On an Efficient Finite Element Method for Navier-Stokes-ω with Strong Mass Conservationv

Transcription

1 Computational Metods in Applied Matematics Vol. (2), No., pp c 2 Institute of Matematics, National Academy of Sciences On an Efficient Finite Element Metod for Navier-Stokes-ω wit Strong Mass Conservationv Carolina C. Manica Monika Neda Maxim Olsanskii Leo G. Rebolz Nicolas E. Wilson Abstract We study 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 well-posedness, unconditional stability, and optimal convergence. Additionally, an improved coice of filtering radius (versus te usual coice of te average mes widt) for te sceme is found, by identifying a connection wit a sceme for te velocity-vorticity-elicity NSE formulation. Several numerical experiments are given tat demonstrate te performance of te sceme, and te improvement offered by te new coice of te filtering radius. 2 Matematical subject classification: 76D5; 76F99; 65M6. Keywords: Navier-Stokes-alpa; Navier-Stokes-omega; approximate deconvolution; Scott-Vogelius elements.. Introduction Recent work on finite element metods (FEM) for te α models of fluid flow as proven teir effectiveness at finding accurate solutions to flow problems on coarser spatial and temporal Partially supported by a researc grant from CNPq, RFBR troug projects 8--45, 9--5, NSF grant DMS94478 Carolina C. Manica Departmento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul ttp://casqueweb.ufrgs.br/ carolina.manica. Monika Neda Department of Matematics, University of Nevada, Las Vegas Monika.Neda@unlv.edu. Maxim Olsanskii Department of Mecanics and Matematics, M. V. Lomonosov Moscow State University, Moscow 9899, Russia Maxim.Olsanskii@mtu-net.ru, ttp:// molsan. Leo G. Rebolz Department of Matematical Sciences, Clemson University, Clemson, SC rebolz@clemson.edu, ttp:// rebolz. Nicolas E. Wilson Department of Matematical Sciences, Clemson University, Clemson, SC newilso@clemson.edu.

2 4 Carolina C. Manica et al. discretizations tan are necessary for successful direct numerical simulations of te Navier- Stokes equations (NSE) [4, 8, 9, 7, 24, 25, 29, 3, 36]. Of tese models, NS-ω stands apart because in addition to its excellent teoretical properties suc as well-posedness and energy and model elicity conservation [22, 26], it can be computed efficiently wit unconditionally stable algoritms [25]. Hence we restrict attention erein to tis model. We study an efficient finite element metod for NS-ω togeter wit van Cittert approximate deconvolution to increase accuracy, a natural element coice for te model tat provides pointwise mass conservation and removes te effect of te pressure error on te velocity error, and a different approac to coosing te filtering radius parameter tat leads to improved results. Wit all of te α models, iger formal accuracy to true fluid flow can be acieved if van Cittert approximate deconvolution is used to model te removed fine scales, by deconvolving te filtered terms. Our study of NS-ω includes tis type of deconvolution as part of te regularization, giving te continuous model u t + ( D N F u) u + q ν u = f, (.) u =, (.2) were u is te velocity field, q te pressure, f a forcing term, and ν te kinematic viscosity. Te function F denotes te Helmoltz filter: F := ( α 2 + I) were α > is te filtering radius, and D N is te N t order van Cittert approximate deconvolution operator: D N := N (I F )n. Te finite element discretization we use for te model (.)-(.2) includes a novel combination of several ideas. Te first idea is te linearization of te regularized terms via te metod of Baker []. Tis allows for te decoupling of te momentum-mass system from te filtering and deconvolution, tus allowing for te iger orders of deconvolution to be used wit minimal effect on computational time, wile maintaining unconditional energy stability. Te second idea is te use of te Scott-Vogelius velocity-pressure element pair [38, 39], wic is a fundamental component of te sceme. Tis element coice provides solutions wit pointwise conservation of mass and as recently been sown to be LBB stable and admits optimal approximation properties under only mild restrictions [4], provides excellent computational results, and decouples te pressure error from te velocity error, and is easily implemented [6, 7, 27, 28]. Since te goal of NS-ω is to find accurate solutions on coarse meses, suc an element coice will elp provide solutions tat are more pysically plausible; even if particular flow features are captured or an L 2 error comparison against known solution values is small, a solution wit poor mass conservation is of little value in most applications. Te problem of poor mass conservation for typical finite elements is well known, and coices of (P 2, P ) elements or discontinuous Galerkin metods are common due to teir local mass conservation property, even toug suc coices ave teir own sub-optimal features. Anoter important feature of using Scott-Vogelius element is tat it decouples te pressure error from te velocity error, wic is important for NS-ω. Since NS-ω is a rotational form model, its pressure approximates Bernoulli pressure, and is tus more complex tan usual pressure and susceptible to large error, especially near walls [23, 3, 32, 33]. Tis element coice keeps te pressure error from adversely affecting te velocity error. Lastly, we offer a different approac to coosing te filtering radius α, based on a connection between te model s discretization and a splitting metod for te recently proposed velocity-vorticity-elicity (VVH) formulation of te NSE [34]. Error analysis erein and for oter α-models as sown tat te coice of α O(), were is te average mes-widt, is

3 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 5 sufficient for optimal asymptotic accuracy [24, 25, 3]. However, on a fixed mes, te coice α = c is often used to improve results, were c is a tuning parameter. To aid in tis searc for a better α, we identify a connection between te NS-ω sceme proposed erein and te VVH splitting metod, wic suggests a coice of α = ν t. Our numerical experiments sow tat tis alternative parameter coice can provide improvement over α = on some test problems. Tis paper is arranged as follows. Section 2 presents notation and preliminaries, and te numerical discretization of NS-ω studied erein. In Section 3, we present a detailed, rigorous analysis of te sceme. We analyze stability and convergence, and te coice of α troug te connection to VVH. Compared to te earlier analysis of discretized Navier- Stokes-α models found in [2, 24, 25, 3], in te present paper we are able to remove te time step restriction and to weaken te dependence of constants in error estimate on te Reynolds number (cf. Remark 3.2). Numerical experiments are given in Section 4. Here we demonstrate te improvement in pysical fidelity offered by te SV element, provide numerical evidence of te decoupling of te velocity error from te pressure error for SV element solutions (and not for Taylor-Hood (TH) element solutions), and oter numerical experiments wic sow tat te proposed metod is able to obtain accurate answers on bencmark problems. 2. Preliminaries Tis section summarizes te notation, definitions, and preliminary lemmas needed. We start by introducing te following notation. Te L 2 (Ω) norm and inner product will be denoted by and (, ). Likewise, te L p (Ω) norms and te Sobolev Wp k (Ω) norms are denoted by L p and W k p, respectively. For te semi-norm in Wp k (Ω) we use W k p. H k is used to represent te Sobolev space W2 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 In te discrete case, we use te analogous norms ( T /m v(t, ) m k dt). v,k := max v n k, v /2,k := max v n+/2 k, n M n M ( M /m ( M /m v m,k := v n m k t), v /2 m,k := v n+/2 m k t). We will consider te case of internal flow, wit Ω being a regular, bounded, polyedral domain in R d (d = 2 or 3) and X = (H (Ω)) d := {v (H (Ω)) d : v Ω = }, Q = L 2 (Ω). We denote te dual space of X by X, wit te norm. Te space of divergence free functions is denoted V := {v X, ( v, q) = q Q}.

4 6 Carolina C. Manica et al. We now formally define te conforming finite element spaces, specifically tose corresponding to Scott-Vogelius elements, tat are subspaces of X and Q. In space dimension d, we define X SV to be te space of continuous element-wise vector functions of polynomial order k d on a mes T, X SV := {v [C (Ω)] d : v T [P k (T )] d, for all T T, v = on Ω}. We require tat te mes be built from a barycenter refinement of a regular triangularization (tetraedralization) of te domain Ω if k d, but if k = d (k = 2 in 3D or k = in 2D), ten a Powell-Sabin mes must be used, [4, 4]. Note tat te velocity space for Scott-Vogelius is te same as for te well-known Taylor- Hood element pair, wile te pressure space only differs from Taylor-Hood s in tat its pressures are discontinuous, defined by Q SV := {q L 2 (Ω) : q T P k for all T T }. Since it is discontinuous, te dimension of te pressure space for Scott-Vogelius elements is significantly larger tan for Taylor-Hood elements. Tis creates a greater total number of degrees of freedom needed for linear solvers using SV elements, owever it is not immediately clear weter tis will lead to a significant increase in computational time if specific preconditioners, suc as Augmented Lagrangian type are used [2]. 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 V SV := {v X SV : ( v, q ) = q Q SV }, wic is also te pointwise divergence free subspace of X SV. Te SV element is very interesting from te mass conservation point of view since its discrete velocity space and its discrete pressure space fulfill an important property, namely X SV Q SV. (2.) 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 most element coices, including Taylor-Hood. It was proved by S. Zang in [4, 4] tat te SV elements are LBB stable under tese restrictions and admit optimal approximation properties. It is well known tat te TH pair is LBB stable for tis case and tat tey satisfy te optimal approximation properties as well, [5, 5]. For te convergence studies, we make use of te following approximation properties: inf v X SV u v C k+ u k+, u (H k+ (Ω)) d, inf v X SV u v C k u k+, u (H k+ (Ω)) d, inf r Q SV p r C s+ p s+, p H s+ (Ω) Assumption 2.. Trougout tis report, we will assume tat k d and tat te mes is created from a barycenter refinement of a regular mes. Hence, Scott-Vogelius elements are LBB stable and admit optimal approximation properties wen used in tis report.

5 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 7 We define a trilinear form and list estimates necessary for te analysis studies. Definition 2.. Define b : X X X R, by b(u, v, w) := (( u) v, w). Lemma 2.. 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, (2.2) b(u, v, w) u v w, (2.3) b(u, v, w) C (Ω) u v w, (2.4) b(u, v, w) C (Ω) v /2 v /2 u w, (2.5) and if u, v, w V and w (H 2 (Ω)) d, ten b(u, v, w) C w 2 v u (2.6) 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 nonrotational form. Te last bound takes more work. Begin wit a simple vector identity and tat te curl is self-adjoint wit u, v, w X (( u) v, w) = (w v, u) = ( (w v), u) (2.7) Continuing wit anoter vector identity for te curl of te cross of two vectors, wic reduces since v, w V to (w v) = v w w v + ( v)w ( w)v, (2.8) (w v) = v w w v. (2.9) Combining tis wit (2.7), we ave by Holder and Poincare s inequalities, (( u) v, w) (v w, u) + (w v, u) C w 2 v u. (2.) Te error analysis uses a discrete Gronwall inequality, recalled from [6], for example. Te specific version given below is given as a remark to Lemma 5. in [6]. Lemma 2.2 (Discrete Gronwall Lemma). Let t, H, and a n, b n, c n, d n (for integers n ) be finite nonnegative numbers suc tat Ten for t > a l + t a l + t l l b n t d n a n + t ( ) l l b n exp t d n ( t l c n + H for l. (2.) ) l c n + H for l. (2.2)

6 8 Carolina C. Manica et al. 2.. Filtering and Deconvolution 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,, 7]. Tey can arise, for example, as approximations to Gaussian filters of ig qualitative and quantitative accuracy [3]. Definition 2.2 (Continuous α-filter). For v (L 2 (Ω)) d and α > fixed, denote te filtering operation on v by v, were v is te unique solution in X α 2 v + v = v. (2.3) We denote by F := ( α 2 + I), so F v = v. We define next te discrete differential filter following, Manica and Kaya-Merdan [3], but also enforcing incompressibility. Definition 2.3 (Discrete differential filter). Given v (L 2 (Ω)) d, for a given filtering radius α >, v = F v is te unique solution in X SV of: Find (v, λ ) (X SV, QSV ) satisfying α 2 ( v, χ ) + (v, χ ) (λ, χ ) + ( v, r ) = (v, χ ) (χ, r ) (X SV We now define te van Cittert approximate deconvolution operators., Q SV ). (2.4) Definition 2.4. Te continuous and discrete van Cittert deconvolution operators D N and DN are N N D N v := (I F ) n v, DNv := (Π F ) n v. (2.5) were Π denotes te L 2 projection Π : (L 2 (Ω)) d X. For order of deconvolution N =,, 2, 3 and v X we ave D v = v, D v = 2v v, D 2v = 3v 3v + v, D 3v = 4v 6v + 4v v. D N was sown to be an O(α 2N+2 ) approximate inverse to te filter operator F in Lemma 2. of []. Te proof is an algebraic identity and olds in te discrete case as well, giving te following. Lemma 2.3. D N and D N are bounded, self-adjoint positive operators. For v (L2 (Ω)) d, and v = D N v + ( ) (N+) α 2N+2 N+ F (N+) v v = D Nv + ( ) (N+) α 2N+2 N+ Lemma 2.4. For v X, we ave te following bounds: F (N+) v v v, v v and v v. (2.6)

7 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 9 Proof. Tis proof can be found in [25]. Lemma 2.5. For v X, we ave te following bounds: D Nv C(N) v. (2.7) Proof. Te proof follows from an inductive argument based on te definition of te deconvolution operator DN and Lemma 2.4. Lemma 2.6. For smoot v te discrete approximate deconvolution operator satisfies N+ v DNv Cα 2N+2 N+ F N+ v + C(α k + k )( F n v k+ ). (2.8) Te proof is based on using similar arguments as in [24]. Te dependence of te F n (v) k+ terms in (2.8) upon te filter radius α, for a general smoot function φ, is not fully understood for te case of deconvolution order N 2 (i.e., for n 2)[2, 2]. In te case of v periodic te F n (v) k+ are independent of α. Also, for v satisfying omogeneous Diriclet boundary conditions, wit te additional property tat j v = on Ω for j [ ] k+ 2, te F n (v) k+ are independent of α. Our analysis of te metod is for general N, and tus for N 2, we make tis assumption of independence. However, our computations are for N =, and our experience as sown tat tere is typically little or no gain for larger N wit polynomials approximating velocities wit degree tree or less. For elements wit iger order polynomials, we would expect a difference 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 [] extrapolation to linearize and maintain unconditional stability. We denote u(t n+/2 ) = u((t n + t n+ )/2) for te continuous variable and u n+/2 = (u n + u n+ )/2 for bot continuous and discrete variables. Algoritm 2.. 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 ten set u (u, v ) (λ, v ) = (u, v ), v X SV (2.9) ( u, r ) = r Q SV, (2.2) := u, and find (un+, q n+ 2 ) (X SV t (un+ u n, v ) + (( DNF ( 3 2 un 2 un ) u n+ 2 ), v ) n=, QSV ) for n =,,..., M satisfying (q n+ 2, v ) + ν( u n+ 2, v ) = (f n+ 2, v ) v X SV, (2.2) ( u n+, r ) = r Q SV. (2.22)

8 Carolina C. Manica et al. 3. Analysis of te sceme In tis section, we sow tat solutions of te NS-ω algoritm are unconditionally energystable, i.e., discrete solutions obey te energy estimate and are optimally convergent. Since NS-ω approximates NSE solutions, we study te convergence analysis of te finite element solutions of NS-ω to te true solutions of te NSE. We sall assume tat te solution to te NSE tat is approximated by te model is a strong solution and, in particular, satisfies w L 2 (, T ; X) L (, T ; (L 2 (Ω)) d ) L 4 (, T ; X), p L 2 (, T ; Q), w t L 2 (, T ; X ) and (w t, v) + b(w, w, v) (p, v) + ν( w, v) = (f, v) v X, (3.) (q, w) = q Q. (3.2) 3.. Stability and well-posedness Lemma 3.. Consider te NS-ω algoritm 2.. A solution u l, l =,... M, exists at eac time-step and it is unique. Te algoritm is also unconditionally energy-stable: te solutions satisfy te á priori bound u M 2 + ν t u n+/2 2 u 2 + t f n+/2 2 ν. (3.3) Proof. Te á priori bound can be obtained by setting v = u n+/2 term in te sceme vanises wit tis coice. Tus, for every n in (2.2). Te nonlinear i.e., 2 t ( un+ 2 u n 2 ) + ν u n+/2 2 2ν f n+/2 2 + ν 2 un+/2 2, t ( un+ 2 u n 2 ) + ν u n+/2 2 ν f n+/2 2. Summing from n =... M gives te bound (3.3). Restricting to any particular time level n, te sceme is linear, and tus since it is also finite dimensional, uniqueness of solutions gives existence as well. Following te proof for te á priori bound, uniqueness is immediate at any time level. Tus solutions exist uniquely for te entire sceme. Remark 3.. Since te kinetic energy KE(u n ) := 2 un 2 and energy dissipation ε(u n ) := ν u n 2 of NS-ω, take te usual form, it olds KE(u M) + ν t ε(u n+/2) = KE(u ) + t (f n+/2, u n+/2 ). (3.4) Tus, if ν = and f =, KE(u M ) = KE(u ). Hence Algoritm 2. is energy conserving.

9 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 3.2. Convergence Analysis Our main convergence result for te discrete NS-ω model described in Algoritm 2. is given next. Teorem 3. (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 (3.5)-(3.6) are finite. Suppose (u, p ) are te VSV and Q SV interpolants of (w(), p()), respectively. Suppose (u, q ) satisfies te sceme (2.2)-(2.22). Ten tere is a constant C = C(w, p) suc tat ( ν t were (w n+/2 (u n+ + u n )/2) 2 ) /2 w u, F ( t,, α) + C k+ w,k+, (3.5) F ( t,, α) + Cν /2 ( t) 2 w tt 2, + Cν /2 k w 2,k+, (3.6) F ( t,, α) := C {(ν + ν ) /2 k w 2,k+ +ν /2 ( ) k w 2 4,k+ + w /2 2 4, + C(N)( t) ( ) 2 w ttt 2, + f tt 2, + (ν + ν ) /2 w tt 2, + ν /2 ( t 2 + α 2N+2 + α k + k ) w /2 2, }. (3.7) Remark 3.2. Tere are tree important points to note from te teorem. First, te velocity error does not depend at all on te pressure error. Second, optimal accuracy can be acieved if α O(), and 2N + 2 k, wic provides a guide for parameter selection. Finally, tere is no time step restriction for te result; typically tere is a strong restriction on te timestep resulting from using te Gronwall inequality, but our analysis allowed for a less restrictive discrete Gronwall inequality and no restriction on te time step. Proof of Teorem 3.. Tis proof closely follows te convergence estimate in [25], except for a few subtle differences tat lead to improved estimates erein. Hence we give a sortened proof of te teorem, wic igligts tese differences. Let b ω (u n+/2, v n+/2, χ n+/2 ) := (( DNF ( 3 2 un 2 un )) v n+/2, χ n+/2 ), and, ten by adding and subtracting terms, we can write b ω (u n+/2, v n+/2, χ n+/2 ) = b(u n+/2, v n+/2, χ n+/2 ) F E(u n+/2, v n+/2, χ n+/2 ), were te linear extrapolated deconvolved filtering error FE is given by F E(u n+/2, v n+/2, χ n+/2 ) := (( u n+/2 D NF ( 3 2 un 2 un At time t n+/2, te solution of te NSE (w, p) satisfies ( ) w n+ w n, v + b ω (w n+/2, w n+/2, v ) + ν( w n+/2, v ) t )) v n+/2, χ n+/2 ). = (f n+/2, v ) + Intp(w n, v ), v V SV (3.8)

10 2 Carolina C. Manica et al. were te pressure term disappears since V SV is now pointwise div-free, as stated in Section 2. Te term Intp(w n, v ) collects te interpolation error, te above linear extrapolated deconvolved filtering error, and te consistency error. It is given by ( ) w Intp(w n n+ w n, v ) = w t (t n+/2 ), v + ν( w n+/2 w(t n+/2 ), v ) t + b ω (w n+/2, w n+/2, v ) b ω (w(t n+/2 ), w(t n+/2 ), v ) F E(w(t n+/2 ), w(t n+/2 ), v ) + f(t n+/2 ) f n+/2, v ). Subtracting (3.8) from (2.2) and letting e n = w n u n we ave t (en+ e n, v ) + b ω (w n+/2, w n+/2, v ) b ω (u n+/2, u n+/2, v ) + ν( e n+/2, v ) = Intp(w n, v ), v V SV. were te pressure term of NS-ω disappears since V SV (3.9) (3.) is now pointwise div-free. De-, and U is in (3.) we obtain compose te error as e n = (w n U n ) (u n Un ) := η n φ n were φ n V SV te L 2 projection of w in V SV 2 ( φn+ 2 φ n 2 ) + ν t φ n+/2. Setting v = φ n+/2 2 = (η n+ η n, φ n+/2 ) + tν( η n+/2, φ n+/2 ) + t b ω (η n+/2, w n+/2, φ n+/2 ) t b ω (φ n+/2, w n+/2, φ n+/2 ) + t b ω (u n+/2, η n+/2, φ n+/2 ) + t Intp(w n, φ n+/2 ) = T + T 2 + T 3 + T 4 + T 5 + T 6. (3.) We point out again tat, tis error equation we will proceed to analyze, is independent of pressure error, wic occurs due to te use of Scott-Vogelius elements and would not appen if, for example, Taylor-Hood elements were cosen. Estimates for te rigt and side terms of (3.) follow similar to tose in te convergence proof in [25], except for te last tree terms. For te last term, te main difference is te use of deconvolution, and so one can follow estimates in [3, 24] to bound tis term. Tus we proceed to estimate T 4 and T 5 terms of (3.). For T 4 term, using (2.6), we get t b ω (φ n+/2, w n+/2, φ n+/2 ) = t ( DNF ( 3 2 φn 2 φn ) w n+/2, φ n+/2 C t w n+/2 φ n+/2 2 D NF ( 3 2 φn 2 φn ) C(N) t w n+/2 φ n+/2 2 ( φ n + φ n ) ν t 2 φn+/2 2 + C(N) t ν ( φ n 2 + φ n 2 ) w n+/ ) For T 5 term, we begin by splitting te first entry of tis term by adding and subtracting w n+/2, followed by rewriting te resulting error term as pieces inside and outside of te finite element space. tb ω (u n+/2, η n+/2, φ n+/2 ) = tb ω (η n+/2, η n+/2, φ n+/2 + tb ω (φ n+/2 ), η n+/2, φ n+/2 ) + tb ω (w n+/2, η n+/2, φ n+/2 ). (3.2)

11 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 3 For te first and last terms of te above equation (3.2), usual estimates give tb ω (η n+/2, η n+/2, φ n+/2 ) ν t 24 φn+/2 2 + C(N) tν ( η n 2 + η n 2 ) η n+/2 2, tb ω (w n+/2, η n+/2, φ n+/2 ) ν t 24 φn+/2 2 + C(N) tν η n+/2 2. Te second term of (3.2) is a bit more delicate, and is majorized as tb ω (φ n+/2 t (φ n+/2 t (φ n+/2, η n+/2, φ n+/2 ) = t( DNF ( 3 2 φn 2 φn ) η n+/2, φ n+/2 η n+/2, D NF ( 3 2 φn 2 φn )) η n+/2, D NF ( 3 2 φn 2 φn )) + t (η n+/2 φ n+/2, DNF ( 3 2 φn 2 φn )) C(N) t η n+/2 φ n+/2 3 2 φn /2 2 φn ( 3 2 φn 2 φn ) C(N) /2 t η n+/2 φ n+/2 3 2 φn 2 φn ν t 2 + C(N) t ν ( φ n 2 + φ n 2 ) η n+/ φn+/2 ) Combining estimates and summing from n = to M (assuming tat φ = ) reduces (3.) to φ M 2 + ν t φ n+/2 2 n= C t{ Cν ( w n+/ η n+/2 2 )( φ n 2 + φ n 2 ) n= ( + (ν + ν ) η n+/2 2 + ν ( η n 2 + η n 2 ) w n+/2 2 n= +ν ( η n 2 + η n 2 ) η n+/2 2) + n= C t{ Cν ( w n+/ η n+/2 2 ) φ n 2 n= n= Intp(w n, φ n+/2) } +(ν + ν ) η n+/2 2 + ν η n 2 w n+/2 2 M +ν η n 4 + n= Intp(w n, φ n+/2) }. (3.3) /2

12 4 Carolina C. Manica et al. From ere, standard tecniques will finis te proof, except for one subtle difference: te alternate Gronwall lemma (e.g., of [6]) can be used since no φ M 2 appears on te rigt and side. Tis implies no timestep restriction is needed, and te constants tat arise will depend on ν instead of ν 3 [2] 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 [34], tat suggests an alternative coice of α tat may aid in tis process. NS-ω can be considered as a rotational form of te 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 [34] for te VVH NSE formulation Algoritm 3.. Step. Given u n, u n, w n and u = 3 2 un 2 un, find w n+ and el n+/2 from w n+ w n ν w n+/2 + 2D(w n+/2 )u el n+/2 = f n+/2 t (3.4) w n+ = (3.5) w n+ = (2u n u n ) on Ω (3.6) Step 2. Given u n, w n and w n+, find u n+ and P n+/2 u n+ u n ν u n+/2 + w n+/2 u n+/2 P n+/2 = f n+/2 t (3.7) u n+ = (3.8) u n+ = φ on Ω (3.9) were φ is te Diriclet boundary condition function and D( ) denotes te deformation tensor, i.e., D(v) = ( v)+( v)t, u, w denote velocity and vorticity, P is te Bernoulli pressure 2 and el is elical density. If f is irrotational and we remove te nonlinear term from te vorticity equation, tis system is analogous to 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 2, i.e., α = ν t. 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.

13 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 5 4. 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 wen α is cosen as we suggest erein. Te tird and fourt tests are done wit Scott- Vogelius elements and Taylor-Hood elements, and compare solutions for a problem wit known analytical solution and te cylinder problem. All tests are done using our Matlab code, and linear solver uses te slas command. Of course, suc a solver is non-optimal for saddle point problems, and for larger 3D problems, a more sopisticated linear solver will be needed to solve te saddle point system. Tis will be addressed in future work. 4.. Experiment : Cannel flow over a forward-backward facing step Our first numerical experiment is for te bencmark 2D 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)/25, ) T. We use te initial condition of u = (y( y)/25, ) 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, 9]. 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 test, we computed Algoritm 2. wit (P 2, P disc ) Scott-Vogelius elements on a barycenter-refinement of a Delauney triangulated mes, yielding 4,467 total degrees of freedom, wit deconvolution order N =, and varying α. Tis mes provided a smallest element widt of.5 units beind te step, and largest of.9 units, at te top of te cannel around x = 3. For comparison, we also directly compute te (non-regularized) NSE (α = N = ). We compute first wit timestep t =.5, and te solutions at T = 4 are sown in Fig. 4.. 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 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 = =.627, give increasingly worse solutions. Tis is 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.25, wit parameter α = ν t. Te results at T = 4 are sown in Figure 4.2, and sow good results 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.

14 6 Carolina C. Manica et al. 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 4.. 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

15 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 7 NS-ω, t =.25, α = ν t NS-ω, t =., α = ν t Figure 4.2. 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 4.2. Experiment 2: Cannel flow around a cylinder Te bencmark problem of 2d cannel flow around a cylinder as been studied in numerous works, e.g., [8, 2, 23, 37], and is well documented in [37]. Te domain is te rectangle [, 2.2] [,.4] representing te cannel wit flow in te positive x direction, wit a circle radius.5 centered at (.2,.2) 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 [ ] T 6 u(, y, t) = u(2.2, y, t) = sin(πt/8)y(.4 y),, y 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 timestep 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 = =.84 and α = ν t, on a barycenter refined mes tat provides 26,656 degrees of freedom for (P 2, P disc ) Scott-Vogelius elements (minimum and maximum element widts were.42 and.7 respectively), and again find tat α = ν t provides a better solution tan α =. Tese results are sown for t = 7 in Fig Te α = ν t solution agrees wit documented DNS results [7, 23], 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.

16 8 Carolina C. Manica et al SV elements, α = = SV elements, α = ν t = Figure 4.3. Te.2above pictures sow te velocity fields and speed contours at t = 7 using Scott-Vogelius. 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 Comparison to Taylor-Hood element solution Using te same problem data as Experiment 2 above, we also compute using (P 2, P ) Taylor-Hood 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 Scott-Vogelius elements (tey differ only in te pressure space being continuous or not), a comparison is of interest. Since Taylor-Hood uses a continuous pressure space, wit te same mes te total number of degrees of freedom is 7,36. All of problem data is kept te same, and results are sown in Figs. 4.4 and 4.5. In Fig. 4.4, we observe tat te Taylor-Hood solution is muc worse tan te Scott-Vogelius solution sown in Fig. 4.3; te Taylor-Hood solution fails to resolve te important beavior beind te cylinder. Figure 4.5 sows mass conservation versus time for te Taylor-Hood and Scott- Vogelius solutions. As expected, te Scott-Vogelius solution is divergence-free up to macine precision. Te mass conservation offered by te Taylor-Hood solution is poor. Even toug, asymptotically, te divergence error is optimal, on coarse meses (were one would ope to use regularization models) te actual error can be bad. However, on average te linear solver took.7 seconds for te SV element wile te linear solver for te TH element only required.49 seconds. It is not surprising tat te Taylor-Hood solution is muc worse tan te Scott-Vogelius solution. It was sown in [23] tat for te rotational form NSE, te Bernoulli pressure error can be large enoug to dramatically reduce velocity error for tis problem. Since NS-ω is also rotational form, tis same effect can be expected (and is seen in comparing Fig. 4.3 to Fig. 4.4). However, for te Scott-Vogelius solution, as sown erein, te velocity error is independent of te pressure error. Tus even toug te pressure error may be large, it as no adverse effect on te velocity error, leaving te good solution seen in Fig. 4.3.

17 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation TH elements, α = ν t Figure 4.4. Te above picture sows te t = 7 solution using Taylor-Hood elements, as velocity vector field and speed contours. Tis solution is incorrect, as it fails to capture any wake beind te cylinder 5 u (t n ) 5 Taylor Hood Scott Vogelius t n Figure 4.5. Sown above are te plots of te L 2 norms of te divergence of te velocity solutions versus time, for te SV and TH solutions, bot wit α = ν t =.22 As expected, te Scott-Vogelius solution is incompressible to near macine precision. Te Taylor-Hood solution, owever, gives poor mass conservation 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 Scott-Vogelius and Taylor-Hood solutions in te above experiment of flow around a cylinder. Te error analysis in Section 3.2 sowed tat in Algoritm 2. wic uses Scott-Vogelius elements, te velocity error is not affected by te pressure error. If Taylor-Hood elements are used, owever, ten te energy error of te velocity can be sown to depend on C(ν ) t inf r Q T H q r, e.g. [25], altoug te scaling by C(ν ) of tis term can be reduced by using grad-div stabilization [23, 3, 35]. To better demonstrate tis effect, we compute Algoritm 2. wit bot Scott-Vogelius and Taylor-Hood elements, for a series of simple test problems wit increasing pressure complexity and te same velocity solution. On te domain, Ω = (, ) 2 and t. = T, we coose u = ( +.t) ( cos(y) sin(x) ), p = x + y + sin(n(x + y)),

18 2 Carolina C. Manica et al. wic will solve te NSE wit an appropriate function f. Solutions are approximated to tis solution on a quasi-uniform barycenter-refined mes tat provides 2,64 degrees of freedom wit (P 2, P disc ) Scott-Vogelius elements (7,258 for velocity and 5,364 for pressure) and 8,82 degrees of freedom wit (P 2, P ) Taylor-Hood (7,258 for velocity and 924 for pressure), kinematic viscosity is set to be ν =., timestep t =.25, α = ν t =.58, N =, mes widt () =.625, and te parameter for pressure complexity n =,, 2, 3. Te results are sown in Table 4., and as expected te error in te Scott-Vogelius velocity solution is unaffected by te increase in pressure complexity. However, te Taylor-Hood velocity solution significanty lost accuracy. Also included in te table is te size of te velocity divergence, measured in L 2 (, T ; L 2 (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 worse wit increasing pressure complexity. n unse u SV 2, u SV 2, unse u T H 2, u T H 2, 7.332E-5.6E E E E-5.67E E E E-5.2E-4.76E-2.533E E E E E-2 Table 4.. Errors in velocity and divergence for Experiment for Scott-Vogelius and Taylor-Hood elements used wit Algoritm 2. References [] G. Baker, Galerkin Approximations for te Navier-Stokes Equations, Harvard University, 976. [2] M. Benzi and M. Olsanskii, An augmented Lagrangian-based approac to te Oseen problem, SIAM J. Sci. Comput., 28 (26), pp [3] L. Berselli, T. Iliescu, and W. J. Layton, Matematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer, 26. [4] A. Bowers and L. Rebolz, Increasing accuracy and efficiency in FE computations of te Leray-deconvolution model, Numerical Metods for Partial Differential Equations, (2) DOI:.2/num [5] S. Brenner and L. Scott, Te Matematical Teory of Finite Element Metods, Springer-Verlag, 994. [6] E. Burman and A. Linke, Stabilized finite element scemes for incompressible flow using Scott-Vogelius elements, Applied Numerical Matematics, 58 (28), no., pp [7] M. Case, V. Ervin, A. Linke, and L. Rebolz, Improving mass conservation in FE approximations of te NSE wit C velocities: A connection between Scott-Vogelius elements and grad-div stabilization, Submitted, available as tecnical report as WIAS preprint 5 at ttp:// preprints 5.pdf, (2). [8] S. Cen, C. Foias, D. Holm, E. Olson, E. Titi, and S. Wynne, Te Camassa-Holm equations as a closure model for turbulent cannel and pipe flow, Pys. Rev. Lett., 8 (998), pp [9] S. Cen, D. Holm, L. Margolin, and R. Zang, Direct numerical simulations of te Navier-Stokes alpa model, Pysica D, 33 (999), pp [] A. Ceskidov, D. Holm, E. Olson, and E. Titi, On a Leray-alpa model of turbulence, Royal Society London, Proceedings, Series A, Matematical, Pysical and Engineering Sciences, (25), pp

19 On an efficient finite element metod for Navier-Stokes-ω wit strong mass conservation 2 [] A. Dunca and Y. Epsteyn, On te Stolz-Adams deconvolution model for te Large-Eddy simulation of turbulent flows, SIAM J. Mat. Anal., 37 (25), no. 6, pp [2] V. Ervin, W. Layton, and M. Neda, Numerical analysis of filter based stabilization for evolution equations, Submitted, available as tecnical report TR-MATH - from ttp:// (2). [3] G. Galdi and W. Layton, Approximating te larger eddies in fluid motion II: A model for space filtered flow, Mat. Metods and Models in Appl. Sci., (2), no. 3, pp. 8. [4] M. Germano, Differential filters for te large eddy numerical simulation of turbulent flows, Pysics of Fluids, 29 (986), pp [5] M. Gunzburger, Finite Element Metods for Viscous Incompressible Flow: A Guide to Teory, Practice, and Algoritms, Academic Press, Boston, 989. [6] J. Heywood and R. Rannacer, Finite element approximation of te nonstationary Navier-Stokes problem. Part IV: Error analysis for te second order time discretization, SIAM J. Numer. Anal., 2 (99), pp [7] D. Holm and B. Guerts, Leray and LANS-α modelling of turbulent mixing, J. of Turbulence, 7 (26), no.. [8] V. Jon, Reference values for drag and lift of a two dimensional time-dependent flow around a cylinder, International Journal for Numerical Metods in Fluids, 44 (24), pp [9] V. Jon and W. J. Layton, Analysis of numerical errors in Large Eddy Simulation, SIAM J. Numer. Anal., 4 (22), no. 3, pp [2] A. Labovsky, W. Layton, C. Manica, M. Neda, and L. Rebolz, Te stabilized extrapolated trapezoidal finite element metod for te Navier-Stokes equations, Comput. Metods Appl. Mec. Engrg., 98 (29), pp [2] W. Layton, A remark on regularity of an elliptic-elliptic singular perturbation problem, Tec. rep., University of Pittsburg, 27. [22] W. Layton, Te existence of smoot attractors for te NS-ω model of turbulence, Journal of Matematical Analysis and Applications, 366 (2), pp [23] W. Layton, C. Manica, M. Neda, M. Olsanskii, and L. Rebolz, On te accuracy of te rotation form in simulations of te Navier-Stokes equations, J. Comput. Pys., 228 (29), no. 5, pp [24] W. Layton, C. Manica, M. Neda, and L. Rebolz, Numerical analysis and computational testing of a ig-accuracy Leray-deconvolution model of turbulence, Numerical Metods for Partial Differential Equations, 24 (28), no. 2, pp [25] W. Layton, C. Manica, M. Neda, and L. Rebolz, Numerical analysis and computational comparisons of te NS-omega and NS-alpa regularizations, Comput. Metods Appl. Mec. Engrg., 99 (2), pp [26] W. Layton, I. Stanculescu, and C. Trencea, Teory of te NS-ω model: A complement to te NS-alpa model, Submitted, available as tecnical report TR-MATH8-8 at ttp:// (28). [27] A. Linke, Collision in a cross-saped domain A steady 2d Navier-Stokes example demonstrating te importance of mass conservation in CFD, Comp. Met. Appl. Mec. Eng., 98 (29), no. 4 44, pp [28] A. Linke, G. Matties, and L. Tobiska, Non-nested multi-grid solvers for mixed divergence free scottvogelius discretizations, Computing, 83 (28), no. 2-3, pp [29] E. Lunasin, S. Kurien, M. Taylor, and E. Titi, A study of te Navier-Stokes-alpa model for twodimensional turbulence, Journal of Turbulence, 8 (27), pp [3] C. Manica and S. K. Merdan, Convergence Analysis of te Finite Element Metod for a fundamental model in turbulence, Tec. rep., University of Pittsburg, 26.

20 22 Carolina C. Manica et al. [3] C. Manica, M. Neda, M. Olsanskii, and L. Rebolz, Enabling accuracy of Navier-Stokes-alpa troug deconvolution and enanced stability, M2AN: Matematical Modelling and Numerical Analysis, 8 (2), pp [32] M. Olsanskii, A low order Galerkin finite element metod for te Navier-Stokes equations of steady incompressible flow: A stabilization issue and iterative metods, Comp. Met. Appl. Mec. Eng., 9 (22), pp [33] M. Olsanskii, G. Lube, T. Heister, and J. Löwe, Grad-div stabilization and subgrid pressure models for te incompressible Navier-Stokes equations, Comp. Met. Appl. Mec. Eng., 98 (29), pp [34] M. Olsanskii and L. Rebolz, Velocity-vorticity-elicity formulation and a solver for te Navier-Stokes equations, Journal of Computational Pysics, 229 (2), pp [35] M. Olsanskii and A. Reusken, Grad-Div stabilization for te Stokes equations, Mat. Comp., 73 (24), pp [36] L. Rebolz and M. Sussman, On te ig accuracy NS-α-deconvolution model of turbulence, Matematical Models and Metods in Applied Sciences, 2 (2), pp [37] M. Scäfer and S. Turek, Te bencmark problem flow around a cylinder flow simulation wit ig performance computers II, in E.H. Hirscel (Ed.), Notes on Numerical Fluid Mecanics, 52, Braunscweig, Vieweg (996), pp [38] M. Vogelius, An analysis of te p-version of te finite element metod for nearly incompressible materials. Uniformly valid, optimal error estimates, Numer. Mat., 4 (983), pp [39] M. Vogelius, A rigt-inverse for te divergence operator in spaces of piecewise polynomials. Application to te p-version of te finite element metod, Numer. Mat., 4 (983), pp [4] S. Zang, A new family of stable mixed finite elements for te 3d Stokes equations, Mat. Comp., 74 (25), no. 25, pp [4] S. Zang, Quadratic divergence-free finite elements on Powell-Sabin tetraedral grids, Calcolo, (2), DOI:.7/s