, generalized-newtonian flows ave te same stress-strain relationsip, but wit a non-constant fluid viscosity dependent upon te velocity 7

17 of te flow σ = ν( D(u) )D(u), (.5) were te viscosity function ν( D(u) ) is cosen to reflect te flow being modeled. Te Cross model specifies te viscosity function as ν( D(u) ) := ν + (ν ν ), (.6) + (λ D(u) ) r were λ > is a time constant, r is a dimensionless rate constant, and ν and ν denote limiting viscosity values at a zero and infinite sear rate, respectively, assumed to satisfy ν ν. We take te approac of [48] to approximate our model problem subject to flow rate and mean pressure conditions. Te problem is formulated as an optimal control problem for wic we analytically justify te use of te metod of Lagrange multipliers to derive an optimality system. We ten sow tat te resulting adjoint system is well-posed. Finally, we consider a complex numerical experiment to test te robustness of an optimization algoritm previously presented in [48]. In Capter 7, we consider te same defective boundary problem but for viscoelastic fluids governed by te Jonson-Segalman modeling equations. Viscoelastic fluids are a type of non- Newtonian fluid tat exibit bot viscous and elastic caracteristics wen undergoing deformation. Tis is reflected in te modeling equations by an extra nonlinear constitutive equation, wic relates te stress tensor σ to te fluid velocity. Some analytical and numerical studies for an optimal control of non-newtonian flows can be found in [,, 45, 49]. Te Jonson-Segalman modeling equations for viscoelastic, creeping flow are given by σ + λ(u )σ + λg a (σ, u) αd(u) =, (.7) σ ( α) D(u) + p = f, (.8) u =. (.9) Here λ denotes te Weissenberg number, defined as te product of relaxation time and a caracteristic strain rate of te fluid, α is a number satisfying < α < wic can be considered as te fraction 8

18 of viscoelastic viscosity, and g a (σ, u) is a nonlinear function of σ and u tat will be explicitly defined in Capter 7. We consider te defective boundary problem for viscoelastic fluids governed by tese equations. Tis includes a fully detailed formulation of te problem itself, te minimization problem, and a derivation of te optimality system. Te numerical algoritm presented in Capter 6 will ten be used to solve te minimization problem, along wit a second, new algoritm. Finally, we consider a numerical test to compare and contrast bot algoritms. Tis work is arranged as follows. Capter contains matematical notation and preliminaries tat will be used trougout te following sections. Capter 3 presents a stability analysis and numerical testing of a finite element metod for te VVH formulation. Capter 4 fully defines a new vorticity boundary condition, and presents a numerical experiment designed to verify its accuracy. Capter 5 derives and analyzes a new reduced order MDM, and presents two numerical tests to verify its efficiency. Capter 6 presents te work on generalized-newtonian flows wit defective boundary conditions, and Capter 7 contains te work on viscoelastic flows wit defective boundary conditions. Finally, Capter 8 contains conclusions from te various works presented erein. 9

19 Capter Preliminaries Trougout te analysis presented in tis work we will assume tat te domain Ω denotes a bounded, connected subset of R d (wit d = or 3), wit piecewise smoot boundary Ω. We will denote te L (Ω) norm and inner product by and (, ), respectively, wile L p (Ω) norms will be denoted by L p. Sobolev W k p (Ω) norms and seminorms will be indicated by W k p and W p k, respectively. We will use te standard notation of H k (Ω) to refer to te sobolev space W k (Ω), wit norm k. Dual spaces will be denoted ( ) wit duality pairing, and norm. For domains oter tan Ω we will explicitly indicate te domain in te space and norm notation. For k R te space H k is defined as H k (Ω) := {v H k (Ω) v = on Ω}. Te zero-mean subspace of L (Ω) is defined as L (Ω) := {q L (Ω) q = }. Ω of te norms For functions v(x, t) defined on Ω (, T ) for some positive end time T, we will make use ( ) /n T v n,k := v(, t) n k dt and v,k := ess sup v(, t) k. <t<t For functions of time, we will use te notation t n := n t were t denotes a cosen time-step. For

20 continuous functions of time f(t), we use te notation f n := f(t n ), and f n+/ := f(t n+ ) = f ( t n+ + t n ). Te average of te nt and (n + )st time level of a discrete function v is denoted v n+/ := vn+ + v n. Our error analysis will require te use of discrete time analogues of te continuous in time norms: v p,k := ( NT ) /p v n p k t n= and v,k := max n N T v n k We will use bold font to denote vector functions and tensor functions. We will also use bold font to denote vector function spaces, e.g. H (Ω) := (H (Ω)) d and H (Ω) := (H (Ω)) d. Trougout our analysis we will frequently employ te following inequality, one result of wic is tat for v H (Ω), te seminorm v is equivalent to v. Lemma.. (Te Poincare-Friedrics inequality). Tere exists a positive constant C P F = C P F (Ω) suc tat v C P F v v H (Ω). Proof. A proof of tis well known inequality can be found in [8]. We will often use te (H (Ω)) = H (Ω) norm, denoted by, to measure te size of

21 a forcing function. Te H (Ω) norm is defined as f := f, v sup v H (Ω) v. We note tat te space H (Ω) is te closure of L (Ω) in. Te continuous velocity, pressure, and stress spaces, denoted X, Q, and Σ, respectively, will be specified in eac capter. Te weakly divergence-free subspace V of X is defined as V := {v X ( v, q) = q Q}. In te discrete setting, we begin by letting τ denote a regular, conforming triangulation or tetraedralization of Ω. Te velocity and pressure finite element spaces defined on τ will be denoted as X and Q, respectively, and will be specified in eac capter. Te divergence-free subspace V of X is defined as V := {v X ( v, q ) = q Q }. We will often make use of te Taylor-Hood (TH) element pair, defined as (X, Q ) = ((P k ) d, P k ), i.e. X := {v H (Ω) v K (P k ) d (K) K τ } Q := {q L (Ω) C (Ω) q K P k (K) K τ }. It is a well-known result tat for k te TH element pair satisfies te discrete inf-sup condition [, 3]. We will also make use of te Scott-Vogelius (SV) element pair (X, Q ) = ((P k ) d, P disc k ) [6], wic uses te same velocity approximation space as TH elements, but allows te pressure approximation space to be discontinuous. An immediate consequence of tis coice of spaces is tat X Q. Hence, wit tis coice of elements, te discretely div-free subspace V X now becomes V := {v X ( v, q ) = q Q } = {v X v = }.

22 Tis makes te SV element pair a natural coice for bot velocity-pressure and vorticity-elicity systems as it results in pointwise enforcement of solenoidal constraints for te velocity and vorticity (as opposed to weak enforcement by TH elements). Te drawback of using discontinuous elements is tat te dimension of Q in te SV element pair is significantly larger tan in te TH element pair, resulting in a linear system wit a greater amount of degrees of freedom wen using SV elements. Figure.: Barycenter refined tetraedra and triangle. In order for te SV element pair to be discretely inf-sup stable, any of te following conditions on te mes τ are sufficient [57, 66, 65, 67]:. In d, k 4 and te mes as no singular vertices. In 3d, k 6 on a quasi-uniform tetraedral mes 3. In d or 3d, wen k d and te mes is generated as a barycenter refinement of a regular, conforming triangular or tetraedral mes 4. Wen te mes is of Powell-Sabin type and k = in d or k = in 3d We note tat a complete classification of conditions for discrete inf-sup stability of SV elements, including te minimum degree for general meses witout special refinements, is an open question. In 3

23 our computations performed wit SV elements we will always use condition 3. Figure. illustrates a barycenter-refined triangle. For our convergence studies in Capter 3, 4, and 5 we will assume our coice of finite element spaces satisfies te following well known approximation properties: inf v X u v C k+ u k+ inf v X u v C k u k+ inf r Q p r C s+ p s+ for any u H k+ (Ω) for any u H k+ (Ω) for any p H s+ (Ω). We note tat tese approximation properties old for bot TH and SV elements. In Capter 5, te trilinear operator b : X X X R defined by b (u, v, w) := (u v, w) will be used. Te following useful properties of b are proven in [46]. Lemma... If u =, ten b (u, v, v) =. Additionally, tere exists a constant C dependent on te size of Ω suc tat b (u, v, w) C u u v w, b (u, v, w) C u v w. Our error analysis will also use te following discrete version of te Gronwall inequality. Lemma..3. Let k, B and a µ, b µ, c µ, γ µ, for integers µ, be nonnegative numbers suc tat n n n a n + k b µ k γ µ a µ + k c µ + B for n. (.) µ= µ= µ= 4

24 Suppose tat kγ µ <, for all µ, and set σ µ = ( kγ µ ). Ten, ( ) ] n n n a n + k b µ exp k σ µ γ µ [k c µ + B µ= µ= µ= for n. (.) Remark..4. If te first sum on te rigt in (.) extends only up to n, ten estimate (.) olds for all k >, wit σ µ =. Proof. A proof of tese results can be found in [4]. 5

25 Capter 3 A Numerical Study for a Velocity-Vorticity-Helicity formulation of te 3D Time-Dependent NSE In tis capter we study a finite element metod for te 3d NSE in velocity-vorticity-elicity form. For Ω R 3, recent work [56] as sown tat te NSE can be equivalently written in VVH form as: Find u : Ω (, T ) R 3 and p : Ω (, T ) R satisfying u t ν u + w u + P = f in Ω (, T ), (3.) u = in Ω (, T ), (3.) u t= = u in Ω, (3.3) u = φ on Ω (, T ), (3.4) 6

26 and find w : Ω (, T ) R 3, η : Ω (, T ) R satisfying w t ν w + D(w)u η = f in Ω (, T ), (3.5) w = in Ω (, T ), (3.6) w t= = u in Ω, (3.7) w = u on Ω (, T ), (3.8) were φ is a Diriclet boundary condition for velocity satisfying φ n = t (, T ). In [55], Ω a numerical algoritm based on a -step linearization of te VVH formulation was proposed. In tis capter, we study tis discretization of te VVH formulation furter by providing a rigorous stability analysis, testing te metod on several bencmark problems, and wit various vorticity boundary conditions. 3. Discrete VVH Formulation For our finite element discretization of te VVH formulation, we will coose velocity and pressure spaces (X, Q ) (H (Ω), L (Ω)) on our mes τ to be te Scott-Vogelius element pair (P k, P disc k ). We will denote te vorticity space by Y, were Y H (Ω) is te space P k. We note tat te only difference between te velocity and vorticity finite element spaces is te value of te finite element functions on te boundary Ω. To simplify te analysis, we require te mes is sufficiently regular so tat te inverse inequality olds, u C i u. (3.9) For te initial conditions for our velocity and vorticity approximations we will use te L projection into V. For φ L (Ω), te L projection of φ into V, denoted by P V (φ), satisfies (P V (φ), v ) = (φ, v ) for any v V. problem: Define te operator A : L (Ω) V as te solution operator to te discrete Stokes ( A ψ, v ) = (ψ, v ), v V. (3.) 7

27 Tis operator will not be used in computations, but is used in te analysis of te proposed algoritm. Te following lemma was proven in [5]. Lemma 3... Assume Ω is suc tat te Stokes problem is H -regular. For any ψ L (Ω) it olds A ψ L + A ψ L 3 C ψ, (3.) and for any f L (Ω), q L (Ω), and φ H (Ω) (f, A ψ) C( f + f ) ψ, (3.) (q, A ψ) C( q + q ) ψ, (3.3) ( φ, A ψ) C( φ + φ, Ω + φ ) ψ. (3.4) Te cosen time discretization is trapezoidal, and te linearization uses second order extrapolation. Te fully discrete -step version of (3.)-(3.8) we study is: find (u, w, P, η ) (X, Y, Q, Q ) satisfying (v, χ, q, r ) (X, X, Q, Q ), Step : t (un+ u n, v ) + ν( u n+, v ) (P n+, v ) +(( 3 wn wn ) u n+, v ) (f n+, v ) =, (3.5) Step : t (wn+ w n, χ ) + ν( w n+, χ ) ( u n+, q ) =, (3.6) +(η n+, χ ) + γν (( A wn+ ) u n+, ( χ ) u n+ ) (3.7) +(D(w n+ )u n+, χ ) ( f n+, χ ) =, ( w n+, r ) =, (3.8) w n+ Ω I ( u n+ ) Ω =, (3.9) were u = P V (u ), w = P V ( u ), and I denotes an appropriate interpolant. As is common practice in trapezoidal scemes for fluid flow, te Lagrange multiplier terms are solved for directly at teir n + / time levels, i.e. no splitting into time n and n + pieces is necessary, and so P n+ 8

28 and η n+ are approximations to teir continuous counterparts at t = t n+/. Note also tat we ave assumed a omogeneous Diriclet boundary condition for velocity, and a Diriclet condition for vorticity tat it be equal to an appropriate interpolant of te curl of te velocity on te boundary. Tis is te simplest case for analysis, but is still quite formidable. Extension to oter common boundary conditions will lead to additional tecnical details, and need to be considered on case by case basis. Due to te difficulties associated wit any analysis involving te vorticity equation, tere are two components in te above sceme tat are for te purposes of analysis only. Te unconditional stability of te velocity does not depend on eiter of tese components of te numerical sceme, but proving unconditional stability of te vorticity requires bot of tem. First, te boundary condition for te discrete vorticity (3.9) is given in terms of te true velocity, wic is not practical. In computations, we use instead te condition w n+ Ω I ( u n+ ) Ω =, (3.) owever analyzing te system wit suc a boundary condition does not appear possible in tis particular formulation. Developing improved formulations for wic suc a vorticity boundary condition does allow analysis is an important open question. We will consider two possibilities of interpolants in our computations: i) a nodal interpolant of te L projection of te curl of te velocity into V, and ii) a nodal interpolant of a local averaging of te curl of te velocity. A new vorticity boundary condition, presented in te next section, is also feasible wit tis discretization. Te second part of te sceme tat is not used in computations is te penalty term in (3.7), i.e. we coose γ = in our computations. In te continuous case tis term is consistent for te omogeneous or periodic boundary conditions on a rectangular box: for sufficiently regular 9

29 solutions, w = u, and A te continuous Stokes solution operator, since u =, ( A w) u = ( A ( u)) u (3.) = (A ( ( u))) u = (A ( u ( u)) u = (A ( u)) u = (A (Au)) u =. Outside of te periodic case, te differential operators will not commute and tus errors will arise at te boundary from tis term; ence te term appears to damp vorticity creation at te boundary, and we do not use it in our computations. However, it does not appear possible to prove a vorticity stability bound witout it. 3.. Stability Analysis Lemma 3.. (Stability). Assume f L (, T ; H (Ω)) and u L (Ω). Ten velocity solutions to (3.5)-(3.7) are unconditionally stable, and satisfy M u M + t ν u n+ t n= M n= ν f n+ + C u := C 4. (3.) Proof. Let v = u n+ in (3.5) and simplify to get t ( u n+ u n ) + ν u n+ = (f n+ n+, u ). Using Caucy Scwarz, Young s inequality, and simplifying yields t ( u n+ u n ) + ν u n+ ν f n+.

30 Multiplying by t and summing from to M ten gives M u M + t ν u n+ t n= wic proves te estimate (3.). M n= M t ν f n+ + u ν f n+ + C u, n= Remark We note tat te unconditional stability of te velocity solution is independent of bot te vorticity boundary condition and te penalty term of te discrete vorticity equation. Lemma Assume f L (, T ; L (Ω)), u H (Ω), u L (, T ; H (Ω)), u t L (, T ; H (Ω)), and u tt L (, T ; H (Ω)). Ten vorticity solutions are also stable, in te sense of A wm M + t ν n= w n+ C(ν, C 4, M, T, f, u) := C 5. (3.3) Remark It appears tat te penalty parameter γ needs to satisfy γ > for te proof to old. Wen γ =, we are reduced to te non-penalty term case, for wic we are unable to prove unconditional stability. Proof. For te vorticity bound, let w n = I ( u n ) were I is a discretely div-free preserving interpolant. Note w n V and w n satisfies te vorticity boundary condition (3.9). Te vorticity solution can ten be decomposed as w n+ = w n+ n+ + w, (3.4) were w n+ V. Letting I ( u) C u for all t, we ave wn+ C u. (3.5)

31 Substituting (3.4) into te vorticity equation (3.7) yields, χ V, t (wn+ w n, χ ) + γν (( A wn+ ) u n+, ( χ ) u n+ ) + ν( w n+, χ ) = ( f n+, χ ) (D(w n+ )u n+, χ ) γν (( A wn+ n+ ) u, ( χ ) u n+ ) (D(w n+ n+ )u, χ ) ν( w n+, χ ) t (wn+ w n, χ ). (3.6) Let χ = A wn+ t (wn+ w n, A and simplify to get wn+ ) + γν ( A wn+ ) u n+ = ( f n+, A wn+ ) (D(w n+ )u n+ + ν wn+, A wn+ ) (D(w n+ n+ )u, A wn+ ) ν( w n+, A wn+ ) γν (( A wn+ n+ ) u, ( A wn+ ) u n+ ) t (wn+ w n, A wn+ ). (3.7) It is straigtforward to sow tat A is a symmetric operator on V and tus t (wn+ w n, A wn+ ) = = = t [(wn+, A (w n, A t [(wn+ A t ( ) + (w n+, A wn+ wn+, A wn+ wn+ ) (w n, A wn )] wn ) ) (w n, A wn )] A ). (3.8) wn Using (3.) on te first RHS term of (3.7) yields ( f n+, A wn+ ) = (f n+, A C( + V f n+ C(ɛ)ν ( f n+ wn+ ) f n+ ) wn+ (3.9) f + n+ ) + νɛ wn+ V. Using vector identities, integration by parts, and tat A wn+ is divergence free, on te first

32 trilinear term in (3.7) gives (D(w n+ )u n+, A wn+ ) = ( (w n+ u n+ ) (u n+ w n+ = ( (w n+ u n+ ), A wn+ ) + (u n+ = (w n+ u n+, A wn+ ) = (w n+, ( A γ ν wn+ wn+ + γν ), A wn+ ) w n+, A wn+ ) ) u n+ ) (3.3) ( A wn+ ) u n+. Te part of te penalty term on te rigt-and side of (3.7) is majorized as γν (( A wn+ n+ ) u, ( A γν ( A wn+ γν ( A wn+ ) u n+ ) u n+ + γν 4 wn+ ) u n+ ) (3.3) ( A wn+ ) u n+ ( A wn+ ) u n+. Te first term on te rigt and side of (3.3) can be bounded using Holder s inequality, (3.5) and Lemma 3..: ( A wn+ ) u n+ C A wn+ u n+ L 3 L 6 CC wn+ u n+ (3.3) CCC u u n+. (3.33) Substituting back into (3.3) we now ave γν (( A wn+ n+ ) u, ( A wn+ ) u n+ ) γν CCC u u n+ + γν 4 ( A wn+ ) u n+. (3.34) 3

33 Te second trilinear term in (3.7) can be bounded using Lemma 3.. and 3.. to obtain (D(w n+ n+ )u, A wn+ ) C wn+ u n+ A wn+ L C(ɛ)C 4 ν w n+ + νɛ wn+ (3.35) C(ɛ)C 4 Cuν + νɛ wn+. (3.36) Using (3.4) gives ν( w n+, A wn+ ) Cν( wn+ C(ɛ)ν( wn+ + +νɛ wn+ C(ɛ)Cuν + νɛ wn+ + wn+ +, Ω +, Ω wn+ wn+ ) wn+ wn+ ). (3.37) Finally, Caucy Scwarz, Young s inequality, and te definition of w n yield t (wn+ w n, A wn+ ) I ( ( un+ u n )) t A wn+ C I ( (u t (t n+ ) + u tt (t ))) wn+ (3.38) C C(ɛ)ν ( I ( u t (t n+ )) + I ( u tt (t )) ) +νɛ wn+ C C(ɛ)ν + νɛ wn+. Substitute into (3.7) using (3.8)-(3.38) to get t ( A ) ( A wn + ν ) γ 4ɛ w n+ ( f Cν n+ f + n+ ) + γν CC C V u u n+ wn+ + C(ɛ)C 4 ν C u + C(ɛ)νC u + C C(ɛ)ν. (3.39) 4

34 Coosing an arbitrarily small ɛ, te penalty parameter γ satisfying ( γ 4ɛ) >, multiplying by t, and summing from to M yield A wm M + t n= 4 ν wn+ M ν tc ( + C(ν, C 4, C u, ν) + A w n= f n+ + f n+ ) V + γν CC Cu t M n= ν u n+. (3.4) Using te result for te velocity stability bound on te last sum of (3.4) finises te proof. 3. Numerical Results We now present two numerical experiments to test te VVH metod studied in tis capter. For all tests, we use (P 3, P disc ) Scott-Vogelius elements, on barycenter-refined tetraedral meses. To solve te linear systems, we use te robust and efficient metod proposed in [56] for tis element coice. Tis is te lowest order element pair tat is LBB stable on tis mes. Te first experiment confirms expected convergence rates, and te second tests te metod on 3D cannel flow over a step. All computations use γ =. In te computations, vorticity appears to be stable wit tis coice, and so it was not necessary to add tis (costly) stabilization term. However, proving discrete stability of vorticity does not seem possible in tis case, and so its use is believed to cover a gap in te analysis only. 3.. Convergence Rates Our first experiment is used to test convergence rates for te problem Ω = (, ) 3, were te true solution is given by ( +.t) cos(πz) u(x, y, z, t) = ( +.t) sin(πz) ( +.t) sin(πx) (3.4) For tis problem we take ν =, and initial condition u = P V (u ), w = P V ( u ). We compute wit end time T =, and monitor error wile decreasing te values of t wit. Uniform meses 5

35 are used in te sense tat eac mes divides Ω into equal size cubes, ten divides eac cube into six tetraedra, and ten performs a barycenter refinement of eac tetraedra. In te tables, denotes te lengt of a side of a cube. For te velocity boundary condition, we use te nodal interpolant of te true solution on te boundary. For te vorticity boundary condition, we compute tree different ways, all using a Diriclet condition for discrete vorticity: using te nodal interpolant of te true vorticity, using te nodal interpolant of te L projection of te curl of te discrete velocity into V, and also using a simple local averaging of te curl of te discrete velocity. Te results are sown in Tables , respectively. Wit our coice of elements and a trapezoidal time discretization, optimal error is O( t + 3 ), and since we tie togeter te spatial and temporal refinements by cutting t in tird wen is cut in alf, O( 3 ) is optimal. All tree vorticity boundary conditions provide similar results: suboptimal rates are observed in te L (, T ; H (Ω)) norm until te last mes refinement, wen te rate jumps to around 3. We also see tat for te velocity in te L (, T ; L (Ω)) norm we see optimal convergence rates, were as te vorticity in te L (, T ; L (Ω)) norm we do not seem to recover any L lift. Here, wile te errors observed using te (more practical) non-exact boundary conditions are expectably larger, te rates of convergence observed do not seem to decrease. A complete convergence teory for te metod currently appears impenetrable witout several assumptions not needed for usual NSE analysis, but progress on tis front will likely lead to answers about boundary-dependence of convergence rates. dof t u u L (,T ;L (Ω)) Rate u u L (,T ;H (Ω)) Rate /, e e- - /4 78,46 /3.337e e /6 6,474 / e e /8 65,99 /9.574e e /,98,746 / e e t w w L (,T ;L (Ω)) Rate w w L (,T ;H (Ω)) Rate / e /4 / e e-.757 /6 /6.533e e-.589 /8 /9 5.99e e-.976 / /8.4978e e-.97 Table 3.: Velocity and Vorticity errors and convergence rates using te nodal interpolant of te true vorticity for te vorticity boundary condition. 6

36 t u u L (,T ;L (Ω)) Rate u u L (,T ;H (Ω)) Rate / / 3.869e e- - /4 /3.337e e /6 / e e /8 /9.574e e / / e e t w w L (,T ;L (Ω)) Rate w w L (,T ;H (Ω)) Rate / / 7.753e /4 / e e-.473 /6 /6.936e e-.366 /8 /9 7.47e e-.74 / /8 3.49e e-.975 Table 3.: Velocity and Vorticity errors and convergence rates using te nodal interpolant of te L projection of te curl of te discrete velocity into V, for te vorticity boundary condition. t u u L (,T ;L (Ω)) Rate u u L (,T ;H (Ω)) Rate / / 3.869e e- - /4 /3.337e e /6 / e e /8 /9.574e e / / e e t w w L (,T ;L (Ω)) Rate w w L (,T ;H (Ω)) Rate / / 6.695e /4 /3 8.45e e /6 /6.989e e-.349 /8 /9 8.e e-.699 / / e e-.945 Table 3.3: Velocity and Vorticity errors and convergence rates using nodal averages of te curl of te discrete velocity for te vorticity boundary condition. 7

37 3.. 3D Cannel Flow Over a Forward-Backward Facing Step Te next experiment tests te sceme on 3d flow over a forward-backward facing step, studied in [43, 5]. In te problem te cannel is modeled by a [, ] [, 4] [, ] rectangular box, wit a step on te bottom of te cannel, beginning 5 units into te cannel. A diagram of te flow domain is sown in Figure 3.. Figure 3.: Flow domain for te 3d step test problem. We compute to end-time T =, ν =, and t =.5. No-slip boundary conditions are used on te top, bottom, and sides of te cannel, as well as on te step, and an inflow=outflow condition is employed for bot. For te initial condition, we use te Re = steady solution. Note tis is consistent wit [5] but in contrast to [43], were a constant inflow profile (u(x,, z) =<,, >) is used; suc a boundary condition is non-pysical, but also not usable in a metod tat solves for vorticity (since it will blow up as at te inflow edges). We compute te solution on a barycenter-refined tetraedral mes, wic provides,8,9 total degrees of freedom. For te vorticity boundary condition on te walls and sides, we tried Diriclet conditions tat it be a nodal interpolant of te local average of te curl of te velocity, simply zero, and te projection of te curl of te velocity into V. Only for te case of nodal averaging did we see te expected results, sown in Figure 3. as a speed contour plot of te sliceplane x=5 wit overlaying streamlines, were eddies form beind te step and sed. Plots of vorticity magnitude and elical density are also provided. For te case of zero vorticity boundary condition latter, te simulation did not capture eddy detacment, and for te projection boundary condition, we saw instabilities occur and a bad solution resulted. 8

38 Figure 3.: Sown above are (top) speed contours and streamlines, (middle) vorticity magnitude, and (bottom) elical density, from te fine mes computation at time t = at te x = 5 mid-sliceplane for te 3d step problem wit nodal averaging vorticity boundary condition. 9

39 Capter 4 Natural vorticity boundary conditions for coupled vorticity equations Tis capter derives new natural boundary conditions for te vorticity equations tat result from te application of te curl operator to te steady NSE momentum equation, given by ν w + (u )w (w )u = f. (4.) A finite element metod for solving te 3d vorticity equations is presented to test te accuracy of te proposed boundary conditions, and results from a simple numerical experiment are presented verifying optimal convergence rates are aceived. We note tat te vorticity boundary conditions presented erein could also easily be derived for te time-dependent vorticity equations, and would apply equally well to te vorticity-elical density equations studied in te previous capter. 4. Derivation Suppose we are given some general Diriclet boundary condition for te velocity in te NSE, i.e. u = g on Ω. We are mainly interested in te case were Ω is a solid wall wit no-slip (g = ) 3

40 boundary conditions, and so leaving g to be general includes tis case. Our first vorticity boundary condition easily follows: w n = ( g) n on Ω. (4.) To deduce two more boundary conditions for w, consider te incompressible NSE written in rotational form (see, e.g. [3] for more on rotational form of NSE), ν w + w u + P = f, were P = u + p is te Bernoulli pressure. Taking te tangential component of bot sides of tis equation gives ν( w) n = (f P w g) n on Ω, (4.3) wic provides two more boundary conditions for w in terms of te primitive NSE velocity and pressure variables. In velocity-vorticity splitting scemes were te NSE momentum equation is used for te velocity (as in te work of Wong and Baker [6] or te sceme presented in te previous capter of tis work), te NSE velocity and pressure are considered as knowns wen solving te vorticity (or vorticity-elical density) equations. We observe tat te boundary condition (4.3) is te natural boundary condition for te following weak formulation of te vorticity equation: Find w H (div) H(curl) satisfying for any v H (div) H(curl). ν( w, v) + ν( w, v) + ((u )w (w )u, v) + ((w g) n) v ds = ( f, v) + ((f P ) n) v ds, (4.4) Ω Ω To avoid computing pressure gradient over Ω we rewrite te last term in (4.4) using integration by parts on Ω. To tis end, we use te surface gradient and divergence, defined as: Γ p = p (n p)n, and div Γ v = tr( Γ v), wic are intrinsic surface quantities and do not depend on an extension of a scalar function p and a vector quantity v off a surface. We also need te following identity, proved in [34], for a smoot, 3

41 closed surface Γ: p div Γ v + v Γ p ds = κ(v n)p ds, (4.5) Γ Γ were κ is te surface mean curvature. From te definition of te surface gradient we immediately get te identity: ( P ) n = ( Γ P ) n. Hence, wit (4.5) we see (( P ) n) v ds = (( Γ P ) n) v ds = (v n) Γ P ds Ω Ω Ω = P div Γ (v n) ds. (4.6) Ω Finally, using v n = we ave div Γ (v n) = ( v) n. Tus, te weak formulation of te vorticity equation now reads: Find w H (div) H(curl) satisfying ν( w, v) + ν( w, v) + ((u )w (w )u, v) + ((w g) n) v ds = ( f, v) + (f n) v ds + P ( v) n ds, (4.7) Ω Ω Ω for any v H (div) H(curl). In te case of no slip boundary conditions (g = ), te system reduces to ν( w, v) + ν( w, v) + ((u )w (w )u, v) = ( f, v) + (f n) v ds + p( v) n ds. (4.8) Ω Ω 4. Numerical Results In tis section we consider a basic 3d numerical test designed to evaluate te accuracy of a numerical sceme implementing our new vorticity boundary conditions. Let I denote some 3

42 interpolation operator on Ω. Te sceme we propose to compute wit is given in two steps: Step : ν( u, v ) + (u u, v ) (p, v ) = (f, v ) ( u, q ) = u Ω = I(g) Step : ν( w, v ) + ν( w, v ) + ((u )w (w )u, v ) + = ( f, v ) + (f n) v ds + (w I( g)) n Ω =. Ω Ω ((w g) n) v ds Ω p( v ) n ds Our 3d numerical experiment is designed to test convergence rates for te problem Ω = (, ) 3, were te true, steady NSE solution is given by u (x, y, z) = sin(πy) u (x, y, z) = cos(πz) u 3 (x, y, z) = e x p(x, y, z) = sin(πx) + cos(πy) + sin(πz), were p denotes te standard NSE pressure. Because we are given nonomogenous boundary conditions for te velocity, if we want to enforce vorticity boundary conditions only wit boundary integrals, we must include te left and side term ((w g) n) v ds in te vorticity equation. We compute wit Reynolds number Re = and force field Ω f = u u + p ν u as determined by te true NSE solution. Velocity and vorticity approximations are computed using Q elements, wile Q elements are used for te pressure. All computations were performed using te software deal.ii [6, 7]. 33

43 For our coice of finite element spaces optimal convergence in te H (Ω) norm is O( ), and ence we compute on a series of uniform exaedral meses, eac of wic is one uniform refinement of te previous mes (i.e. is cut in alf wit eac successive mes). Velocity and vorticity errors are sown in Tables 4. and 4.. Bot velocity and vorticity errors are optimal in te energy norm H (Ω). u u rate u u rate /.376E-.877 /4.8537E E-.97 /8.343E E-.98 /6.9438E E-.99 Table 4.: Velocity errors and convergence rates for te first 3d numerical experiment w w rate w w rate / /4.599E /8.6973E E-.99 / E E-. Table 4.: Vorticity errors and convergence rates for te first 3d numerical experiment 34

44 Capter 5 A New Reduced Order Multiscale Deconvolution Model Tis capter proposes a new reduced order multiscale deconvolution model (MDM). Tis model leads to a natural, efficient numerical sceme tat is bot unconditionally stable and optimally accurate. Numerical tests are provided tat confirm te effectiveness of te sceme. 5. Derivation Recall te NSE, given by u t + u u + p ν u = f (5.) u =. (5.) In tis capter we will only consider omogenous Diriclet boundary conditions for te velocity u. Our model formulation will use two different incompressible Helmoltz filters, te first of wic is 35

45 defined by α u + α λ + u = u, (5.3) u =, (5.4) (u u) Ω =, (5.5) were α denotes te lengt scale associated wit te filtered velocity u. For convenience, we will denote te solution operator to (5.3)-(5.5) by F α ( ), i.e. F α u = u. Te second incompressible Helmoltz filter is given by γ ũ + γ ρ + ũ = u, (5.6) ũ =, (5.7) (u ũ) Ω =, (5.8) were γ is a second, intermediate lengt scale associated wit te filtered velocity ũ, satisfying < γ α. In practice, α is generally cosen to be te size of te smallest flow structures to be resolved, and γ is a parameter determining te modeling error relative to te NSE. In [] it was sown tat ũ as an explicit form depending on u given by ũ = α α u ( γ γ )ũ, so we will define our multiscale, approximate deconvolution operator G γ as G γ u := α α α u + ( )ũ = γ γ γ u α γ γ ũ. (5.9) Using tis definition we immediately see tat we can write te intermediate filtered quantity φ explicitly as u ũ = G γ u. 36

46 Because F α is invertible we can write u = F α u, and using te definition of te α-filter (5.3) on te time derivative term, te NSE momentum equation (.) becomes ( α u t + α λ t + u t ) + F α u (F α u) + p ν (F α u) = f. (5.) It was sown in [] tat we ave te following local error estimate for te accuracy in approximating te operator F α by G γ : u G γ u Hk (Ω) = u ũ H k (Ω) γ u H k+ (Ω). Tis estimate suggests tat multiscale, approximate deconvolution can be used to approximate te inverse of te α-filter (i.e. F α φ G γ φ). Tus, denoting v := u and q := p + α λ t in (5.), we arrive at v t α v t + G γ v G γ v + q ν G γ v = f. Employing te same multiscale, approximate deconvolution approximation in te conservation of mass equation (5.) yields G γ u = G γ v =. Note tat by (5.7) and te definition of te deconvolution operator G γ, te new conservation of mass constraint is satisfied via enforcing v =. Te reduced order multiscale deconvolution model (RMDM) wit incompressible filters for omogenous Diriclet boundary conditions is ten v t α v t + G γ v G γ v + q ν G γ v = f (5.) v = (5.) γ ṽ + γ ρ + ṽ = v (5.3) ṽ = (5.4) (v ṽ) Ω =. (5.5) 5. Te Discrete Setting Define finite dimensional spaces X X and Q Q to be te Scott-Vogelius (SV) mixed finite element pair (X, Q ) := (P k (τ ), P disc k (τ )). Te following discrete filter is defined analogously to its continuous counterpart by taking 37

47 its variational formulation and restricting to finite dimensional spaces. Definition 5... Given φ L (Ω), define φ to be te solution of te problem: Find ( φ, ρ ) (X, Q ) satisfying γ ( φ, χ ) + ( φ, χ ) (ρ, χ ) = (φ, χ ) χ X, (5.6) ( φ, q ) = q Q. (5.7) We will denote te solution operator of tis discrete filter by F, i.e. F φ := φ. Te equivalent discretely divergence-free representation of te filter is: Given φ L (Ω), find φ V satisfying γ ( φ, χ ) + ( φ, χ ) = (φ, χ ) χ V. Te following lemma, found in [58], contains useful bounds for discretely filtered functions. Lemma 5... For φ L (Ω), φ φ. For φ X, tere exists a constant C dependent on te size of Ω suc tat φ C φ. For φ V, φ φ. Te next lemma provides a bound on te difference between continuously filtered and discretely filtered functions. Lemma For φ H k (Ω) V we ave te bound φ φ + γ ( φ φ ) C(γ k+ + k ) φ k+. (5.8) For k, we ave te improved bound φ φ + γ ( φ φ ) C( k+ + γ k ) φ k+. (5.9) 38

48 Proof. Multiplying te γ-filter equation (5.6) by arbitrary χ V and integrating over Ω yields γ ( φ, χ ) + ( φ, χ ) = (φ, χ ). Subtracting te discrete γ-filter equation (5.6) and denoting e = φ φ gives, for any χ V, γ ( e, χ ) + (e, χ ) =. Standard finite element analysis and interpolation estimates produce te bound φ φ + γ ( φ φ ) C( k+ + γ k ) φ k+. In [47] it was sown tat for k 3 we ave te estimate γ φ k+ C φ k+, and ence we get te bound φ φ + γ ( φ φ ) C(γ k+ + k ) φ k+. Additionally, for k =,, it was sown in [47] tat we ave te improved estimate φ k φ k, wic finises te proof. 5.. An Unconditionally Stable Algoritm for te RMDM Te fully discrete algoritm we study for te RMDM is backward Euler in time and finite element in space. Algoritm Given two filtering radii α γ >, initial velocity v V, a forcing function f L (, T ; H (Ω)), end time T >, and timestep t >, set M = T t and compute for n =,,..., M, and for all χ X and r Q, α t ( vn+ v, n χ ) + t (vn+ v, n χ ) + ν( ( α γ vn+ (q n+, χ ) + b ( α γ vn α γ γ ṽ n α, γ vn+ α γ γ ṽ n ), χ ) α γ ), χ = (f n+, χ ), (5.) γ ṽ n ( v n+, r ) =. (5.) 39

49 In te following stability analysis we will assume tat tere exists a const C suc tat α = C γ, (5.) i.e. te coarse mes and fine mes filtering radii will always be tied togeter by te constant C. Lemma 5..5 (Stability). Solutions to Algoritm 5..4 satisfy α v M + v M + ν t M v n+ n= (ν t(c ) + C α ) v + C v + ν t M were C depends on data but can be considered independent of α, γ, t, ν, and. Proof. Coosing χ = α γ v n+ pressure terms, and leaves α ( v n+ v n t, ( α α γ γ γ vn+ n= f n+ Cν, (5.3) ṽ n in (5.) immediately eliminates bot te nonlinear and α γ γ +ν (α α γ γ vn+ ) ṽ n ) + ( v n+ v n t, α γ vn+ γ ṽ n ) = (f n+, α γ vn+ α γ γ α γ γ ) ṽ n ṽ n ). (5.4) We begin by bounding te rigt and side term of (5.4) in te usual manner (f n+, α γ vn+ α γ γ ṽ n ) f n+ ν + ν (α γ vn+ α γ γ ṽ n ). (5.5) Decomposing te first term in (5.4) produces α ( v n+ v n t, ( α γ vn+ α γ = α4 γ t ( vn+ v n, v n+ γ ) ṽ n ) ) α (α γ ) γ t ( v n+ v, n ṽ n ). (5.6) 4

50 Using Caucy Scwarz and Young s inequalities and rearranging terms yields α ( v n+ v n t, ( α γ vn+ α γ ) γ ṽ n ) α4 γ t ( v n+ v n ) α (α γ ) γ t ( v n+, ṽ n α (α γ ) ) + γ ( v n t, ṽ n ). (5.7) Because te operator F is bot self-adjoint and positive in te L inner product in V, we ten get te lower bound α ( v n+ v n t, ( α α4 γ t ( v n+ γ vn+ α γ γ ) ṽ n ) v n ) α (α γ ) γ t α4 γ t ( v n+ v n ) + α (α γ ) γ t ( F / vn+, F / ( F / vn vn ) + α (α γ ) γ F t v n F / vn+ ). (5.8) Using te same tecniques we can produce te following lower bound for te second term in (5.4) ( v n+ v n t, α γ vn+ α γ ) γ ṽ n α γ t ( v n+ v n ) α γ γ t ( F / vn F / vn+ ). (5.9) Expanding te viscous term in (5.4) gives ν (α γ vn+ α γ γ ṽ n ) = να4 v n+ γ 4 + ν(α γ ) γ 4 ṽ n να (α γ ) γ 4 ( v n+, ṽ n ). (5.3) Using te Caucy-Scwarz inequality yields ν (α γ vn+ ṽ n α γ γ ṽ n ) να4 γ 4 v n+ + ν(α γ ) γ 4 να (α γ ( ) v n+ γ 4 + ṽ n ), (5.3) wic, after simplifying, leads to te lower bound ν (α α γ γ ṽ n ) να v n+ γ ν(α γ ) γ γ vn+ ṽ n. (5.3) 4

51 Using Lemma 5.. ten gives ν (α α γ γ ṽ n ) να v n+ γ ν(α γ ) γ v n = ν v n+ + ν(α γ ) ( v n+ γ v n ). (5.33) γ vn+ Applying te bounds in (5.5)-(5.33) to (5.4) yields α 4 tγ ( v n+ v n ) + α (α γ ( ) F / tγ vn F / vn+ ) + α tγ ( v n+ v n ) + α γ ( F / tγ vn F / vn+ ) + ν v n+ + ν(α γ ) γ ( v n+ v n ) ν f n+. (5.34) Summing from n = to M and multiplying by t gives ( α 4 v M γ α (α γ ) F / γ + ν t(α γ ) v M γ + ν t M vm v n+ n= ( α 4 + γ v α (α γ ) F / γ ) ( α + v M γ α γ F / γ v ν t(α γ ) γ v ) ( α + γ v α γ F / γ Using te Caucy Scwarz inequality and Lemma 5.. we see tat similarly F / F / φ φ. Terefore, multiplying by and reducing, we see vm ) M + tν f n+ v n= ). (5.35) φ = (φ, φ) φ, and α v M + ν t(α γ ) γ M v M + v M + ν t n= v n+ ν t(α γ ) γ v + α 4 γ v + α γ v + tν Substituting γ = C α and simplifying finises te proof. M n= f n+. (5.36) Remark Wit te current analysis, te assumption α = C γ is necessary because of te presence of γ on te rigt side of estimate (5.36). 4

52 5.3 Error Analysis We now prove convergence of Algoritm 5..4 to solutions of te model (5.)-(5.5). Teorem 5.3. (Convergence Estimate). Let α γ > denote two fixed filtering radii, and assume v to be te model solution to (5.)-(5.5) satisfying te given problem data of Algoritm If we assume te model solution satisfies te smootness conditions v L (, T ; H k+ (Ω)), v t, v tt L (, T ; H (Ω)) wit k, ten for any timestep t >, te error in te numerical solution from Algoritm 5..4 satisfies v M v M + α (v M v M ) + ν t M n= (v n+ v n+ ) Cν exp(ν ) ( k (α α + ν + ν + ν α ) + k+ ( + α 4 + ν α 4 ) + t (α ν ) ), were C is a constant dependent only on data, independent of α, γ, t,, and ν. Proof. Trougout tis proof we will use C to represent a generic constant, possibly different at eac instance, tat is independent of ν,, α, γ, and t. Beginning wit (5.)-(5.), multiply by χ V and integrate over Ω to get α ( v t, χ ) + (v t, χ ) + b (G γ v, G γ v, χ ) + ν( G γ v, χ ) = (f, χ ). (5.37) After adding and subtracting terms, we ten ave for n =,,..., M and for any χ V α t ( vn+ v n, χ ) + t (vn+ v n, χ ) + ν ( α γ vn+ α γ ) γ ṽ n, χ ( α + b γ vn α γ γ ṽ n, α γ vn+ α γ ) γ ṽ n, χ = (f n+, χ ) + G(v, χ, n), (5.38) 43

53 were G(v, χ, n) is defined as ( v G(v, χ, n) := α n+ v n ) ( v v t (t n+ n+ v n ) ), χ t + v t (t n+ ), χ t + ν [( α γ vn+ α γ ) γ ṽ n, χ ( G γ v n+ ) ], χ [ b ( G γ v n+, G γ v n+ ), χ b ( G γ v n, G γ v n+ )], χ [ b ( G γ v n, G γ v n+ ), χ b (G γ v n, α γ vn+ α γ )] γ ṽ n, χ [b (G γ v n, α γ vn+ α γ ) γ ṽ n, χ ( α b γ vn α γ γ ṽ n, α γ vn+ α γ )] γ ṽ n, χ. (5.39) From (5.38) subtract (5.), restricting χ V, and denote e n+ = v n+ v n+ to get α ( α t ( en+ e n, χ ) + b γ vn α γ γ ṽ n, α γ vn+ α γ ) γ ṽ n, χ + ( α t (en+ e n, χ ) b γ vn α γ γ ṽ n α, γ vn+ α γ ) γ ṽ n, χ + ν ( α γ en+ α γ ) γ ẽ n, χ = G(v, χ, n). (5.4) Decompose e n+ = (v n+ w n+ ) + (w n+ v n+ ) =: η n+ + φ n+, were w n+ is an arbitrarily cosen element of V. Ten we can rewrite (5.4) as α t ( φn+ φ n, χ ) + t (φn+ φ n, χ ) + ν ( α = α t ( ηn+ η n, χ ) t (ηn+ η n, χ ) ν + G(v, χ, n) b ( α b ( α γ vn α γ γ γ en α γ γ ṽ n α, γ φn+ α γ γ ( α ) φn, χ γ ηn+ α γ ) η γ n, χ ẽ n, α γ vn+ α γ ) γ ṽ n, χ γ en+ α γ ) γ ẽ n, χ. (5.4) Continue by coosing χ = α γ φn+ α γ φn γ. Using similar analysis tecniques to tose employed in Lemma 5..5, we can lower bound te time 44

54 derivative terms on te left side of (5.4) as α ( φ n+ φ n t, α ( φ n+ α4 γ t γ φn+ α γ γ ) φn φ n ) + α (α γ ) γ t ( F / φn F / φn+ ), (5.4) and, ( φ n+ φ n t, α γ φn+ α γ ) φn γ ( α γ φ n+ φ n t ) + α γ γ t ( F / φn F / φn+ ). (5.43) Using te bounds (5.4)-(5.43) on (5.4) in conjunction wit our coice of χ = α γ φ n+ α γ γ φn gives α ( γ φ n+ φ n t ) + α γ ( F / γ t φn ( + α4 γ φ n+ φ n t ) + α (α γ ) γ t ( + ν α γ φn+ α γ ) φn γ α t (η n+ η n, α t γ φn+ α γ γ ( ( α ν γ ηn+ α γ ) ( α η γ n, ( α b γ φn α γ α φn γ, ( α b γ ηn α γ η γ n, α b ( α b ( α γ vn α γ γ γ vn α γ γ γ vn+ α γ γ γ vn+ α γ γ F / φn+ ( F / φn ( η n+ η n, ) φn + G(v, χ, n) γ φn+ α γ γ ) F / φn+ ( α )) φn ) γ φn+ α γ γ ṽ n, α γ φn+ α γ ) φn γ ṽ n, α γ φn+ α γ φn γ ṽ n α, γ φn+ α γ α φn γ, γ φn+ α γ φn γ ) ) )) φn ṽ n α, γ ηn+ α γ η γ n, α γ φn+ α γ ) φn γ. (5.44) 45

55 Te first two terms on te rigt side of (5.44) are majorized as in [3]: and (η n+ η n, α t γ φn+ α γ ) φn γ t n+ 4C P F η ν t t dt + ν ( α t 6 n α t ( ( α η n+ η n, γ φn+ α γ γ t n+ 4α4 η ν t t dt + ν t 6 n ( α γ φn+ α γ )) φn γ φn+ γ α γ γ ) φn, (5.45) ) φn. (5.46) Te viscous term on te rigt side of (5.44) is bounded using Caucy-Scwarz and Young s inequalities, as well as Lemma.: ( α ( ν γ ηn+ α γ ) ( α η γ n, ( ) ν α γ ηn+ α γ ( α η γ n α γ ν ( η n+ α γ φn+ α γ γ + α γ ( γ ν η n α γ φn+ α γ γ ν ( α 6 γ φn+ α γ γ γ φn+ α γ γ )) φn γ φn+ α γ γ ) φn ) φn ) φn ) ( φn + Cν α4 γ 4 η n+ + η n ). (5.47) Using Lemma.., Young s inequality and Lemma 5.., majorize te first nonlinear term in (5.44) 46

56 to get ( α b γ φn α γ α φn γ, ( α C γ φn α γ γ ( α γ vn+ α γ γ γ vn+ α γ γ / ( φn α ṽ n 4Cν α γ φn α γ γ + ν ( α 6 γ φn+ α γ γ Cν (α γ ) ( α φ n φ n 4 γ 4 + ν ( α 6 γ φn+ ṽ n, α γ φn+ α γ ) φn γ ) / φn γ φn α γ γ ) ( α γ φn+ α γ γ ) ) φn ( φn α γ φn α γ ) ( φn γ α α γ γ ) φn γ 4 v n+ + (α γ ) γ 4 ṽ n ) φn γ vn+ α γ γ Cν α8 (α γ ) 4 γ 6 φ n v n+ 4 + Cν (α γ ) 4 (α γ ) 4 γ 6 φ n v n 4 + ν 6 φn + ν ( α 6 γ φn+ α γ ) φn γ ν ( α 6 γ φn+ α γ γ ) ṽ n ) ) φn + ν ( 6 φn α6 + Cν γ 6 φn v n+ 4 + v n 4). (5.48) 47

57 Using Lemma.. and Young s inequality on te second nonlinear term in (5.44) gives ( α b γ ηn α γ η γ n, α γ vn+ α γ γ ( ) C α γ ηn α γ ( α η γ n γ vn+ α γ γ C α γ ( ) γ η n α γ vn+ α γ ( α γ ṽ n C α (α γ ) γ 4 η n ( v n+ α γ φn+ α γ γ + C (α γ )(α γ ( ) γ 4 η n v n α ν ( α 6 γ φn+ α γ γ ṽ n, α γ φn+ α γ γ ) ( α γ φn+ ṽ n ) φn γ φn+ α γ γ ) φn α γ γ γ φn+ α γ γ ) φn ) φn + Cν α4 (α γ ) γ 8 η n v n+ ) φn ) φn + Cν (α γ ) (α γ ) γ 8 η n v n ν ( α 6 γ φn+ α γ ) ( φn γ α8 + Cν γ 8 ηn v n+ + v n ). (5.49) By Lemma.., te tird nonlinear term in (5.44) vanises. Finally, te fourt nonlinear term is bounded in te same manner as in (5.49) ( α b γ vn α γ γ ν ( α 6 γ φn+ α γ γ ṽ n α, γ ηn+ α γ η γ n, α γ φn+ α γ ) φn γ ) ( φn α8 + Cν γ 8 vn η n+ + η n ). (5.5) Applying te bounds (5.45)-(5.5) to equation (5.44) yields α ( φ n+ γ φ n t ) + α γ γ t ( + α4 φ n+ γ t + ν ( α 6 γ φn+ α γ γ + 4α4 ν t t n+ t n α8 + Cν ( + G ( F / φn φ n ) + α (α γ ) γ t ( F / F / φn+ φn ) φn ν 6 φn + 4C P F ν t ) t n+ t n F / φn+ η t dt ) ( η t dt + Cν α4 η n+ γ 4 + η n ) ( α6 + Cν γ 6 φn v n+ 4 + v n 4) γ 8 ηn ( v n+ + v n ) + Cν v, α γ φn+ α γ γ α8 γ 8 vn ( η n+ + η n ) ) φn, n. (5.5) 48

58 We will now continue by majorizing ( G v, α γ φn+ α γ γ ) φn, n, as defined in (5.39) wit χ = α γ φ n+ α γ γ φn. Te first term of (5.39) is bounded using standard estimates based on Taylor series ( v α n+ v n t ν ( α 6 ( α v t (t n+ ), γ φn+ α γ )) φn γ γ φn+ α γ γ ) φn + t α 4 ν v tt L (t n,t n+,h (Ω)). (5.5) Te second term of G is bounded in te same manner ( v n+ v n v t (t n+ ), α t γ φn+ α γ γ ν ( α 6 γ φn+ α γ γ ) φn ) φn + t CP F ν v tt L (t n,t n+,h (Ω)). (5.53) Using Caucy-Scwarz and Young s inequalities on te tird term of (5.39) yields ( α ( ν γ vn+ α γ γ = ν(α γ ) γ = ν(α γ ) γ + ν(α γ ) γ ν ( α 6 ṽ n ) ( α G γ v n+, ( (ṽn+ ṽ n), ( (ṽn+ ) ṽ n, γ φn+ + Cν (α γ ) γ 4 ( (ṽn ṽ n), α γ γ ( α ( α γ φn+ α γ γ γ φn+ α γ γ )) φn γ φn+ α γ )) φn γ ( α ) φn γ φn+ α γ γ )) φn ( (ṽn+ ṽ n ) + (ṽn ṽ n) ). )) φn Using estimates based on Taylor series as well as Lemma 5..3 to bound te difference between 49

59 continuous and discrete filters, we ten see ( α ( ν γ vn+ α γ ) γ ṽ n ν ( α 6 γ φn+ α γ γ ( α G γ v n+, γ φn+ α γ )) φn γ ) φn + Cν α4 γ 4 t v t L (t n,t n+,h (Ω)) + Cν α4 γ 6 (γ k+ + k ) v n H k+ (Ω). (5.54) To bound te fourt term of G, we use Lemma.., Young s inequality, and Taylor s teorem to arrive at ( b G γ v n+, G γ v n+, α γ φn+ α γ ) φn γ b (G γ v n, G γ v n+, α γ φn+ α γ ) φn γ ( = b G γ (v n+ v n ), G γ v n+, α γ φn+ α γ ) φn γ ( α γ φn+ α γ ) φn γ Gγ (v n+ v n ) Gγ v n+ ν ( α 6 γ φn+ α γ ) φn γ + Cν Gγ (v n+ v n ) Gγ v n+ ν ( α 6 γ φn+ α γ ) φn γ α8 + Cν (v n+ γ 8 v n ) v n+ ν ( α 6 γ φn+ α γ ) φn γ α8 + Cν γ 8 t v t L (t n,t n+,h (Ω)) v n+. (5.55) Te same tecniques are used to bound te fift term in (5.39) ( b G γ v n, G γ v n+, α γ φn+ α γ γ ( b G γ v n, α γ vn+ α γ γ ( = b G γ v n, α γ ) (ṽn γ ṽn+ ν ( α 6 γ φn+ α γ γ ν ( α 6 γ φn+ α γ γ ) φn ṽ n, α γ φn+ α γ ) φn γ, α γ φn+ α γ ) φn γ ) φn + Cν (α γ ) ) (ṽn γ 4 ṽn+ Gγ v n ) φn α8 + Cν γ 8 t v t L (t n,t n+,h (Ω)) vn. (5.56) 5

60 For te sixt term in (5.39), we start by using Lemma.. and Lemma 5.. ( b G γ v n, α γ vn+ α γ γ ṽ n, α γ φn+ α γ ) φn γ ( α b γ vn α γ γ ṽ n, α γ vn+ α γ γ ( = α γ ) (ṽn b γ ṽ n, α γ vn+ α γ γ ( b G γ v n, α γ (ṽn ṽ n), α γ γ φn+ α γ γ C α γ ) ( (ṽn γ ṽ n α γ φn+ α γ γ ( ( ) α γ vn+ α γ γ ṽ n + G γ v ) n C α (α γ ) ) ( (ṽn γ 4 ṽ n α γ φn+ α γ γ ṽ n, α γ φn+ α γ ) φn γ ṽ n, α γ φn+ α γ φn γ ) φn ) φn ) ) ( φn v n+ + v n ). Using Young s inequality and Lemma 5..3 we ten get ( b G γ v n, α γ vn+ α γ γ ṽ n, α γ φn+ α γ ) φn γ ( α b γ vn α γ γ ṽ n, α γ vn+ α γ γ ṽ n, α γ φn+ α γ ) φn γ ν ( α 6 γ φn+ α γ ) φn γ + Cν α4 (α γ ) ) (ṽn γ 8 ṽ n ( v n+ + v n ) ν ( α 6 γ φn+ α γ ) φn γ α8 ( + Cν γ γ k+ + k) v n ( H k+ (Ω) v n+ + v n ). (5.57) 5

61 Using te bounds (5.5)-(5.57) on equation (5.5), we get α γ t + α4 ( φ n+ φ n ) + α γ γ t ( φ n+ ( F / φn φ n ) + α (α γ ) γ t ( F / F / φn+ φn ) F / φn+ γ t + ν ( α 4 γ φn+ α γ ) φn γ ν t n+ 6 φn + 4C P F η ν t t dt t n t n+ ( + 4α4 η ν t t dt + Cν α4 η n+ t γ 4 + η n ) ( α6 + Cν γ 6 φn v n+ 4 + v n 4) n ) ( α8 + Cν γ 8 ηn v n+ + v n ) ( α8 + Cν γ 8 vn η n+ + η n ) + Cν t α 4 v tt L (t n,t n+,h (Ω)) + Cν t v tt L (t n,t n+,h (Ω)) + Cν t α 4 γ 4 v t α4 L (t n,t n+,h (Ω)) + Cν γ 6 (γ k+ + k ) v n H k+ (Ω) + Cν t α 8 γ 8 v t L (t n,t n+,h (Ω)) v n+ + Cν t α 8 γ 8 v t L (t n,t n+,h (Ω)) vn α8 ( + Cν γ γ k+ + k) v n H k+ (Ω) ( v n+ + v n ). (5.58) 5

62 Multiplying troug by t and summing from n = to M yields α γ ( φ M + α (α γ ) γ ν t 8 ) φ M n= + Cν t α4 γ 4 ( F / + α γ γ φ ( F / φ F / φm F / φm ) + ν t ) + α4 M n= γ ( α T T φ n + Cν η t dt + Cν α 4 η t dt M n= ( η n+ + η n ) + Cν t α6 + Cν t α8 M ( γ 8 η n v n+ + v n ) n= + Cν t α8 M ( γ 8 v n η n+ + η n ) n= M + Cν t 3 α 4 + Cν t 3 α 4 γ 4 + Cν t 3 α 8 + Cν t 3 α 8 n= M n= γ 6 M v tt L (t n,t n+,h (Ω)) + Cν t 3 n= M n= ( φ M γ φn+ ) φ α γ γ ) φn φ n ( v n+ 4 + v n 4) v tt L (t n,t n+,h (Ω)) M v t L (t n,t n+,h (Ω)) + Cν tα4 γ 6 (γ k+ + k ) M γ 8 n= M γ 8 n= v t L (t n,t n+,h (Ω)) v n+ v t L (t n,t n+,h (Ω)) vn n= v n H k+ (Ω) + Cν t α8 M γ (γ k+ + k ) v n ( v n+ H k+ (Ω) + v n ). (5.59) n= Note tat our assumption of v V implies φ =. Using tis and assumptions on te regularity 53

63 of te model solution, (5.59) becomes α φ M α γ F / γ γ + ν t M n= + Cν α 4 T ( α γ φn+ φm α γ + α4 φ M α (α γ ) F / γ γ γ η t dt + Cν t α4 γ 4 φm ) φn ν t M T φ n 8 + Cν M n= n= ( η n+ + η n ) + Cν t α6 η t dt γ 6 M n= φ n + Cν t α8 M γ 8 η n + Cν t α8 M ( γ 8 v n η n+ + η n ) + Cν T t α 4 n= + Cν T t + CνT t α 4 n= α4 ( + CνT γ4 γ γ 6 k+ + k) + Cν T t α 8 γ 8 + Cν T α8 γ ( γ k+ + k). (5.6) Wit regularity assumptions and interpolation estimates, (5.6) becomes α φ M α γ F / γ γ + ν t + Cν T M n= φm + α4 φ M α (α γ ) F / γ γ φm ) φn ν t M φ n 8 + Cν t α6 M γ 6 φ n ( α γ φn+ α γ γ n= ( k+ + α 4 k + α8 γ 8 k + t α 4 + t + t α 8 γ 8 + α8 ( α 4 + CνT γ 4 k + t α 4 γ 4 + α4 γ 6 (γ k+ + k ) ) + Cν t α8 n= ) γ (γ k+ + k ) γ 8 k M We can lower bound te viscous term in te same fasion as in Lemma 5..5 to get ν t M n= ( α ν t = ν t γ φn+ M n= M n= α γ γ ) φn n= ( φ n+ + α γ γ ( ) φ n+ φ n ) v n. (5.6) φ n+ + ν t(α γ ) φ M γ 4. (5.6) 54

64 Combining (5.6) and (5.6), and using Lemma 5..5 and (φ, φ) φ gives φ M + α φ M + ν t ( ( + Cν k α 4 + α8 ( α + Cν ( k 4 γ 4 + α4 γ 6 M n= γ 8 + α8 α8 + ν γ γ 8 ) + α4 γ 8 k+ + α4 γ 4 t M M φ n+ ν t φ n 8 + Cν t α6 γ 6 φ n n= n= ) ) )) + ( k+ + α8 + t (α α8 γ ). (5.63) γ 8 Subtracting ν t 8 φ M M n= φ n from bot sides yields + α φ M + 3ν t 8 M n= ( ( + Cν k α 4 + α8 γ 8 + α8 α8 + ν γ γ 8 ( ) ) α + Cν ( k 4 γ 4 + α4 γ 6 + α4 γ 8 k+ + α4 γ 4 t φ n+ Cν t α6 γ 6 M n= φ n ) ) )) + ( k+ + α8 + t (α α8 γ. (5.64) γ 8 Applying te discrete Gronwall lemma we see, for any t >, φ M + α φ M M + ν t Cν exp (ν α6 γ 6 ( + k+ + α8 γ n= φ n+ ) [ (α k 4 + α8 γ 8 + α8 γ α4 + ν γ 8 α8 α4 α4 + ν + ν + ν γ8 γ4 γ 6 ) + t (α α8 α4 + ν γ8 γ 4 ) )]. (5.65) Finally, using our assumption tat α = C γ gives us te bound φ M + α φ M M + ν t n= φ n+ (5.66) Cν exp(ν ) ( k (α α + ν + ν + ν α ) + k+ ( + α 4 + ν α 4 ) + t (α ν ) ). Using te triangle inequality now finises te proof. 55

65 5.4 Numerical Results Tis section presents two bencmark numerical experiments cosen to evaluate te accuracy of Algoritm 5..4 on flow problems wit complex beaviours. In bot experiments, a resolved solution is generated by computing te NSE directly on a fine triangular mes, using te fully implicit Crank-Nicolson finite element metod (as in [46]). Solutions are ten generated for Algoritm 5..4 using various values for α and γ. We ope to see an increase in accuracy in RMDM solutions generated wit α > γ as opposed to tose generated wit α = γ (i.e. tose wit no deconvolution). All computations were performed using te open-source software FreeFem++ [39] D Cannel flow over a backward-facing step Fine mes -44, 456 dof Coarse mes - 5, 97 dof Figure 5.: Fine mes used for te resolved NSE solution and te coarse mes used for te RMDM approximations. Te first experiment we present is a well-known bencmark flow problem consisting of D cannel flow over a backward-facing step. Once te flow passes over te step, te sear-layer separates from te bottom wall, causing complex flow beaviour beind te step. It as been well documented tat te subsequent lengt until te flow reattaces to te wall is dependent upon bot 56

66 te Reynolds number and te expansion ratio of te cannel (see e.g. [5] and [44]). Te domain Ω consists of a 4 cannel wit a step of eigt and lengt running along te bottom of te cannel, as seen in Figure 5.. Our coice of cannel parameterization yields an expansion ratio of /9. For our experiment we employed a step eigt of =. Flow entering te cannel on te left is assumed to satisfy te parabolic Diriclet boundary condition 4(y )( y)/8 u =, wile zero-traction boundary conditions are enforced on flow exiting te cannel on te rigt. Noslip boundary conditions are assumed on all oter boundaries. All computations were performed using Reynolds number Re =, a time-step of t =., and wit te flow started from rest at T =. Te reference NSE solution was computed using TH (P, P ) elements on a mes providing 44, 456 combined degrees of freedom for te velocity and pressure (see Figure 5.). Te computed reference solution reaces a steady state by T =, wit a steady-state reattacment lengt of approximately 7.5 units, wic agrees wit te experimental results found in [5]. A plot of te velocity streamlines and speed contours of te solution at T = can be seen in Figure 5.. Solutions to Algoritm 5..4 were computed using TH elements on a coarse mes providing 5, 97 combined degrees of freedom for te velocity and pressure (see Figure 5.). We began by computing Algoritm 5..4 wit α = γ =.5, i.e. wit no deconvolution. A plot of te streamlines and speed contours of te solution at T = is sown in Figure 5.. It is clear tat te RMDM witout deconvolution produces a solution tat is incorrect. Te tree eddies present beind te step ave yet to merge into one large eddy. However, wen γ is lowered to.6, te RMDM solution looks very close to te reference solution. From Figure 5. we can clearly see tat bot te number of eddies present beind te step and te reattacment lengt bot seem to be correct. Hence, by reducing γ we can produce a substantially more accurate approximation D Cannel flow wit a contraction and two outlets Te second bencmark experiment we present is cannel flow wit a contraction, one inlet on te left of te cannel, and two outlets at te top and te rigt of te cannel (as seen in Figure 57

67 5.3). Tis problem was first studied by Heywood et al. [4]. Flow entering te cannel satisfies te parabolic profile u = 4y( y). Zero-traction boundary conditions are enforced on bot outflow boundaries, and no-slip boundary conditions are enforced on all wall boundaries. All computations were performed wit Re =, flow starting at rest at T =, and an end time of T = 4. Te reference NSE solution was computed using TH (P, P ) elements on a fine mes providing 6, 378 combined degrees of freedom for te velocity and te pressure (see Figure 5.3), and wit a time-step of t =.5. Speed contours of te resolved solution are sown in Figure 5.4. Note tat te flow speeds up in te contraction, and seems to oscillate up and down on te rigt side of te cannel. Additionally, te flow seems to remain in a single stream once it passes te contraction, altoug by T = 4, we also see te appearance of smaller flow structures near te rigt outflow boundary. Solutions to Algoritm 5..4 were computed using TH elements wit a time-step of t =. on te coarse mes pictured in 5.4, providing, 8 degrees of freedom for te velocity and pressure. Figure 5.4 sows a plot of te speed contours of te solution of our proposed metod for te RMDM wit α = γ =.5 (no deconvolution). It is clear from te plots tat tis velocity solution does not accurately capture te sape of te flow after te contraction. We also see flow leaving te cannel troug te top outlet, wic is not present in te resolved NSE solution plot. However, wen we recompute wit α =.5 and γ =.3, te velocity solution (pictured in Figure 5.4) muc more accurately captures te sape of te flow after te contraction. Additionally, we see very little fluid exiting te cannel troug te top outlet, wic agrees wit our true resolved solution. Bot solutions seem to partially capture te formation of smaller flow structures near te rigt outlet. Again, we see tat te solution to Algoritm 5..4 wit α > γ is muc closer to te resolved NSE solution tan tat computed wit no deconvolution. 58

68 Resolved NSE solution RMDM α =.5, γ =.5 RMDM α =.5, γ =.6 Figure 5.: Fine mes used for te resolved NSE solution and te coarse mes used for te RMDM approximations. 59

69 Contraction domain Fine mes - 6, 378 dof Coarse Mes -, 8 dof Figure 5.3: Diagram of te contraction domain, along wit te fine and coarse meses used in te computations for te contraction problem. 6