2009 Elsevier Science. Reprinted with permission from Elsevier.

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1 P. Hansbo and M. Juntunen Weakly imposed Diriclet boundary conditions for te Brinkman model of porous media flow. Applied Numerical Matematics, volume 59, number 6, pages doi: /j.apnum lsevier Science Reprinted wit permission from lsevier.

2 Applied Numerical Matematics ) Weakly imposed Diriclet boundary conditions for te Brinkman model of porous media flow P. Hansbo a,, M. Juntunen b a Department of Matematical Sciences, Calmers University of Tecnology and Göteborg University, S-4196 Göteborg, Sweden b Department of ngineering Pysics and Matematics, Institute of Matematics, Helsinki University of Tecnology, P.O. Box 1100, 0015 TKK, Finland Available online 5 August 008 Abstract We use low order approximations, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Brinkman s equation of porous media flow, applying an edge stabilization term to avoid locking. In order to andle te limiting case of Darcy flow, wen only te velocity component normal to te boundary can be prescribed, we impose te boundary conditions weakly using Nitsce s metod [J. Nitsce, Über ein Variationsprinzip zur Lösung von Diriclet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abandlungen aus dem Matematiscen Seminar der Universität Hamburg ) 9 15]. We sow tat tis leads to a stable metod for all coices of material parameters. Finally we present some numerical examples verifying te teoretical predictions and sowing te effect of te weak imposition of boundary conditions. 008 IMACS. Publised by lsevier B.V. All rigts reserved. Keywords: Brinkman model; Stokes Darcy model; Stabilized metods; Finite element; Interior penalty metod; Nitsce s metod 1. Introduction Te Brinkman equations model creeping flow in porous media and can be seen as a mixture of Darcy s equations and Stokes equations. Te beavior of solutions to te Brinkman equations will be controlled by te ratio of permeability in Darcy) to viscosity Stokes), and it is desirable from a numerical point of view to develop metods tat can andle te wole range of possible ratios, from te pure inviscid Darcy problem to te Stokes problem wit full infinite) permeability. In doing so, we are led to formulate te Darcy equations in mixed form using velocities and pressure as variables, as is done in te Stokes case. One problem tat ten arises is te fact tat a good metod for te Stokes problem may perform badly, or not even work, in te case of a mixed form of te Darcy problem, see Mardal, Tai, and Winter [9]. In [6], tis problem was overcome by using a stabilized metod tat was sown to be convergent for bot Darcy and Stokes; te same metod will be used in te present study. Anoter inconvenient fact, from te point of view of numerical implementation, is tat te Darcy equations do not admit te same boundary conditions as te Stokes equations: in te Darcy case only te velocity normal to te boundary can be prescribed, wereas no-slip boundary conditions are usually employed for Stokes. In tis paper we suggest a remedy to tis last inconvenience: * Corresponding autor. -mail addresses: peter.ansbo@me.calmers.se P. Hansbo), mika.juntunen@tkk.fi M. Juntunen) /$ IMACS. Publised by lsevier B.V. All rigts reserved. doi: /j.apnum

3 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) te use of weakly prescribed Diriclet boundary conditions for te velocities using Nitsce s metod [10]. Te same approac was also used for Oseen s problem by Burman, Fernández, and Hansbo [5], but wit a different focus. Weakly imposed Diriclet boundary conditions, first proposed for flow problems by Freund and Stenberg [7], ave been sown to be advantageous for convection diffusion problems wit outflow layers, already in [7] and later in te work of Burman [3], in tat it will lead to discontinuous jumps in te solution at te boundary rater tan forcing a continuous numerical solution to mimic discontinuities. It as also been promoted by Bazilevs and Huges [1] as an alternative to wall function models in turbulent cannel flow, allowing for limited slip at te boundary. In tese cases, te balance between a first order term convection) and a second order term diffusion/viscosity) is te factor tat favors weak boundary conditions; in our case it is te balance between a zero order term in Darcy) and a second order term in Stokes). Te idea is tus more general and its full potential awaits exploitation. We will consider te following Brinkman model of porous flow σ u μ u) + p = f in Ω, u = 0 inω, 1) were Ω is an open, bounded subset of R d, u denotes te average fluid velocity in te porous medium, σ te viscosity divided by te permeability, μ te effective viscosity, p te pressure, and f is a given forcing term. We assume tat Ω as polygonal boundary Ω and tat te boundary is divided into two non-overlapping sets Ω = Γ D Γ N.Te respective boundary conditions are u = u 0 on Γ D, μ n u pn = g on Γ N, ) were g = g n n + μg t, wit g n a scalar and g t a vector in te plane perpendicular to n. In oter words we prescribe only te normal component of te Neumann condition in Darcy limit μ = 0. Furtermore we assume tat te measure of Γ D is not zero i.e. we always ave some Diriclet boundary. Te side condition u = 0 requires tat some care is taken in te coice of approximating spaces in order to avoid over-constraining te problem. Here we sall use a stabilized sceme proposed for Stokes by Huges and Franca [8], and for Darcy by Burman and Hansbo [6] cf. also Burman [4] for a discussion of related metods). In tis paper we apply tis mixed stabilized metod to te Brinkman equations wit weakly imposed boundary conditions and prove optimal a priori estimates in te energy norm. We also give an a posteriori error estimate and adaptive algoritm for energy norm control of te computational error. Finally, we give some numerical examples sowing te performance of te metod and te adaptive algoritm.. Finite element formulation In order to formulate our finite element metod we first introduce te weak formulation of problem 1). We introduce te Hilbert spaces and W u0 = { v [ H 1 Ω) ] d s.t. v ΓD = u 0 }, L 0 {q = L Ω) s.t. Ω } q dx = 0. We denote te product space W u0 L 0 by W u 0 and define te following norm on W u0, u,p) W = σ u 0,Ω + μ u 1,Ω + u 0,Ω + p 0,Ω. Consider te bilinear form B [ u,p),v,q) ] = μ u, v) 0,Ω + σ u, v) 0,Ω p, v) 0,Ω q, u) 0,Ω. 3) Te weak formulation of 1) now takes te form, find u,p) W u0 suc tat B [ u,p),v,q) ] = f, v) 0,Ω + g, v) 0,ΓN v,q) W 0. 4)

4 176 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Let T be a conforming, sape regular triangulation of Ω. Wit K we denote an element of te triangulation and wit an edge/face of te triangulation. By K and we denote te size of an element or edge/face, respectively, and by we denote te size of te largest element in T. We introduce te two classical finite element spaces of piecewise linears and piecewise constants V = { v s.t. v K P 1 K), v C 0 Ω) }, { } Q = q s.t. q K P 0 K), q dx = 0. Ω Te velocity field will be sougt in W =[V ] d and te pressure field in Q. In analogy wit te notation above we use te notation W := W Q. We introduce te following bilinear and linear forms on wic we will base our finite element metod: B [ u,p),v,q) ] = μ u, v) 0,Ω + σ u, v) 0,Ω p, v) 0,Ω q, u) 0,Ω J p, q) and μ n u, v) 0,ΓD μu, n v) 0,ΓD + μγ μ 1 u, v ) 0,Γ D + p, v n) 0,ΓD + u n,q) 0,ΓD + γ σ 1 u n, v n ) 0,Γ D 5) L [ v,q) ] := f, v) 0,Ω + g, v) 0,ΓN u 0, n v) 0,ΓD + u 0,μγ μ 1 v ) 0,Γ D + u 0 n,q) 0,ΓD + u 0 n,γ σ 1 v n ), 6) 0,Γ D were J p, q) = δ [p], [q] )0,, 7) T )\ Ω wit [ ] denoting te jump over te element edge taken on te interior edges only). Above, and in wat follows, in te inner product = x) i.e. correspond to te element under integration, not to te global maximum. We propose te following finite element formulation: find u,p ) W suc tat B [ u,p ), v,q )] = L [ v,q )] v,q ) W. 8) Tis finite element formulation is simply te standard Galerkin formulation wit Nitsce boundary conditions and te penalizing term J p, q) added. In te following we will assume tat te pressure is in H 1 Ω): ten te penalizing term is consistent and we ave te following Lemma.1. If u,p)is a weak solution to 1) wit u,p) W H 1 Ω) L 0 ten B [ u u,p p ), v,q )] = 0 v,q ) W. 9) Proof. Immediate by noting tat if p H 1 Ω) ten te trace of p is well defined and ence J p, q ) = 0 for all q Q. 3. Stability Since it is a well known fact tat te above coice of finite element spaces results in an ill posed discrete problem if used in a standard Galerkin metod, te crucial point is to sow tat our stabilization operator J p, q) introduces sufficient coupling between te degrees of freedom in te pressure field suc tat an inf sup condition is satisfied. Tis was done for Darcy in [6], using te standard way of andling Diriclet boundary conditions. Here, we extend te analysis of [6] to te Brinkman model wit weakly imposed Diriclet boundary conditions. In te analysis, we will use te following norms: u,p) := u,p) W + J p, p) + μ u 1/,,Γ D + u n 1/,,Γ D, 10) u,p) := u,p) + μ u 1/,,Γ D, 11)

5 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) were v 1/,,Γ D := 1 v,v ) 0,Γ D and v 1/,,Γ D := v,v) 0,ΓD. Note tat te norms contain te L -norm of u; tis term is superfluous for Stokes since we already control te H 1 -norm of te velocities, but of vital importance for Darcy. In fact, te control of te divergence is wat allows us to prove optimal error estimates in te energy norm for sufficiently regular solutions. In wat follows we will use te following well known estimates: n v, w) 0, Ω n v 1/,, Ω w 1/,, Ω v, w V, 1) n v 1/,, Ω C I v 0,Ω v V, 13) v 1/,, Ω C II v 0,Ω v V. 14) Note tat due to estimate 13) te norms and are equivalent on te finite element subspace. In te following, we will let C denote a generic positive constant wose value may cange from instance to instance. Te main result of tis section is te following teorem, assuring te well-posedness of our discretization. Teorem 3.1. Assume γ μ >C I and γ σ > 0. Ten te finite element formulation 8) satisfies te following inf sup condition α u,p ) sup B [u,p ), v,q )] v,q ) W v,q, u,p ) W. ) Proof. Te idea of te proof is to acquire control of te different terms of te energy norm wit different coices of test functions and finally combine te coices using te linearity of te bilinear form. Step 1. Taking first v,q ) = u, p ) we obtain B [ u,p ), u, p )] = μ u 0,Ω + σ u 0,Ω + J p,p ) μ n u, u ) 0,Γ D + γ μ μ u 1/,,Γ D + γ σ u n 1/,,Γ D. Using Young s inequality and estimates 1) and 13) we get B [ u,p ), u, p )] 1 C ) I μ u ɛ 0,Ω + σ u 0,Ω + J p,p ) + γ μ ɛ)μ u 1/,,Γ D + γ σ u n 1/,,Γ D, were ɛ>0 is a parameter from Young s inequality. Our assumption is tat γ μ >C I. Terefore we can coose C I <ɛ<γ μ and we ave B [ u,p ), u, p )] C 1 μ u 0,Ω + σ u 0,Ω + J p,p ) + C μ u 1/,,Γ D + γ σ u n 1/,,Γ D. 15) Step. We are still missing te control of te pressure and te divergence of te velocity. To gain control over te pressure we note tat as a consequence of te surjectivity of te divergence operator tere exists a function v p [H 1 0 Ω)]d suc tat v p = p and v p 1,Ω C p 0,Ω. 16) Let π v p denote te Scott Zang interpolant cf. []) of v p onto [V 0 ]d, were V0 := { v s.t. v K P 1 K), v C 0 Ω), v = 0on Ω }. By te stability of te interpolant we ave π v 1,Ω p c p 0,Ω. 17)

6 178 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) We now coose te test function to be v,q ) = π v p, 0). Adding 0 = p 0,Ω p, v p )0,Ω and recalling tat π v p vanises on te boundary we obtain B [ u,p ), π v p, 0 )] = μ u, π ) v p 0,Ω σ u,π v p )0,Ω + p 0,Ω + p, π )) v p v p 0,Ω + μ u, n π v p. )0,Γ D Integrating te fourt term by parts on eac element K we get B [ u,p ), π v p, 0 )] = μ u, π ) v p 0,Ω σ u,π v p )0,Ω + p 0,Ω + K T 1 [ p ], π v p v p ) n )0, K + μ u, n π v p ) 0,Γ D. Splitting te inner products using Scwarz inequality, followed by Young s inequality, we ave B [ u,p ), π v p, 0 )] 1 ɛ μ u 0,Ω ɛ μ π v p 0,Ω 1 ɛ σ u 0,Ω ɛ σ π v p 0,Ω + p 0,Ω 1 ɛ μ u ɛ 1/,,Γ D μ n π v p 1/,,Γ D 1 δɛ J p,p ) ɛ T )\ Ω π v p v p ) n 0,. Using estimate 13) and te stability of te interpolate 17) we get B [ u,p ), π v p, 0 )] 1 ɛ μ u 0,Ω 1 ɛ σ u 1 0,Ω + ɛ ) μ c + σ c + μ cc I ) p 1 ɛ μ u 1/,,Γ D 1 δɛ J p,p ) ɛ T )\ Ω 0,Ω π v p v p ) n 0,. To conclude we need te following trace inequality, cf. [11], w n 0, K C 1 w 0,K + ) w 1,K w [ H 1 K) ] d, 18) from wic we deduce, using 16), π ) v p v p n 0, C v p 1,Ω C p 0,Ω. T )\ Ω K T Using te inequality above we obtain B [ u,p ), π v p, 0 )] 1 ɛ μ u 0,Ω 1 ɛ σ u 1 0,Ω + ɛ ) μ c + σ c + μ cc I + C) p 1 ɛ μ u 1 1/,,Γ D δɛ J p,p ). Setting 0 <ɛ</μ c + σ c + μ cc I + C) we finally ave B [ u,p ), π v p, 0 )] C 3 μ u 0,Ω C 3σ u 0,Ω + C 4 p 0,Ω C 3 μ u 1/,,Γ D C 3 δ J p,p ). 19) Step 3. Te divergence of te velocity is already contained in te H 1 norm of te velocity if μ>0 but we want to ave te control of te divergence even if μ = 0. Te control of u 0,Ω is obtained coosing v,q ) = 0, u ), wic leads to 0,Ω

7 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) B [ u,p ), 0, u )] = u 0,Ω + J p, u ) u, u n ) 0,Γ D u 0,Ω ξ δ u 1/,, 1 ξ J p,p ) T )\ Ω ξ u 1 u n. 1/,,Γ D ξ 1/,,Γ D Using estimate 14) we get B [ u,p ), 0, u )] 1 ξ ) C II1 + δ) u 0,Ω 1 ξ J p,p ) 1 u n. 0) ξ 1/,,Γ D Step 4. Now we ave control over all te terms in te energy norm separately. Finally we take v,q ) = βu π v p,βp u ). Combining te results 15), 19) and 0) we get B [ u,p ), v,q )] βc 1 C 3 )μ u 0,Ω + β C 3)σ u 0,Ω + C 4p 0,Ω + 1 ξ ) C II1 + δ) u 0,Ω + β 1 ξ C ) 3 J p,p ) δ + βc C 3 )μ u + βγ 1/,,Γ D 0 1 ) u n. ξ 1/,,Γ D Te fourt term on te rigt-and side is positive if te parameter from Young s inequality is ξ</c II 1 + δ)). Wit tis te rest of te terms are positive if β>c 3 /C 1, β>c 3, β>c II 1 + δ) + C 3 /δ, β>c 3 /C, and β>c II 1 + δ)/γ 0. Ten we ave tat B [ u,p ), v,q )] u,p ) and te claim follows since tere exists C>0 suc tat u,p ) C v,q ). Note tat te Nitsce stability parameter γ σ forcing te Darcy problems boundary conditions as no lower bound. Tis olds even on Darcy limit μ = 0. In te rest of te paper we assume tat te stability requirement is satisfied, i.e. we make te following assumption. Assumption 3.. Te real parameter γ μ satisfies γ μ >C I. 4. rror analysis 4.1. A priori estimates First of all, we note tat applying te trace inequality 18) we easily derive te following approximation property for couples of functions u,p) [H Ω)] d H 1 Ω), u π u,p π p ) C u,ω + p 1,Ω ), 1) were π u,π p) W denotes te interpolates. Witout proof we also state te continuity of te bilinear form. Lemma 4.1. For all u,p),v,q) W it olds B [ u,p),v,q) ] C u,p) v,q). ) Te main result in tis section is te following lemma. Lemma 4.. Assume tat te solution u,p) to te problem 1) resides in [H Ω)] d H 1 Ω) L 0 Ω); ten te finite element solution 8) satisfies te error estimate u u,p p ) C u,ω + p 1,Ω ).

8 180 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Proof. In view of te approximation property 1) we only need to sow te inequality for u π u,p π p). By te stability, see Teorem 3.1, tere exist a pair v,q ) W suc tat v,q ) = 1 and u π u,p π p ) α 1 B [ u π u,p π p ), v,q )]. Using te Galerkin ortogonality, see Lemma.1, we obtain u π u,p π p ) α 1 B [ u π u,p π p ), v,q )]. Furtermore, using te continuity of te bilinear form, q. ), and recalling tat te energy norms are equivalent in te finite element subspace, we ave u π u,p π p ) α 1 C u π u,p π p ). Now te claim follows by te approximation property 1). 4.. A posteriori estimate In tis section we propose and prove te a posteriori estimate in te energy norm. In wat follows we will need two meses. Te original mes T and mes T / derived from te original mes by splitting te elements. By pair u,p ) we denote te solution on te mes T and by u /,p / ) te solution on te mes T /. Te proof is based on te following saturation assumption. Assumption 4.3 Saturation assumption). Tere exists 0 <β<1 suc tat u u /,p p /) / β u u,p p ). 3) We note tat tis assumption is asymptotic in nature and cannot be expected to old is tere are unresolved features in te flow. However, tis does not mean tat unresolved features will not be detected being a residual-based estimate it will still signal large errors were te residual is large), only tat tere will be a lack of sarpness of te estimate outside te asymptotic range. Te residual based elementwise estimator is defined as [ K u,p )] := K μ + σ μ u σ u p + f 0,K + u 0,K K [ + μ n u ] 0, K\ Ω + [ p ] 0, K\ Ω + μ g t n u + n u n ) n + μ 1 0, K Γ N u 0 u 0, K Γ D + g n + p μ n u n + 1 u 0, K Γ N 0 n u n, 4) 0, K Γ D were te approximate gradient of pressure p W is defined as te solution to p, v ) 0,Ω = p, v ) 0,Ω p, v n ) 0, Ω v W. 5) We ten ave te following result. Teorem 4.4. Under te Assumptions 3. and 4.3 it olds u u,p p ) [ C u,p )] ) 1/. 6) Proof. K T K Step 1. By te triangle inequality and te saturation Assumption 4.3 we ave u u,p p ) 1 u / u,p / p ) 1 β /. 7)

9 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Hence it is sufficient to bound u / u,p / p ) /. By te stability, Teorem 3.1, we know tat tere exists v /,q / ) W / suc tat α u / u,p / p ) / B /[ u / u,p / p ), v /,q /)] 8) and v /,q / ) / = 1. Let v,q ) W be an interpolate to v /,q / ) W /. To simplify te notation we denote w := v / v and r := q / q. By scaling arguments one obtains { σ + μ ) K w 0,K + w 0,K + r 0,K + μ K n w 0, K K T / + K r 0, K + μ 1 K w K + 1 K w K} n C v /,q /) C. 9) / Combining qs. 7) and 8), and using te interpolate to split te bilinear form into two parts, we get C u u,p p ) B /[ u / u,p / p ), v /,q /)] = B /[ u / u,p / p ),w,r) ] + B /[ u / u,p / p ), v,q )] = W 1 + W. 30) Step. We bound te terms W 1 and W separately, starting wit W 1. Since u /,p / ) is te solution to te problem, we ave W 1 = L /[ w,r) ] B /[ u,p ),w,r) ] = f, w) 0,Ω + g, w) 0,ΓN + u 0,γ μ μ 1 w ) u 0,Γ D 0,μ n w) 0,ΓD + u 0 n,γ σ 1 w n ) 0,Γ D + u 0 n,r) 0,ΓD μ u, w ) 0,Ω σ u, w ) 0,Ω + p, w ) 0,Ω + u,r ) 0,Ω + J / p,r ) + μ n u, w ) 0,Γ D + μu, n w ) 0,Γ D γ σ μ 1 u, w ) 0,Γ D p, w n ) 0,Γ D u n,r ) 0,Γ D γ σ 1 u n, w n ) 0,Γ D. 31) Integrating te term μ u, w) 0,Ω by parts in eac element, and using te definition of approximate gradient of pressure 5) we get W 1 = f + μ u σ u p, w ) 0,Ω + u,r ) 0,Ω [ μ u n ], w ) 0, \ Ω + [ p ], w n ) 0, + J / p,r ) + g + p n μ n u, w ) u 0,Γ N 0 u,μ n w ) 0,Γ D \ Ω + u 0 u,γ μ μ 1 w ) 0,Γ D + u 0 n u n,r ) 0,Γ D + u 0 n u n,γ σ 1 w n ) 0,Γ D. 3) We split q. 3) into smaller pieces Z 1 := f + μ u σ u p, w ) 0,Ω, Z := u,r ) 0,Ω + J / p,r ), Z 3 := [ μ u n ], w ) 0, + [ p ], w n ) 0,, \ Ω Z 4 := g + p n μ n u, w ) 0,Γ N, \ Ω Z 5 := u 0 u,μ n w ) 0,Γ D + u 0 u,γ μ μ 1 w ) 0,Γ D, Z 6 := u 0 n u n,r ) 0,Γ D + u 0 n u n,γ σ 1 w n ) 0,Γ D.

10 18 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Next we bound te terms Z i using te Scwarz inequality for bot inner products and sums. Z 1 σ + μ ) 1 1/ K f + μ u σ u p 0,K) K T / Z Z 3 + K T / K T / T / )\ Ω ) σ + μ ) 1/ K w 0,K, 33) u ) 1/ 0,K T / )\ Ω μ [ n u ] 0, [ p ] 0, K T / r 0,K ) 1/ ) 1/ ) 1/ + J / p,p ) 1/ J / r, r) 1/, 34) T / )\ Ω T / )\ Ω ) 1/ μ 1 w 0, Te term Z 4 is first split into normal and tangential components, see q. ). 1 w n 0,) 1/. 35) Z 4 = g n n + μg t μ n u + μ n u n ) n μ n u n ) n, w w n)n + w n)n ) 0,Γ N = g n + p μ n u n, w n ) 0,Γ N + μg t μ n u + μ n u n ) n, w w n)n ) 0,Γ N = Z 4,n + Z 4,t, Z 4,n Z 4,t Z 5 T / ) Γ N T / ) Γ N μ μ 1 T / ) Γ D + Z 6 + μ 1 T / ) Γ D 1 T / ) Γ D 1 T / ) Γ D 36) g n + p μ n u n ) 1/ 1/ 1 0, 0,) w n, 37) T / ) Γ N g t n u + n u n ) n 0, u 0 u ) 1/ 0, u 0 u 0, u 0 n u n 0, u 0 n u n 0, ) 1/ T / ) Γ N μ 1 μ n w 0, T / ) Γ D ) 1/ ) 1/ ) 1/ w 0,) 1/, 38) T / ) Γ D μ 1 w 0,) 1/, 39) r 0, T / ) Γ D ) 1/ ) 1/ T / ) Γ D 1 w n 0,) 1/. 40) By te interpolation estimate 9) and since u,p ) as same values on bot meses, we find tat [ W 1 C u,p )] ) 1/ [ C u,p )] ) 1/. 41) K T / K K T K Step 3. Now we ave bounded te term W 1 and next we bound te term W. Since bot u,p ) and u /,p / ) are solutions to te problem on different meses, we get W = L /[ v,q )] B /[ u,p ), v,q )] L [ v,q )] + B [ u,p ), v,q )]. 4) Below we will denote wit subscripts and / te mes tat we are currently integrating on, e.g., ) 0,ΓD,. Since u,p ) and v,q ) ave same values on bot meses, we ave

11 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) W = J p,q ) J / p,q ) + u u 0,γ μ μ 1 v ) 0,Γ D, + u n u 0 n,γ σ 1 v n ) 0,Γ D, u u 0,γ μ μ 1 v ) 0,Γ D,/ u n u 0 n,γ σ 1 v n ) 0,Γ D,/ = J / p,q ) u u 0,γ μ μ 1 v ) 0,Γ D,/ u n u 0 n,γ σ 1 v n. ) 0,Γ D,/. 43) Using te Scwarz inequality we obtain [ W C J / p,p ) 1/ J / q,q ) 1/ + + μ 1 T / ) Γ D 1 T / ) Γ D u u 0 0, ) 1/ u n u 0 n 0, μ 1 v ) 1/ 0, T / ) Γ D ) 1/ 1 T / ) Γ D v n ) 1/ ] 0,. 44) Since v /,q / ) / = 1 te stability of te interpolate gives [ W C u,p )] ) 1/ [ C u,p )] ) 1/. 45) K T / K K T K Step 4. Now all te pieces are ready and we only need to combine qs. 30), 41) and 45) to get te desired result. 5. Numerical examples In tis section we illustrate te metod and analytical results wit numerical examples. We concentrate on sowing tat te results derived in te previous sections old wit viscosity μ 0, including Darcy limit μ = 0. Furtermore, we compare te results wit te traditional boundary conditions. Our model problem is u μ u + p = 0 u = 0 inω. inω, First we compute te convergence rate of te error in te energy norm wit different values of viscosity μ. Our domain Ω is te unit square wit Diriclet boundary conditions computed from te known exact solution; p = sinx) siny) cos1) 1 ) cos1) 1 ) and ) cosx) siny) u = p =. sinx) cosy) Since te pressure p is armonic, te solution is independent of te viscosity. In all te subsequent computations Nitsce stability parameters are γ μ = 10 and γ σ = 1. In Fig. 1 are te convergence rates of te metod. We see tat te convergence rate is O) wit all te values of viscosity, even on Darcy limit, as predicted by Lemma 4.. In Fig. are te a posteriori estimators computed for te same problem wit te same uniformly refined meses. We see tat also te a posteriori error estimator converges wit te same rate as te exact error. Only at te Darcy limit μ = 0we see a sligt reduction in te convergence rate. Next we compare te Nitsce metod to te traditional boundary conditions. Here our domain is te unit square and we use te following boundary conditions u = 0 on { x 0, 1), y = 0 } and { x 0, 1), y = 1 }, g n = 1, g t = 0 on { x = 0, y 0, 1) }, g n = 0, g t = 0 on { x = 1, y 0, 1) }.

12 184 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Fig. 1. Te convergence of te error in te energy norm for various values of viscosity parameter. Te convergence rates are given in te legend. Notice tat even te Darcy limit case μ = 0) converges wit te optimal rate of O). Fig.. Te convergence of te a posteriori estimator for various values of viscosity parameter. Te convergence rates are given in te legend. Te convergence rates are te same as for te exact error, see te figure above. In Fig. 3 are te velocities wit different values of viscosity. We see tat te traditional metod cannot produce slip boundary conditions of te Darcy problem unless te viscosity is equal to zero. On te oter and, Nitsce s metod moves continuously towards te slip boundary conditions as te viscosity diminises. In Fig. 4 we ave te velocity profile in x-direction at line x = 0.5. We notice tat te traditional metod as oscillations in te velocity near te boundaries in te case of small viscosity. In Fig. 5 we ave te velocity in y-direction at same line x = 0.5. Te velocity in y-direction sould be zero, but wit traditional boundary conditions and wit small viscosity tere is also oscillation in te velocity in y-direction. Tese oscillations are due to te fact tat te problem is very close to Darcy

13 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Fig. 3. Velocity fields of te model problem, on te left using te traditional and on te rigt using Nitsce boundary conditions. From top to bottom viscosity μ as values 0.1, 0.001, and 0. Te lengts of te vectors are scaled differently on eac row. For size of te velocity, see te velocity profile figures below. Notice te difference in te solutions near te boundaries on te middle row. problem but te traditional way of prescribing te boundary conditions does not allow slip in tangential direction before te viscosity is equal to zero. We tink tat te previous example clearly illustrates te sortcoming of te traditional boundary conditions in te Brinkman problem; te wole range of viscosity cannot be used. From te non-pysical oscillations it is obvious tat te results are not accurate or reliable near te boundaries wit te traditional no-slip boundary conditions if te viscosity is small. Refining te mes puses te inaccuracy closer to te boundary but will not remove te problem. Nitsce s metod andles tese difficulties even on a coarse mes. Finally we test te adaptive refinement based on te elementwise a posteriori error estimator. Te domain is te unit square and we use te Diriclet boundary conditions computed from te known exact solution. Te exact solution in te polar coordinates) is

14 186 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Fig. 4. Te velocity profile to x-direction at line x = 0.5 wit different values of viscosity μ. Te solid line is computed wit Nitsce boundary conditions and te dased line wit te traditional boundary conditions. Notice te oscillations wit te traditional metod near te boundary wit small μ. p = r β sinβθ) + sinπβ/4)β/ + sinπβ/4)) + 3β + β and ) u = p = βr β 1 sinθ βθ), cosθ βθ) were β>0 is a parameter. Wit te parameter β we can adjust te smootness of te solution so tat p H β+1 Ω) and u [H β Ω)]. Wit β>1 we assume we ave a solution but te a priori result, Lemma 4., is no longer applicable. In Fig. 6 we ave te first tree rounds of adaptive refinement for β = 1.3 and μ = 1. We see tat te

15 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Fig. 5. Te velocity profile to y-direction at line x = 0.5 wit different values of viscosity μ. Te solid line is computed wit Nitsce boundary conditions and te dased line wit te traditional boundary conditions. Notice te oscillations wit te traditional metod near te boundary wit small μ. a posteriori estimator detects te singularity at te origin and refines tere. In te same Fig. 6 we also ave te same problem wit μ = In tis case te a posteriori indicator as problems in seeing te singularity at te origin and refines also on te boundaries. In bot cases te ratio between te estimated and te exact error stays constant wic is peraps te most important feature for an a posteriori estimator. Looking at Figs. and 6 we see tat te a posteriori indicator is less sarp near te Darcy limit. Tis is because te residual inside te elements in te a posteriori estimator is of te form u + p 0,K if te viscosity and load are zero or very small). More precisely, tere are no powers of in te coefficient; terefore te approximate negative gradient of pressure p as to be close to te velocity. Te problem is worse near te boundaries since te approximate

16 188 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) Fig. 6. First tree rounds of adaptive refinement based on te a posteriori estimator. On te left we ave μ = 1 and on te rigt μ = Tere is a singularity at te origin. Wit smaller viscosity te a posteriori estimator as problems finding te singularity. Tis is because te reconstructed gradient of te pressure as larger role wit small viscosity and near te boundaries te reconstruction is not as sarp as inside te domain. Still, te ratio between te exact error and te estimator stays almost constant wic is important for te estimator. gradient is essentially computed from te pressure jumps, and on te boundary we do not ave a value for te jump. Wit larger viscosity te reconstruction is not as crucial anymore since we ave powers of in te coefficient reducing te residual if te elements are small. References [1] Y. Bazilevs, T.J.R. Huges, Weak imposition of Diriclet boundary conditions in fluid mecanics, Computers & Fluids 36 1) 007) 1 6. [] S. Brenner, L. Scott, Te Matematical Teory of Finite lement Metods, Springer, New York, 1994.

17 P. Hansbo, M. Juntunen / Applied Numerical Matematics ) [3]. Burman, A unified analysis for conforming and nonconforming stabilized finite element metods using interior penalty, SIAM Journal on Numerical Analysis 43 5) 005) [4]. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes and Darcy s problem, Numerical Metods for Partial Differential quations 4 1) 008) [5]. Burman, M.A. Fernández, P. Hansbo, Continuous interior penalty finite element metod for Oseen s equations, SIAM Journal on Numerical Analysis 44 3) 006) [6]. Burman, P. Hansbo, A unified stabilized metod for Stokes and Darcy s equations, Journal of Computational and Applied Matematics 198 1) 007) [7] J. Freund, R. Stenberg, On weakly imposed boundary conditions for second order problems, in: M. Morandi Cecci, et al. ds.), Proceedings of te Nint Int. Conf. Finite lements in Fluids, Venice, 1995, pp [8] T.J.R. Huges, L. Franca, A new finite element formulation for CFD: VII. Te Stokes problem wit various well-posed boundary conditions: Symmetric formulations tat converge for all velocity/pressure spaces, Computer Metods in Applied Mecanics and ngineering 65 1) 1987) [9] K. Mardal, X. Tai, R. Winter, A robust finite element metod for Darcy Stokes flow, SIAM Journal on Numerical Analysis 58 5) 00) [10] J. Nitsce, Über ein Variationsprinzip zur Lösung von Diriclet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abandlungen aus dem Matematiscen Seminar der Universität Hamburg ) [11] V. Tomée, Galerkin Finite lement Metods for Parabolic Problems, Springer, Berlin, 1997.