A connection between grad-div stabilized FE solutions and pointwise divergence-free FE solutions on general meshes

Transcription

1 A connection between grad-div stabilized FE solutions and pointwise divergence-free FE solutions on general meshes Sarah E. Malick Sponsor: Leo G. Rebholz Abstract We prove, for Stokes, Oseen, and Boussinesq finite element discretizations on general meshes, that grad-div stabilized Taylor-Hood velocity solutions converge to the pointwise divergence-free solution (found with the iterated penalty method) at a rate of, where is the grad-div parameter. However, pressure is only guaranteed to converge when (Xh, Xh ) satisfies the LBB condition, where Xh is the finite element velocity space. For the Boussinesq equations, the temperature solution also converges at the rate. We provide several numerical tests that verify our theory. This extends work in [6] which requires special macroelement structure in the mesh. Introduction It has recently been shown for several types of incompressible flow problems that on meshes with special structure that grad-div stabilized Taylor-Hood solutions converge to the Scott-Vogelius solution as the graddiv parameter approaches infinity [, 7, 6]. Large grad-div stabilization parameters can be helpful in enforcing mass conservation in solutions, reducing error by reducing the effect of the pressure on the velocity error, and making the Schur complement easier to precondition and solve. However, a large parameter essentially enforces extra constraints on the velocity solution. Hence it is not clear if a negative effect can arise from a large parameter. By proving what the limit solution is, we can know the quality of a solution found with a large parameter. The results of our paper extend this idea to general meshes where a Scott-Vogelius solution may not exist. In such cases, convergence will be to the pointwise divergence-free solution (which is found with the iterated penalty method). In this general case, the velocity solution still converges, but the pressure solution only converges when (Xh, Xh ) satisfies the LBB condition, where Xh is the finite element velocity space; we consider Xh to be globally continuous piecewise polynomials. We consider the Stokes, Oseen, and Boussinesq systems, and for each case, we rigorously prove the convergence rates for velocity and pressure, and also temperature for Boussinesq systems. We numerically test our theory, and all computational results are consistent with our analysis. This paper is arranged as follows: Section provides notation and preliminaries to allow for a smooth presentation to follow. In Section 3, we study the Stokes equations. The main result is that the grad-div stabilized Taylor-Hood velocity solutions converge to the divergence-free solution with a rate of as (the grad-div parameter) approaches, and a modified pressure converges to the divergence-free pressure if (Xh, Xh ) satisfies the LBB condition. In Section 4, we extend the results of Section 3 to the Oseen equations and obtain analogous results. Finally, in Section 5, we further extend the results to non-isothermic Stokes flows and again find analogous results. Department of Mathematical Sciences, Clemson University, Clemson, SC 9634 (smalick@g.clemson.edu), partially supported by NSF grant DMS59. Department of Mathematical Sciences, Clemson University, Clemson, SC 9634 (rebholz@clemson.edu). Copyright SIAM Unauthorized reproduction of this article is prohibited 367

2 Mathematical preliminaries We assume that the domain Ω Rd (d =, 3) is open and connected with Lipschitz boundary Ω. The L (Ω) norm and inner product are denoted by k k and (, ), respectively. We define the following function spaces X := H (Ω)d, Q := L (Ω). In X, it is well known that the Poincare -Friedrich inequality holds: there exists a CP F = CP F (Ω) such that φ X, kφk CP F k φk. (.) The dual of X is denoted by H (Ω), with norm kφk := sup v X (φ, v). k vk We define the skew-symmetric trilinear operator b : X X X R by b (u, v, w) := (u v, w) (u w, v). From [5], we know that b (u, v, w) M k ukk vkk wk, (.) where M depends on the size of Ω. Let Th be a shape-regular, simplicial and conforming triangulation of Ω. Let ht = diam(t ) denote the diameter of a simplex T and define h := maxt Th. We denote by Pk the space of piecewise polynomials with respect to the triangulation Th with degree not exceeding k, and by Pk := [Pk ]d the analogous vector-valued polynomial space. Throughout the paper, we consider finite dimensional spaces Xh, Qh, defined by Xh = X Pk (Th )d and Qh = Q (Pk (Th )), which are LBB-stable: inf sup qh Qh vh Xh ( vh, qh ) β >. kqh k We define the space of discretely divergence-free functions Vh := {vh Xh : ( vh, qh ) = qh Qh }, and also the space of pointwise divergence-free functions, Vh := {vh Xh : k vh k = }. Note that in general, and for most common element choices like Taylor-Hood, Vh 6= Vh. We also define the space orthogonal to Vh in Xh in the X inner product: Rh = (Vh ) := {rh Xh, ( rh, vh ) = vh Vh }. In Rh, we observe that the L (Ω) gradient norm and the divergence norm are equivalent [3], i.e. there exists CR satisfying CR k vh k vh Rh. (.3) In the case that Vh = Vh, the constant CR is independent of h [4], however in general it can depend inversely on h [3]. For our study of the Boussinesq system, we will also use a temperature space Sh = {sh H (Ω) (Pk (Th )), sh ΓDirichlet = }. 368

3 3 A connection for the Stokes equations We first consider the Stokes equations, which are given by ν4u + p = f, u =, u Ω =, where u represents velocity, p represents pressure, f is an external forcing, and ν is the kinematic viscosity. We will compare the grad-div stabilized Taylor-Hood scheme and the pointwise divergence-free scheme to determine a connection between the two methods. These two methods are given below: Algorithm 3.. Given, find (wh, qh ) (Xh, Qh ) satisfying ν( wh, vh ) (qh, vh ) + ( wh, vh ) = (f, vh ) vh Xh, ( wh, rh ) = rh Qh. (3.a) (3.b) Algorithm 3.. Find uh Vh satisfying ν( uh, vh ) = (f, vh ) vh Vh. (3.) We will use the iterated penalty method to solve (3.). Step : ν( uh, vh ) + (α( uh ), vh ) = (f, vh ) Step k : ν( ukh, vh ) + (α( ukh ), vh ) = ν( uk h, vh ) vh Xh (3.3a) vh Xh (3.3b) 3 [8], then there exists If the iterated penalty method converges, which is known to occur if, e.g., α > dνcr k an N such that k uh k tol f or k N. If tol is sufficiently small, then it is reasonable to assume uh = un h. PN i After convergence, we recover pressure via ph := i= (α( uh )), thus satisfying ν( uh, vh ) (ph, vh ) = (f, vh ) vh Xh. Lemma 3.. Suppose α is sufficiently large so that Algorithm 3. converges, and so that (3.4) CR α <. Then, kph k ν kf k. Proof. The initial step of the iterated penalty method is defined in (3.3a). The system is known to be well-posed (and can be proven using the Lax-Milgram Theorem). Choosing v = uh and using the definition of k k and Young s inequality gives νk uh k + αk uh k = (f, uh ) ν ν kf k + k uh k. Reducing now implies that ν ν k uh k + αk uh k kf k. (3.5) 369

4 PN Now, we consider our earlier definition of ph := k= (α( ukh )), where N is the number of iterations of the iterated penalty method. From previous work in [8], we know that k ukh k CαR k uk h k k N. For ease of notation, let r := CR α. Then < r < kph k α N X < (by the assumption that CR α < ), and so k ukh k k= = α( k uh k + rk uh k + + rk k uh k) = αk uh k( + r + + rn ) rn + = αk uh k r αk uh k = αk uh k. Using (3.5), we obtain the desired result, kph k ν kf k. Theorem 3.3. Let (wh, q ) be the solution to Algorithm 3. for a fixed and let uh be the solution to Algorithm 3.. Then, k (wh uh )k ν CR kf k. Thus, as, the sequence of grad-div stabilized Taylor-Hood velocity solutions, {wh }, converges to uh. Proof. Denote e := wh uh and let ph be the recovered pressure from Algorithm 3.. We subtract (3.4) from (3.a) to obtain ν( e, vh ) (qh ph, vh ) + ( e, vh ) = vh Xh. (3.6) Orthogonally decompose e Vh as e = e + er where e Vh, er Rh. Choosing vh = e in (3.6). This yields νk e k = which implies e =. Next choose vh = er in (3.6). This gives νk er k + (ph, er ) + k er k =, and after applying Cauchy-Schwarz and Young inequalities to the pressure term, we obtain νk er k + k er k = (ph, er ) k er k + kph k. Further reduction yields k er k νk er k + k er k kph k, and after taking the square root of both sides, we obtain k er k kph k. Using (.3), we have that k er k CR k er k CR kph k. and after applying Lemma 3., we obtain kf k. k er k ν CR (3.7) 37

5 Thus, using e = and (3.7), we conclude k ek = k er k ν CR kf k. (3.8) Corollary 3.. Under the same assumptions as Theorem 3.3, if we further assume (Xh, Xh ) satisfies the LBB condition (with parameter β ), then we obtain convergence of a modified pressure: k(qh wh ) ph k ν β CR kf k. Proof. We can rewrite (3.6) as ν( e, vh ) = ((qh wh ) ph, vh ) vh Xh. Dividing both sides by, with 6=, we have vh Xh ((qh wh ) ph, vh ) ν( e, vh ) =. Taking the infimum over vh Xh and applying the assumed inf-sup condition, we obtain β k(qh wh ) ph k νk ek. Finally, applying Theorem 3.3 yields kf k. k(qh wh ) ph k ν β CR 3. Numerical Results We now test the theory above on both barycenter and uniform meshes on Ω = (, ) with h = 6 using (P, P ) elements. On uniform meshes, LBB on the spaces is not known to hold. In this case, we find convergence of velocity, but no pressure convergence. On barycenter meshes with P elements, however, it is known that (Xh, Xh ) satisfies the LBB condition, so in this case we expect convergence of both velocity and pressure. We chose the true solution to be utrue = cos(y), sin(x) ptrue = sin(x + y). We enforce Dirichlet velocity boundary conditions to be the true solution at the boundary, set ν =., and calculate the forcing using f = ν4utrue + ptrue. Solutions were computed using Algorithm 3. with varying and Algorithm 3. (using α = ). Table shows results using a barycenter mesh, and here we observe convergence of both velocity and the modified pressure. Results using a uniform mesh are shown in Table. Here we observe O( ) convergence of the velocity but no convergence of the modified pressure. 37

6 3 4 k (wh uh )k.354e-.844e E E E E E k(qh wh ) ph k.676e E E-6 7.5E-7 7.5E E-9.85E Table : Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a barycenter mesh. 3 4 k (wh uh )k.9e-3.59e-4.845e E-5.885E-5.E-6.83E k(qh wh ) ph k.458e-3.457e-3.457e-3.457e-3.457e-3.457e-3.457e Table : Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a uniform mesh. 4 A connection for the Oseen equations We next consider the Oseen equations, which are given by: σu + U u ν4u + p = f, u =, u Ω =, where u represents velocity, p represents pressure, f is an external forcing, σ can be considered as either a friction coefficient or inversely proportional to a time step if f is appropriately modified (e.g. for backward n Euler σ = 4t and f = f + 4t u, with un being the solution at the previous time step), ν is the kinematic viscosity, and U H (Ω) is given. We will compare the grad-div stabilized Taylor-Hood scheme and the pointwise divergence-free scheme to determine a connection between the two methods. These two methods are given below: Algorithm 4.. Given, find (wh, qh ) (Xh, Qh ) satisfying σ(wh, vh ) + ν( wh, vh ) + b (U, wh, vh ) (qh, vh ) + ( wh, vh ) = (f, vh ) vh Xh ( wh, rh ) = rh Qh (4.a) (4.b) Algorithm 4.. Find uh Vh satisfying σ(uh, vh ) + ν( uh, vh ) + b (U, uh, vh ) = (f, vh ) vh Vh. (4.) We use the iterated penalty method to solve (4.): Step : Find uh Xh satisfying σ(uh, vh ) + ν( uh, vh ) + b (U, uh, vh ) + (α( uh ), vh ) = (f, vh ) vh Xh (4.3) 37

7 Step k: Find ukh Xh satisfying k σ(ukh, vh ) + ν( ukh, vh ) + b (U, ukh, vh ) + (α( ukh ), vh ) = σ(uk h, vh ) + ν( uh, vh ) + b (U, uk h, vh ) vh Xh (4.4) 3 [8], then there exists If the iterated penalty method converges, which is known to occur when α > dνcr k an N such that k uh k tol f or k N. If tol is sufficiently small, then it is reasonable to assume uh = un h. PN After convergence, we recover pressure via ph := i= (α( uih )) to satisfy σ(uh, vh ) + ν( uh, vh ) + b (U, uh, vh ) (ph, vh ) = (f, vh ) vh Xh. (4.5) 3 ), and so Lemma 4.. Suppose α is sufficiently large so that Algorithm 4. converges (e.g. if α > dνcr k C C k that α <, where C is a data-dependent constant which satisfies k uh k α k uh k (existence of such a constant is shown in [8]). Then, kph k ν kf k. Proof. The system is known to be well-posed and can be proven (by the Lax-Milgram Theorem). Choosing v = uh in (4.3) and using the definition of k k and the Cauchy-Schwarz and Young s inequalities gives σkuh k + νk uh k + αk uh k = (f, uh ) ν ν kf k + k uh k. Reducing implies that ν ν k uh k + αk uh k kf k. (4.6) PN Now, we consider our earlier definition of ph := k= (α( ukh )), where N is the number of steps until convergence (N is guaranteed to be finite since α is chosen to be sufficiently large). From [8], we know that C such that C k ukh k k uk h k. α σkuh k + For ease of notation, let r := C α. This yields kph k α N X k ukh k = αk uh k k= rn + r. Appyling (4.6) gives ν kph k kf k α α C. From the assumptions on C and α, we can further reduce to obtain kph k ν kf k. Theorem 4.3. Let (wh, q ) be the solution to Algorithm 4. for a fixed, let uh be the solution to Algorithm 4.. Then, ν / k (wh uh )k ( + M CU )CR kf k. Thus, as, the sequence of grad-div stabilized Taylor-Hood velocity solutions, {wh }, converges to uh. 373

8 Proof. Denote e := wh uh, and let ph be the recovered pressure from Algorithm 4.. Subtracting (4.5) from (4.a), we have σ(e, vh ) + ν( e, vh ) + b (U, e, vh ) + ( e, vh ) (qh ph, vh ) = vh Xh. (4.7) Choose vh = e in (4.7) to obtain σkek + νk ek + k ek = (qh ph, e). Now, orthogonally decompose e Vh as e = e + er where e Vh, er Rh and apply Cauchy-Schwarz and Young inequalities to the pressure term (which reduces because e Vh ). This yields σker k + νk er k + k er k kph k + k er k, which implies k er k kph k. Applying Lemma 4. and taking to the other side of the equation, we obtain k er k ν kf k. Taking square roots and using (.3), we have k er k ν kf k CR. (4.8) The next step is to bound k e k. Choose vh = e in (4.7). As, b (U, e, e ) = b (U, er, e ), this gives σke k + νk e k + b (U, er, e ) =, and after taking the nonlinear term to the other side, we obtain σke k + νk e k kb (U, er, e )k. Using (.) and the assumptions on U, we have σke k + νk e k M CU k er kk e k, which further implies νk e k M CU k er k. (4.9) Thus, using (4.8) and (4.9), we can conclude that k ek = k e k + k er k ( + M CU )k er k ν / ( + M CU )CR kf k. Corollary 4.. Under the same assumptions as Theorem 4.3, if we further assume (Xh, Xh ) satisfies the LBB condition (with parameter β ), then we obtain convergence of a modified pressure: k(qh wh ) ph k ν β CR kf k ( + M CU )(σcp F + ν + M CU ). 374

9 Proof. Rewriting (4.7) yields ((qh wh ) ph, vh ) = σ(e, vh ) + ν( e, vh ) + b (U, e, vh ) Taking the pressure term to the other side and dividing both sides by, with 6=, we have vh Xh, ((qh wh ) ph, vh ) σ(e, vh ) + ν( e, vh ) + b (U, e, vh ) =. Applying (??), we have ((qh wh ) ph, vh ) σ(e, vh ) + ν( e, vh ) M CU k ek +. which simplifies to ((qh wh ) ph, vh ) σ(e, vh ) + ν( e, vh ) + M CU k ek. Taking the infimum over vh Xh and applying the assumed inf-sup condition, we obtain β k(qh wh ) ph k (σcp F kek + νk ek + M CU k ek). Using (.), we have k(qh wh ) ph k β (σcp F + ν + M CU ) k ek. After applying Theorem 4.3, we have k(qh 4. wh ) ph k ν β CR kf k ( + M CU )(σcp F + ν + M CU ). Numerical Results We now test the theory above on both uniform and barycenter meshes of Ω = (, ) with h = 6 using (P, P ) elements. On barycenter meshes with P elements, it is known that (Xh, Xh ) is LBB stable, so in this case we expect convergence of both velocity and pressure. On uniform meshes, however, LBB on the spaces is not known to hold. In this case, we find convergence of velocity, but no pressure convergence. We again chose the true solution to be cos(y) utrue =, sin(x) ptrue = sin(x + y). We enforce Dirichlet velocity boundary conditions to be the true solution at the boundary, σ =., ν =., U = utrue, and calculated the forcing using the true solution and f = σutrue + U utrue ν4utrue + ptrue. Solutions were computed using Algorithm 4. with varying and Algorithm 4. (using α = ). Results using a barycenter mesh are shown in Table 3. Here we observe O( ) convergence of both velocity and the modified pressure. Table 4 shows results using a uniform mesh, and we observe convergence of velocity but no convergence of the modified pressure. 375

10 3 4 k (wh uh )k.36e+.88e E E E E E k(qh wh ) ph k.73e- 5.7E E E-5 8.9E E E Table 3: Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a barycenter mesh. 3 4 k (wh uh )k.8e-.5e-.84e E-3.88E-3.4E-4.54E k(qh wh ) ph k.458e-.458e-.457e-.457e-.457e-.457e-.457e Table 4: Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a uniform mesh. 5 A connection for the Boussinesq equations We now consider the Boussinesq equations for the flow of heated silicon oil, which are given by: 4u + p = Ra, T u =, 4T + u T = g, u Ω =, where u represents velocity, p represents pressure, T represents temperature, f is an external forcing, g encompasses the Dirichlet boundary conditions on the temperature, and Ra is the Rayleigh number. We will compare grad-div stabilized Taylor-Hood and the pointwise divergence-free solution to determine a connection between the two methods. The two methods are given below: Algorithm 5.. Given, find (wh, qh, Λh ) (Xh, Qh, Sh ) satisfying ( wh, vh ) (qh, vh ) + ( wh, vh ) = Ra, vh vh Xh, Λh ( Λh, sh ) ( wh, rh ) = + b (wh, Λh, sh ) = (g, sh ) rh Qh, sh Sh. (5.a) (5.b) (5.c) Algorithm 5.. Find (uh, Th ) (Vh, Sh ) satisfying ( uh, vh ) = Ra ( Th, sh ) + b (uh, Th, sh ) = (g, sh ), vh vh Vh, Th (5.a) sh Sh. (5.b) 376

11 We use an iterated-penalty-quasi-newton method to solve (5.a) - (5.b): Step : Find (uk, Tk ) where ( uk, vh ) + α( uk, vh ) Ra, vh = vh Xh, Tk ( Tk, sh ) + b (uk, Tk, sh ) + b (uk, Tk, sh ) b (uk, Tk, sh ) = sh Sh, (5.3a) (5.3b) where uk and Tk are the initial guesses (typically ). Step n: Find (unk, Tkn ) where ( unk, vh ) + α( unk, vh ) Ra, vh = ( un, vh ) + α( un, vh ) vh Xh, (5.4a) k k Tkn ( Tkn, sh ) + b (unk, Tk, sh ) + b (uk, Tkn, sh ) b (uk, Tk, sh ) = sh Sh, (5.4b) where uk and Tk are the solutions to the previous step. Assuming that the iterated penalty method converges, then there exists an N such that k ukh k tol f or k N. If tol is sufficiently small, it is reasonable to assume uh = un h. In our computation, we choose tol = 9. We end the outer iteration when kuk uk k < tol. Pk After convergence, we recover pressure via ph := i= (α( uih )) to satisfy ( uh, vh ) (ph, vh ) = Ra, vh vh Xh, (5.5a) Th ( Th, sh ) + b (uh, Th, sh ) = (g, sh ) sh Sh. (5.5b) Remark 5.3. Throughout this section, we use the small data condition: Ra CP F Cg M <. A small data condition is also used in []. Lemma 5.. If the data satisfies the small data condition Ra CP F Cg M <, then solutions to Algorithm 5. exist uniquely. Proof. Existence of solutions can be proven using the exact same techniques as in []. For uniqueness, assume two solutions, say (w, q, Λ ) and (w, q, Λ ). Note k Λ k = (g, Λ ). Denote ew := w w, eq := q q, and eλ := Λ Λ. We subtract to obtain ( ew, v) (eq, v) + ( ew, v) = Ra,v v X, eλ ( ew, r) = ( eλ, s) + b (w, eλ, s) + b (ew, Λ, s) = (g, s) r Q, s S. Choose v = ew, r = eq, and s = eλ. Then,v eλ (5.6a) Ra k eλ kkew k (5.6b) k ew k + k ew k = Ra k ew k + CP4 F Ra k eλ k (5.6c) 377

12 and k eλ k + b (ew, Λ, eλ ) =. (5.7) k eλ k M k ew kk Λ k M Cg k ew k (5.8a) (5.8b) Further reducing (5.7) yields Combining (5.6c) and (5.8b), we obtain k ew k CP4 F Ra M Cg k ew k. Thus, uniqueness holds if CP4 F Ra M Cg <, i.e. if the small data condition above holds. We will continue with an a priori bound on Λh. Lemma 5.. Suppose the data satisfies the small data condition Ra CP F Cg M < so that Algorithm 5. has a unique solution (wh, qh, Λh ). Then k Λh k Cg, where Cg depends only on g. Proof. Let sh = Λh in (5.c). Then we have k Λh k = (g, Λh ). Applying (.) and the Cauchy-Schwarz inequality, we obtain k Λh k CP F kgk k Λh k. Simplifying, we are left with k Λh k CP F kgk = Cg. Theorem 5.4. Suppose the data is sufficient to allow for Algorithm 5. and Algorithm 5. to have unique solutions, (wh, qh, Λh ) and (uh, Th ) respectively. Let ph be the pressure recovered from Algorithm 5.. Further, suppose kph k Cp < and the data satisfies Ra CP F Cg M <. Then, k (wh uh )k CR Cp, and k (Λh Th )k CR Cp M Cg. Thus, as, the sequence of grad-div stabilized Taylor-Hood velocity solutions {wh } converges to uh and the sequence of temperature solutions {Λh } converges to Th. Proof. We subtract (5.5a) from (5.a) to obtain (5.9a) and (5.5b) from (5.c) to obtain (5.9b). Denoting eu := wh uh and et := Λh Th, we find u u ( e, vh ) (qh ph, vh ) + ( e, vh ) = Ra, vh vh Xh, (5.9a) et ( et, sh ) + b (uh, et, s) + b (eu, Λh, s) = Choose sh = et in (5.9b). Then sh Sh. (5.9b) k et k + b (eu, Λh, et ) =. 378

13 Taking the b term to the other side and applying (.), we obtain k et k M k eu kk Λh kk et k. After simplifying and using Lemma 5., we are left with k et k M Cg k eu k. (5.) Next, we choose vh = eu in (5.9a), which yields u u u k e k + k e k = (qh ph, e ) + Ra, eu. et Using Cauchy-Schwarz and Young s inequalities, (.), (5.b), and Lemma 5., we obtain k eu k + k eu k kph kk eu k + Ra ket kkeu k kph k + kph k + = k eu k + Ra CP F k et kk eu k k eu k + Ra CP F Cg M k eu k, which further yields k eu k ( Ra CP F Cg M ) + k eu k kph k. Thus, we obtain k eu k kph k, and using (.3) and the assumption on ph, k eu k CR Cp. Furthermore, from (5.), k et k CR Cp M Cg. Corollary 5.. Under the same assumptions as Theorem 5.4, if we further assume (Xh, Xh ) satisfies the LBB condition (with parameter β ), then we obtain convergence of a modified pressure: k(qh wh ) ph k CR Cp ( + Ra CP F Cg M )β. Proof. Equation (5.9a) can be rewritten as ( eu, vh ) (qh wh ) ph, vh ) = Ra, vh vh Xh, et and after rearranging and dividing both sides by, with 6=, we have vh Xh, ν( e, vh ) Ra ( et, vh ) ((qh wh ) ph, vh ) = u k e k + Ra ket kkvh k k eu k + Ra CP F k et k. 379

14 Using (5.) and then Theorem 5.4, we obtain ((qh wh ) ph, vh ) k eu k( + Ra CP F Cg M ) CR Cp ( + Ra CP F Cg M ). Taking the infimum over vh Xh and applying the assumed inf-sup condition, we obtain β k(qh wh ) ph k CR Cp ( + Ra CP F Cg M ). This finishes the proof. 5. Numerical Results The test problem we consider models a heated cavity of silicon oil using Ω = (, ) and Ra = 5. We enforce u Ω =, and for temperature we set T = on the right side of the box, T = on the left side of the box, and weakly enforce the top and bottom be insulated via T n =. Since, we explicitly enforce T = on the left side, take g =. We test the theory on both barycenter and uniform meshes with h = 6, using (P, P, P ) elements. On barycenter meshes with P elements, it is known that (Xh, Xh ) satisfies the LBB condition, so in this case we expect convergence of velocity, temperature, and modified pressure. On uniform meshes, however, LBB on these spaces is not known to hold. In this case, we find convergence of velocity and temperature, but no modified pressure convergence. Solutions were computed using Algorithm 5. with varying and Algorithm 5. (using α = ). Results using a barycenter mesh are shown in Table 5. Here we observe O( ) convergence of the velocity, pressure, and temperature. In Figure, we see the sequence of grad-div stabilized Taylor-Hood solutions converging to the pointwise divergence-free solution. Table 6 shows results using a uniform mesh, and here we observe convergence of velocity and temperature, but no pressure convergence. In Figure, we see the sequence of grad-div stabilized Taylor-Hood velocity and temperature solutions converging to the pointwise divergence-free solution, but the pressure solution does not converge. 38

15 3 4 k (wh uh )k 4.766E+ 4.68E+ 4.66E+.38E E- 5.38E E k(qh wh ) ph k.539e+.54e+.4e E+.73E+.9E-.935E k (Λh Th )k.44e-3.44e-3.83e E-4.3E-4.443E-5.457E Table 5: Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a barycenter mesh. Velocity ( = ) Velocity ( = ) Velocity ( = 4 ) Velocity (div-free) Pressure ( = ) Pressure ( = ) Pressure ( = 4 ) Pressure (div-free) Temperature ( = ) Temperature ( = ) Temperature ( = 4 ) Temperature (div-free) Figure : Visuals of three grad-div stabilized Taylor-Hood velocity, pressure, and temperature solutions followed by the pointwise divergence-free solutions, using a barycenter mesh. 38

16 k (wh uh )k.4e+.3e+.6e+.46e+.759e+ 7.59E+.3E+.465E-.485E-.487E k(qh wh ) ph k 5.4E+ 5.4E+ 5.3E E+ 4.96E+.94E+ 9.48E+ 8.64E+ 8.65E+ 8.37E k (Λh Th )k.46e-.44e-.43e-.39e-.8e- 6.95E-3.7E-3.4E-4.5E-5.5E Table 6: Differences between grad-div stabilized Taylor-Hood solutions and pointwise divergence-free solutions on a uniform mesh. Velocity ( = ) Pressure ( = ) Velocity ( = ) Pressure ( = ) Temperature ( = ) Velocity ( = 4 ) Pressure ( = 4 ) Temperature ( = 4 ) Velocity (div-free) Pressure (div-free) Temperature ( = ) Temperature (div-free) Figure : Visuals of three grad-div stabilized Taylor-Hood velocity, pressure, and temperature solutions followed by the pointwise divergence-free solutions, using a uniform mesh. 38

17 6 Conclusions We have proven that for Stokes, Oseen, and Boussinesq systems, grad-div stabilized Taylor-Hood velocity solutions (and temperature solutions for Boussinesq) converge to the pointwise divergence-free solution at a rate of as. Furthermore, if (Xh, Xh ) satisfies the LBB condition, where Xh is the finite element velocity space, then a modified pressure solution will also converge at the same rate. We verified these results by testing on a barycenter mesh, where this LBB condition is satified, and on a uniform mesh, where it is not known to be satisfied. The numerical results were consistent with the theory in all tests. Thus, our work generalizes the work of [7, ], which proved similar results, but only for special meshes having specific macroelement structure. References [] M. Case, V. Ervin, A. Linke, and L. Rebholz. A connection between Scott-Vogelius elements and grad-div stabilization. SIAM Journal on Numerical Analysis, 49(4):46 48,. [] A. Cibik and S. Kaya. A projection-based stabilized finite element method for steady state natural convection problem. Journal of Mathematical Analysis and Applications, 38: ,. [3] K. Galvin, A. Linke, L. Rebholz, and N. Wilson. Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Computer Methods in Applied Mechanics and Engineering, 37:66 76,. [4] V. Girault and P.-A. Raviart. Finite element methods for Navier Stokes equations: Theory and algorithms. Springer-Verlag, 986. [5] William Layton. Introduction to the Numerical Analysis of Incompressible Viscous Flows. SIAM, Philidephia, USA, 8. [6] A. Linke, M. Neilan, L. Rebholz, and N. Wilson. Improving efficiency of coupled schemes for NavierStokes equations by a connection to grad-div stabilized projection methods. Submitted, 6. [7] A. Linke, L. Rebholz, and N. Wilson. On the convergence rate of grad-div stabilized Taylor-Hood to ScottVogelius solutions for incompressible flow problems. Journal of Mathematical Analysis and Applications, 38:6 66,. [8] L. Rebholz and M. Xiao. On reducing the splitting error in Yosida methods for the Navier-Stokes equations with grad-div stabilization. Computer Methods in Applied Mechanics and Engineering, 94:59 77,