BUBBLE STABILIZED DISCONTINUOUS GALERKIN METHOD FOR STOKES PROBLEM

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1 BUBBLE STABILIZED DISCONTINUOUS GALERKIN METHOD FOR STOKES PROBLEM ERIK BURMAN AND BENJAMIN STAMM Abstract. We propose a low order discontinuous Galerkin method for incompressible flows. Stability of the discretization of the Laplace operator is obtained by enriching the space element wise with a non-conforming quadratic bubble. This enriched space allows for a wider range of pressure spaces. We prove optimal convergence estimates and local conservation of both mass and linear momentum independent of numerical parameters. 1. Introduction Discontinuous Galerkin (DG methods for incompressible flow has been studied in Hansbo and Larson [9] in the framework of Nitsche s method and inf-sup stable velocity pressure pairs. The analysis was extended by Toselli to the hp-framework using mixed or equal order stabilized formulations in [12]. Local discontinuous Galerkin methods with equal order velocity and pressure spaces stabilized using penalty on the interelement pressure jumps was proposed by Cockburn et al. [6]. The Navier-Stokes equations discretized using DG has recently been given a full analysis in the framework of domain decomposition on non-matching meshes using DG techniques by Girault et al. [8]. In this paper we extend our previous work on low order discontinuous Galerkin methods for scalar second order elliptic problems to the case of incompressible flow problems [2, 3]. Using piecewise affine discontinuous finite elements enriched with non-conforming bubbles we can eliminate all stabilization terms from the formulation without compromising the adjoint consistency. The upshot is that linear momentum is conserved locally and optimal convergence in the L 2 -norm may be proven using a duality argument. This is a consequence of the fact that the vectors of the velocity gradient matrix are functions in the lowest order Raviart- Thomas space for our choice of velocity finite element space (see also [1]. Several choices for the pressure space are possible without introducing a penalty term. Indeed we can use either globally continuous, piecewise affine functions, piecewise constant, discontinuous functions or functions from a direct sum of these two sets of functions and still satisfy the inf-sup condition uniformly with respect to the mesh size. Depending on the choice of pressure space slightly different results may be obtained. When the pressure space consists of continuous functions we prove that the stresses are continuous. Using spaces with discontinuous functions for the pressure on the other hand leads to a method that also enjoys local mass conservation and Date: January 7, Key words and phrases. discontinuous Galerkin methods, Stokes problem, bubble stabilization. 1

2 2 E. BURMAN AND B. STAMM for which the divergence of the non-pressure stress has optimal convergence. We give a unified analysis for these three choices of pressure space and we then discuss the differences of these approaches from numerical and analytical point of view. 2. Notation Let Ω be a convex polygon (polyhedron in three space dimensions in R d, d =2, 3, with outer normal n. Let K be a subdivision of Ω R d into non-overlapping d- simplices κ and denote by N K the number of simplices of the mesh. Suppose that each κ K is an affine image of the reference element κ, i.e. for each element κ there exists an affine transformation T κ : κ κ. Let F i denote the set of interior faces ((d 1-manifolds of the mesh, i.e. the set of faces that are not included in the boundary Ω. The set F e denotes the faces that are included in Ω and define F = F i F e. Define by N F = card(f and N Fi = card(f i the number of faces resp. interior faces of the mesh. Assume that K is shape-regular, does not contain any hanging node and covers Ω exactly. For an element κ K, h κ denotes its diameter and for a face F F, h F denotes the diameter of F. Set h = max κ K h κ and let h be the function such that h κ = h κ and h F = h F for all κ K and F F. For a subset R Ω or R F, (, R denotes the L 2 (R scalar product, R = (, 1/2 R the corresponding norm, and s,r the H s (R norm. The element-wise counterparts will be distinguished using the discrete partition as subscript, for example (, K = κ K (, K. For s 1, let H s (K be the space of piecewise Sobolev H s functions and denote its norm by s,k. Further let us denote H s (Ω = [H s (Ω] d and H s (K =[H s (K] d. Moreover the following space is defined { } L 2 0(Ω = v L 2 (Ω v dx =0. Let v =(v 1,...,v d H 1 (K, then we define v κ [L 2 (κ] d d by ( v i,j = xj v i,1 i, j d, for each κ K. Based on the scalar L 2 -product we define (v, w K = d (v i,w i K and ( v, w K = i=1 Ω d ( xj v i, xj w i K i,j=1 for all v, w H 1 (K. Further let us define the jump and average operators. Fix F F i and thus F = κ 1 κ 2 with κ 1,κ 2 K. Let v H 1 (K, v H 1 (K and denote by v 1, v 2 resp. v 1, v 2 the restriction of v, v to the element κ 1, κ 2, i.e. v i = v κi resp. v i = v κi, i =1, 2, and denote by n 1, n 2 the exterior normal of κ 1 resp. κ 2. We then define the average and jump operators by and {v} = 1 2 (v 1 + v 2, {v} = 1 2 (v 1 + v 2, {v } = 1 2 (v 1 + v 2, { v } = 1 2 ( v 1 + v 2, [v] = v 1 n 1 + v 2 n 2, [v] = v 1 n 1 + v 2 n 2, [v ] = v 1 n 1 + v 2 n 2, [ v ] = v 1 n 1 + v 2 n 2,

3 BUBBLE STABILIZED DG METHOD 3 where a b R d d is defined by (a b i,j = a i b j,1 i, j d, for all a, b R d. On outer faces F F e we define them by {v} = v, [v] = vn, {v} = v, [v] = v n, {v } = v, [v ] = v n, { v } = v, [ v ] = vn, where n is the outer normal of the domain Ω. Further we introduce some additional notation of the jump and average operators. Let n F {n 1, n 2 } be arbitrarly chosen but fixed and introduce the vectorial quantities, indexed by v, { v } v = { v }n F and [v ] v = [v ]n F. This notation is necessary to define some projection, see Lemma 3.8. Observe that using these definitions that (1 [v ] : { w } = [v ] v { w } v for all v H 1 (K, v, w H 1 (K and where a : b = d i,j=1 a ijb ij for all a, b R d d. Additionally remark that and thus [v ] = [v ] v n F, [v] = [v ]n F n F (2 [v ] F = [v ] v F, [v] F c [v ] F for some mesh independent constant c>0. Lemma 2.1 (Integration by parts. Let v H 1 (K and v, w H 1 (K, then (v, w K = ( v, w K +({v}, [w] F + ([v], {w} Fi, ( v, w K = ( v, w K +({ v }, [w ] F + ([ v ], {w } Fi. Proof. Element-wise integration by parts and applying the definitions of the jump and average operators leads to the result. Respecting the equalities (1 leads to the following corollary. Corollary 2.2 (Integration by parts. Let v H 1 (K and v, w H 1 (K, then ( v, w K = ( v, w K +({ v } v, [w ] v F + ([ v ], {w } Fi. Let us denote by V p h p 0 defined by 3. Bubble stabilized finite element space the standard discontinuous finite element space of degree V p h = { v h L 2 (Ω vh κ P p (κ, κ K }, where P p (κ denotes the set of polynomials of maximum degree p on κ. Consider then the enriched finite element space V bs = V 1 h { v h L 2 (Ω vh (x =α x x, α V 0 h }, where x =(x 1,...,x d denotes the physical variables. Let us additionally define some functional spaces that consists of functions only defined on the skeleton of the mesh: W 0 h = { v h L 2 (F vh F P 0 (F, F F }. Define also the vectorial versions V p h =[V p h ]d, V bs =[V bs ] d and W 0 h =[W 0 h ]d.

4 4 E. BURMAN AND B. STAMM Let v [L 2 (F] m, m {1, d, d 2 }, and define by v the L 2 -projection of v onto [W 0 h ]m, i.e. ( v, w h F =(v, w h F, w h [W 0 h ]m Properties of the bubble stabilized finite element space. Let us discuss some important properties of the space V bs. Lemma 3.1. For v h V bs we have that v h V 0 h. Moreover the application : V bs /V 1 h V 0 h is bijective. Proof. Observe that w h = 0 for all w h Vh 1 0 and that (αx x = 2dα Vh where d is the dimension of Ω. Let us denote by RT 0 (κ the local lowest order Raviart-Thomas space on κ. The following Lemma holds. Lemma 3.2. For all v h V bs with v h =(v h,1,..., v h,d there holds v h,i κ RT 0 (κ, κ K, and for all κ K and r h =(r h,1,...,r h,d with r h,i RT 0 (κ, there exists v h V bs such that v h,i κ = r h,i for all 1 i d. Proof. For the scalar case we refer to the proof of Lemma 3.4 in [2] and the vectorial version is constructed componentwise. Corollary 3.3. For v h V bs we have that { v h } v W 0 h, [ v h ] W 0 h. Moreover the applications { } v : V bs W 0 h and [ ] : V bs W 0 h are surjective. Lemma 3.4. There is a constant c>0 independent of h such that for all v h V bs there holds h 1 2 [vh ] 2 F ( h c 1 2 [vh ] 2 F + v h 2 K. Proof. We refer to the proof of Lemma 4.1 in [2] for the scalar case. Its vectorial conterpart follows from the componentwise construction. Lemma 3.5 (Poincaré inequality. There is a constant c>0 independent of h such that for all v h V bs there holds ( v h 2 K c P h 1 2 [vh ] 2 F + v h 2 K. Proof. We refer to the proof of Corollary 4.2 in [2] for the scalar case. Its vectorial counterpart follows from the componentwise construction. Observe that Lemma 3.4 and 3.5 are only valid on discrete spaces and thus does not hold for functions in H 1 (K. Thus we define a norm for H 1 (K by v 2 = v 2 K + h 1 2 [v] 2 F for all v H 1 (K. Nevertheless observe that there exists a constant c d > 0 such that (3 c d v h 2 v h 2 K + h 1 2 [vh ] 2 F v h 2 for all v h V bs.

5 BUBBLE STABILIZED DG METHOD Technical lemmas. Let us cite some well known results. For the proofs we refer to [4]. Lemma 3.6 (Inverse inequality. Let v h [V bs ] m, m {1,d}, then there exists a constant c I > 0 independent of h such that c 1 I h 2 v h K h v h K c I v h K. Lemma 3.7 (Trace inequality. Let v [H 1 (K1] m1 and v h [V bs ] m1, m 1 {1,d} resp. w [H 1 (K1] m2 and w h [V bs ] m2, m 2 {d, d 2 }, then there holds ( {v} F + [v] F c T h 1 2 v K + h 1 2 v K, {v h } F + [v h ] F c T h 1 2 vh K and {w } F + [w ] F c T ( h 1 2 w K + h 1 2 w K, {w h } F + [w h ] F c T h 1 2 wh K where c T > 0 is a constant independent of h Projections. Moreover denote by π p : L 2 (Ω V p h the L2 -projection onto V p h defined by π k (v h w h dx = v h w h dx w h V p h Ω Ω and by π p : L 2 (Ω V p h its vectorial counterpart defined by (π pv i = π p v i,1 i d. Then π p and π p satisfy the following approximation result: Let v H p+1 (K resp. v H p+1 (K, then (4 (5 v π p v K + h (v π p v K ch p+1 v p+1,k, v π p v K + h (v π p v K ch p+1 v p+1,k. Denote the Crouzeix-Raviart space by CR and its vectorial counterpart by CR. Additionally let us denote by i c : H 1 (Ω CR the vectorial Crouzeix-Raviart interpolant satisfying the following approximation result: if v H 2 (Ω, then (6 v i c v K + h (v i c v K ch 2 v 2,K. Further denote by C h : L 2 (Ω V 1 h,c the Clément interpolant [5] where V 1 h,c = { vh C 0 ( Ω vh κ P p (κ, κ K } is the continuous finite element space of degree 1. Remind that the Clément interpolant satisfies (7 v C h v K + h (v C h v K ch γ+1 v γ+1,k for all v H γ+1 (K, γ {0, 1}. Note that one can prove that the Clément interpolant conserves the mean of a function over Ω, i.e., Ω C hv(x dx = v(x dx, Ω and thus C h (L 2 0(Ω L 2 0(Ω. Note also that C h is H 1 -stable, i.e. (C h v K c v K. We denote by C h the vectorial version of C h sharing all properties. Furthermore, we present the following projection which will be used in the analysis.

6 6 E. BURMAN AND B. STAMM Lemma 3.8. Let a h Vh 0 and b h, c h Wh 0 be fixed. Then, there exists a unique function φ h V bs such that π 0 φ h = a h, (8 { φ h } v F = b h F F F, {φ h } F = c h F F F i. Moreover φ h satisfies the following stability result (9 h 1 φ h 2 K + φ h 2 c ( h 1 a h 2 K + h 1 2 bh 2 F + h 1 2 ch 2 Fi. Proof. Let us first establish the a priori estimate. Firstly by the trace inequality observe that (10 h 1 2 [φh ] 2 F c h 1 φ h 2 K and using (5 that (11 h 1 φ h 2 K h 1 π 0 φ h 2 K + h 1 (φ h π 0 φ h 2 K h 1 a h 2 K + c φ h 2 K for some constant c > 0. Secondly integrate by parts, use Lemma 3.1 and Corollary 3.3 φ h 2 K = ( φ h,π 0 φ h K +({ φ h } v, [φ h ] v F + ([ φ h ], {φ h } Fi = ( φ h, a h K +([φ h ] }{{} v, b h F + ([ φ h ], c h Fi. }{{}}{{} I II III Applying (11 and the Cauchy-Schwarz, the inverse or the trace and Young s inequality for each term yields respectively I c I φ h K h 1 a h K 1 4 φ h 2 K + c2 I h 1 a h 2 K II c T h 1 φ h K h 1 2 bh F 1 4 φ h 2 K + 1 4c h 1 a h 2 K + c c 2 T h 1 2 bh 2 F, III c T φ h K h 1 2 ch Fi 1 4 φ h 2 K + c 2 T h 1 2 ch 2 F i, and thus φ h 2 K c ( h 1 a h 2 K + h 1 2 bh 2 F + h 1 2 ch 2 F i, which, combined with (3, (10 and (11, completes the a priori estimate. To conclude the proof, it now suffices to observe that (8 is a square linear system of size d (N K + N F + N Fi. Hence, existence and uniqueness of a solution of the linear system are equivalent. Let us denote by Aw = b the square linear system and assume that there is a vector w 1 and w 2 such that Aw i = b, i =1, 2. Further let us denote the difference between them by e = w 1 w 2 and therefore Ae = 0. The a priori estimate (9 implies that e = 0 and thus the solution is unique and hence the matrix is regular. 4. Bubble stabilized discontinuous Galerkin method for Stokes problem Consider the steady Stokes problem: find u H 1 (Ω and p L 2 0(Ω such that u + p = f in Ω, (12 u = 0 in Ω, u = g on Ω, where f H 1 (Ω and g H 1/2 ( Ω such that Ω g n ds = 0. This setting ensures a unique solution to the model problem (12, see [7].

7 BUBBLE STABILIZED DG METHOD Bubble stabilized discontinuous Galerkin method. Let Q h L 2 0 (Ω be some scalar finite element space that will be precised later. We introduce the bubble stabilized Galerkin method by: find (u h,p h V bs Q h such that (13 a(u h, v h +b(p h, v h b(q h, u h =F (v h,q h (v h,q h V bs Q h where the linear form F (, and the bilinear forms a(, and b(, are defined by a(u h, v h = ( u h, v h K ( { u h }, [v h ] F ([u h ], { v h } F, b(p h, v h = (p h, v h K +({p h }, [v h ] F, F (v h,q h = (f, v h K (g n,q h Fe (g, v h n Fe. Remark 4.1. The discrete solution u h and p h of (13 satisfies the following local linear momentum conservation property { u h }n κ ds + {p h }n κ ds = f dx κ for all κ K and where n κ denotes the outer normal of κ. κ Lemma 4.2 (Consistency. Let the couple (u,p H 2 (Ω H 1 (Ω be the exact solution of (12 and let (u h,p h V bs Q h be the approximation defined by (13. Then, there holds that for all (v h,q h V bs Q h. a(u u h, v h +b(p p h, v h b(q h, u u h = 0 Proof. By integration by parts, Lemma 2.1. Details are left to the reader. 5. Convergence analysis Assume for simplicity homogeneous boundary conditions, i.e. g = 0, in this section. We will specify the choice for the pressure space Q h to get inf-sup stable approximations. Let us first introduce three possible choices for Q h. The first alternative to define the pressure approximation space is as the continuous piecewise linear finite element space defined by Q 1 h,c = { v h C 0 ( Ω L 2 0(Ω vh κ P 1 (κ, κ K } and the second one is the space of piecewise constant functions Q 0 h,d = { v h L 2 0(Ω vh κ P 0 (κ, κ K }. Further let us also consider the direct sum of the above defined spaces Q 1 h,c Q0 h,d. Remark that h q h K and h 1 2 [q h ] Fi is a norm for q h Q 1 h,c resp. q h Q 0 h,d and thus q h 2 q = h q h 2 K + h 1 2 [qh ] 2 F i is a norm for q h Q 1 h,c Q0 h,d. For the following we chose Q h = Q 1 h,c Q0 h,d and observe that the cases Q h = Q 1 h,c and Q h = Q 0 h,d are covered by the analysis as well. Thus let us define the following triple norm for all v h q h V bs Q h. v h,q h 2 = v h 2 + q h 2 q κ

8 8 E. BURMAN AND B. STAMM Proposition 5.1 (Inf-sup condition. There exists a constant c>0 independent of h such that there holds (u h,p h V bs Q h a(u h, v h +b(p h, v h b(q h, u h c u h,p h sup. 0 (v h,q h V bs Q h v h,q h Proof. For the proof of Proposition 5.1 we introduce the following two lemmas. Lemma 5.2. There exists a constant c>0 independent of h such that there exists for each fixed couple (u h,p h V bs Q h a couple (v h,q h V bs Q h with c u h,p h 2 a(u h, v h +b(p h, v h b(q h, u h. Lemma 5.3. There exists a constant c>0 independent of h such that for each (u h,p h and (v h,q h defined by Lemma 5.2 there holds v h,q h c u h,p h. Indeed combining Lemma 5.2 and 5.3 yields a(u h, v h +b(p h, v h b(q h, u h c u h,p h 2 c u h,p h v h,q h. Proof of Lemma 5.2. Firstly fix u h V bs and p h Q h. Observe that chosing v h = u h and q h = p h in (13 yields a(u h, u h (14 = u h 2 K 2( { u h }, [u h ] F u h 2 K 2 h 1 2 { uh } F h 1 2 [uh ] F u h 2 K 2c T u h K h 1 2 [uh ] F 1 2 u h 2 K c 1 h 1 2 [uh ] 2 F using Corollary 3.3, the Cauchy-Schwarz, the trace and Young s inequality. Further let w h V bs be the projection defined by Lemma 3.8 with a h = 0, b h = h 1 [u h ] v and c h = 0. By its properties, integration by parts, Lemma 3.1, Corollary 3.3 and (2 we get (15 a(u h, w h = ( u h, w h K + ([ u h ], {w h } Fi ([u h ] v, { w h } v F = ( u h, π 0 w h K + ([ u h ], {w h } Fi ([u h ] v, { w h } v F = h 1 2 [uh ] 2 F and again by integration by parts b(p h, w h = ( p h, π 0 w h K ([p h ], {w h } Fi = ([p h ], {w h } Fi since p h Vh 0 and by the property of the projection w h. Further since p h Q h we write p h = p h,c + p h,d with p h,c Q 1 h,c and p h,d Q 0 h,d and thus b(p h, w h = ([p h,d ], {w h } Fi =0 since [p h,c ] = 0 and by the property of the projection w h. Let z h be the projection defined by Lemma 3.8 with a h = h 2 p h, b h = 0 and c h = h[p h ] and observe that b(p h, z h = ( p h, π 0 z h K ([p h,d ], {z h } Fi = h p h 2 K + h 1 2 [ph ] 2 F i = p h 2 q.

9 BUBBLE STABILIZED DG METHOD 9 Also, applying integration by parts combined with the Cauchy-Schwarz and Young s inequality, yields a(u h, z h = ( u h, π 0 z h K + ([ u h ], {z h } Fi ([u h ] v, { z h } v F = ( u h, h 2 p h K ([ u h ], h[p h ] Fi c u h K ( h p h K + h 1 2 [ph ] Fi c u h 2 K 1 2 p h 2 q which implies that (16 a(u h, z h +b(p h, z h 1 2 p h 2 q c 2 u h 2 K. Defining v h = u h +(c w h + 1 4c 2 z h, q h = p h and respecting (14-(16 yields a(u h, v h +b(p h, v h b(q h, u h c u h,p h 2 using (3. Proof of Lemma 5.3. Observe that by (9 we get w h 2 c u h 2 and z h 2 c p h 2 q, and therefore using the definition of v h implies v h,q h 2 = v h 2 + p h 2 q 4 u h 2 +4 w h 2 +4 z h 2 + p h 2 q c ( u h 2 + p h 2 q=c u h,p h 2. Thus the numerical scheme is inf-sup stable for all three choices Q h = Q 1 h,c, Q h = Q 0 h,d and Q h = Q 1 h,c Q0 h,d. Let us define the following auxilliary norm v,q 2 a = h 1 v 2 K + v 2 K + h 1 2 {v} 2 Fi + h 1 2 { v } 2 F + q 2 K + h 1 2 {q} 2 F for all v H 2 (K and q L 2 0(Ω which will be used for the continuity result. We denote further by A + B the (in general not direct sum of the functional spaces A and B. Then we prove the following continuity result. Proposition 5.4 (Continuity. Let v H 2 (K +CR, v h V bs, q H 1 (K and q h Q h. There exists a constant c>0 independent of h such that a(v, v h +b(p, v h b(q h, v c v,q a v h,q h.

10 10 E. BURMAN AND B. STAMM Proof. Applying the Cauchy-Schwarz inequality, Lemma 3.4 and (2 yields a(v, v h = ( v, v h K ( { v }, [v h ] F ( v 2 K + h 1 2 { v } 2 F v,q a v h,q h, b(q, v h = (q, v h K +({q}, [v h ] F 1 2 ( v h 2 K + h 1 2 [vh ] 2 F ( q 2 K + h 1 2 {q} 2 F 1 2 ( v h 2 K + h 1 2 [vh ] 2 F ( q 2 K + h 1 1 ( 2 {q} 2 2 F v h 2 K + h [vh ] 2 2 F v,q a v h,q h, b(q h, v = ( q h, v K ([q h ], {v} Fi ( h q h 2 K + h 1 1 ( 2 [qh ] 2 2 Fi h 1 v 2 K + h {v} 2 2 Fi c v,q a v h,q h. Assume that Q h = αq 1 h,c βq0 h,d for (α, β {(1, 0, (0, 1, (1, 1} and define in a standard manner the following quantities (17 η u = u i c u, ξ u = u h i c u, and η p = p P α p, ξ p = p h P α p, where the projection P α : L 2 (Ω Q h is defined by P α = (1 α π 0 + αc h. Observe that ξ u V bs and ξ p Q h since the projection P α preserves the property of zero mean. Further P α satisfies the error estimate (18 v P α v K + h (v P α v K ch αγ+1 v αγ+1,k for all v H γ+1 (K, γ {0, 1}. Proposition 5.5 (Approximability. Let η u H 2 (Ω + CR and η p H γ+1 (Ω + Q h, with γ {0, 1} and Q h = αq 1 h,c βq0 h,d for (α, β {(1, 0, (0, 1, (1, 1}, be defined by (17. Then, there exists a constant c>0 independent of h such that η u,η p + η u,η p a ch u 2,K + ch 1+αγ p 1+αγ,K. Proof. This is a direct consequence of the trace inquality and the error estimates (6 and (18. Theorem 5.6 (Convergence in Energy norm. Let u H 2 (Ω, p H γ+1 (Ω, γ {0, 1}, be the exact solution of problem (12 and let u h V bs, p h Q h, with Q h = αq 1 h,c βq0 h,d for (α, β {(1, 0, (0, 1, (1, 1}, be the approximation defined by (13. Then, there exists a constant c>0 independent of h such that u u h + p p h K + p p h q ch u 2,K + ch 1+αγ p 1+αγ,K. Proof. Let us first establish the a priori estimate for the triple norm. Split the error in a standard manner in two parts (19 u u h,p p h η u,η p + ξ u,ξ p. By Proposition 5.5 it follows that η u,η p ch u 2,K + ch 1+αγ p 1+αγ,K.

11 BUBBLE STABILIZED DG METHOD 11 For the second term of the right hand side of (19 observe that applying the infsup condition of Proposition 5.1, the consistency of Lemma 4.2, the continuity of Proposition 5.4 and the approximability of Proposition 5.5 yields a(ξ u, v h +b(ξ p, v h b(q h, ξ u ξ u,ξ p sup 0 (v h,q h V bs Q h v h,q h Therefore we conclude that a(η u, v h +b(η p, v h b(q h, η u = sup 0 (v h,q h V bs Q h v h,q h c η u,η p a ch u 2,K + ch 1+αγ p 1+αγ,K. (20 u u h + p p h q ch u 2,K + ch 1+αγ p 1+αγ,K. In order to prove the error estimate for the quantity p p h K we follow the standard technique, see [7, 11]. Let H 1 0 (Ω = { v H 1 (Ω v Ω = 0 }. Since p p h L 2 0 (Ω there exists v p H 1 0 (Ω such that v p = p p h and v p K c p p h K. Applying integration by parts yields p p h 2 K =(p p h, v p K = ( (p p h, v p K + ([p p h ], {v p } Fi. Further split this equation (21 p p h 2 K = I + II with I = ( (p p h, v p C h v p K + ([p p h ], {v p C h v p } Fi, II = ( (p p h, C h v p K + ([p p h ], {C h v p } Fi and where C h denotes the vectorial Clément interpolation operator. Firstly observe using the Cauchy-Schwarz inequality, the trace inequality and the approximation properties of C h that (22 I c ( h (p p h K + h 1 2 [p ph ] Fi v p K c p p h q p p h K. Secondly using the consistency of Lemma 4.2 implies II = b(p p h, C h v p =a(u u h, C h v p (23 = ( (u u h, C h v p K ([u u h ], { C h v p } F ( (u u h 2 K + h 1 2 [u uh ] 2 F c ( (u u h K + h 1 2 [u uh ] F C h v p K c ( (u u h K + h 1 2 [u uh ] F p p h K 1 2 ( C h v p 2 K + h 1 2 { Ch v p } 2 F using that C h v p H0 1 (Ω, the Cauchy-Schwarz inequality, the trace inquality, the H 1 (Ω-stability of C h and the stability of v p. Combining (20-(23 leads to the result. Remark 5.7 (Optimal L 2 -convergence. Optimal convergence in the L 2 -norm can be shown using Nitsche s trick. The details are left to the reader. 1 2

12 12 E. BURMAN AND B. STAMM Proposition 5.8. Let (u h,p h V bs Q h be the solution of (13 with Q h = αq 1 h,c βq0 h,d for (α, β {(1, 0, (0, 1, (1, 1}. Then, there holds (24 h f + u h p h K + (1 β h 1 2 [ uh ] Fi + h 1 2 [uh ] F ch f π 0 f K. Additionally, if f H 1 (K there holds (25 h f + u h p h K + (1 β h 1 2 [ uh ] Fi + h 1 2 [uh ] F ch 2 f K. Proof. Let w h be the function defined by Lemma 3.8 with a h = h 2 ( p h π 0 f u h, b h = h 1 [u h ] v and c h = (1 β h[ u h ]. By its properties, integration by parts, Lemma 3.1, Corollary 3.3 and (2 we get a(u h, w h = ( u h, w h K + ([ u h ], {w h } Fi ([u h ] v, { w h } v F = ( u h, π 0 w h K + ([ u h ], {w h } Fi ([u h ] v, { w h } v F = ( u h, a h K + (1 β h 1 2 [ uh ] 2 F i + h 1 2 [uh ] 2 F. On the other hand since p h = αp h,c + βp h,d, using again integration by parts and the properties of the projections we get b(p h, w h = ( p h, w h K ([p h ], {w h } Fi =( p h, π 0 w h K β([p d,h ], {w h } Fi = ( p h, a h K β(1 β([p d,h ], h[ u h ] Fi =( p h, a h K since β(1 β = 0 for the considered set of values of β. Moreover for this set we have that (1 β = (1 β 2 and since a(u h, w h +b(p h, w h = (f, w h K we get h 2 π 0 f + u h p h 2 K + (1 β2 h 1 2 [ uh ] 2 F i + h 1 2 [uh ] 2 F = a(u h, w h +b(p h, w h (π 0 f, w h K =(f π 0 f, w h K f π 0 f K w h K. Using further the stability estimate (9 of w h yields w h K ch (h 2 π 0 f + u h p h 2 K + (1 β2 h 1 2 [uh ] 2 F + h [ uh ] 2 2 Fi and therefore we get h π 0 f + u h p h K + (1 β h 1 2 [ uh ] Fi + h 1 2 [uh ] F ch f π 0 f K. Finally we observe that f + u h p h K f π 0 f K + π 0 f + u h p h K which yields (24 and additionally noting that for f H 1 (K there holds f π 0 f K ch f K proves (25. Corollary 5.9. If f is piecewise constant, i.e. f V 0 h, then [u h ] F =0, [ u h ] Fi =0, if α =1,β =0, [u h ] F =0, u u h K =0, if α =0,β =1, [u h ] F =0, if α =1,β =1. Corollary If f is piecewise constant, i.e. f Vh 0, and Q h = Q 1 h,c then, the discrete solution u h and p h of (13 satisfies the following local linear momentum conservation property u h n κ ds + p h n κ ds = f dx κ κ κ for all κ K and where n κ denotes the outer normal of κ.

13 BUBBLE STABILIZED DG METHOD 13 Q h h u u h K u u h p p h K (u u h K Q 1 h,c E E E E E-04 ( E-03 ( E-04 ( E-03 ( E-05 ( E-03 ( E-04 ( E-03 ( E-06 ( E-03 ( E-05 ( E-03 (0.98 Q 0 h,d E E E E E-04 ( E-02 ( E-02 ( E-02 ( E-04 ( E-02 ( E-02 ( E-03 ( E-05 ( E-03 ( E-03 ( E-03 (1.00 Q 1 h,c Q0 h,d E E E E E-05 ( E-03 ( E-03 ( E-03 ( E-05 ( E-03 ( E-03 ( E-03 ( E-06 ( E-03 ( E-04 ( E-04 (1.00 Table 1. Smooth problem: Different error quantities of the numerical solution for all three choices of the pressure approximation space with respect to the mesh size h. The quantities in the brackets correspond to the convergence rates. Q h h h 1 2 [u h ] F h 1 2 [[u h ]] F h 1 2 [[ u h ]] Fi Q 1 h,c E E E E-03 ( E-04 ( E-04 ( E-03 ( E-05 ( E-05 ( E-03 ( E-05 ( E-06 (2.94 Q 0 h,d E E E E-02 ( E-04 ( E-02 ( E-02 ( E-05 ( E-02 ( E-03 ( E-05 ( E-02 (0.99 Q 1 h,c Q0 h,d E E E E-03 ( E-04 ( E-03 ( E-03 ( E-05 ( E-03 ( E-03 ( E-05 ( E-03 (0.98 Table 2. Smooth problem: Different error quantities of the numerical solution for all three choices of the pressure approximation space with respect to the mesh size h. The quantities in the brackets correspond to the convergence rates. 6. Numerical tests Let us introduce the numerical examples tested in this section. We consider two numerical tests proposed in [11]. i Problem with smooth solution Consider problem (12 with Ω = (0, 1 2 and f(x imposed such that the exact solution is given by ( (x 4 u(x = 1 2x x2 1 (4x3 2 6x2 2 +2x 2 (4x 3 1 6x2 1 +2x 1(x 4 2 2x3 2 + x2 2, p(x =x 1 + x 2 1. Observe that (u,p satisfies the regularity assumption and thus Theorem 5.6 and Proposition 5.8 are valid. A sequence of structured meshes is considered.

14 14 E. BURMAN AND B. STAMM (0,1 x 2 (0,0.5 Ω (2,0 (12,0 x 1 Figure 1. Geometry of the backstep channel problem. ii Backstep channel problem Consider problem (12 with f = 0 and (8(1 x 2 (x 2 0.5, 0 if x 1 =0, g(x = ((1 x 2 x 2, 0 if x 1 = 12, 0 otherwise, on the domain defined by Figure 1. The domain is not convex and thus the solution does not lie in H 2 (Ω. In consequence Theorem 5.6 is no longer valid. But observe that Proposition 5.8 is independent of the geometry resp. regularity assumption and therefore Proposition 5.8 still holds. A sequence of unstructured and globally uniform meshes is considered. We consider the approximations defined by (13 using a pressure approximation space Q h = αq 1 h,c βq0 h,d for (α, β {(1, 0, (0, 1, (1, 1}. For the computations we use FreeFem++ [10] Smooth problem. The convergence results for test problem i with all three choices of the pressure approximation space are illustrated in Table 1 and 2. We observe the optimal convergence as predicted by Theorem 5.6 and the superconvergence of Proposition Backstep channel problem. The convergence results for test problem ii with all three choices of the pressure approximation space is illustrated in Table 3. Since the exact solution is not knows only the non-conforming error-quantities are given. Observe that Proposition 5.8 is still valid, in contrast to Theorem 5.6 since Ω is non-convex, and since f = 0 the quantity h 1 2 [u h ] F is for any choice of pressure space equal to zero (machine precision. Acknowledgements The authors acknowledge the financial support provided through the Swiss National Science Foundation under grant References [1] F. Brezzi and L. D. Marini. Bubble stabilization of discontinuous Galerkin methods. In W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, and Y. Vassilevski, editors, Advances in numerical mathematics, Proc. International Conference on the occasion of the 60th birthday of Y.A. Kuznetsov, pages Institute of Numerical Mathematics of The Russian Academy of Sciences, Moscow, [2] E. Burman and B. Stamm. Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47(1: , [3] E. Burman and B. Stamm. Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment. C. R. Math. Acad. Sci. Paris, 346(1-2: , 2008.

15 BUBBLE STABILIZED DG METHOD 15 Q h h (u u h K h 2 1 [u h ] F h 2 1 [[u h ]] F h 1 2 [[ u h ]] Fi Q 1 h,c E E E E-01 ( E-01 ( E-01 ( E-01 ( E-01 ( E-01 ( E-01 ( E-01 ( E-01 ( Q 0 h,d E E E E-01 ( E-01 ( E+00 ( E-01 ( E-01 ( E+00 ( E-01 ( E-01 ( E-01 (0.87 Q 1 h,c Q0 h,d E E E E E-01 ( E-01 ( E-01 ( E+00 ( E-01 ( E-01 ( E-01 ( E-01 ( E-02 ( E-02 ( E-01 ( E-01 (0.86 Table 3. Backstep channel problem: Different error quantities of the numerical solution for all three choices of the pressure approximation space with respect to the mesh size h. The quantities in the brackets correspond to the convergence rates and 0 corresponds to zero in machine precision. [4] P. G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM, Philadelphia, PA, Reprint of the 1978 original [North-Holland, Amsterdam; MR (58 #25001]. [5] P. Clément. Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9(R-2:77 84, [6] B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab. Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal., 40(1: (electronic, [7] V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, Theory and algorithms. [8] V. Girault, B. Rivière, and M. F. Wheeler. A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp., 74(249:53 84 (electronic, [9] P. Hansbo and M. G. Larson. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche s method. Comput. Methods Appl. Mech. Engrg., 191(17-18: , [10] F. Hecht, O. Pironneau, A. L. Hyaric, and K. Ohtsuka. FreeFEM++ Manual, [11] B. Rivière and V. Girault. Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Engrg., 195(25-28: , [12] A. Toselli. hp discontinuous Galerkin approximations for the Stokes problem. Math. Models Methods Appl. Sci., 12(11: , Erik Burman, Department of Mathematics, Mantell Building, University of Sussex, Falmer, Brighton, BN1 9RF, United Kingdom address: E.N.Burman@sussex.ac.uk Benjamin Stamm, EPFL, IACS-CMCS, Station 8, 1015 Lausanne, Switzerland address: benjamin.stamm@epfl.ch