STABILIZED HP-DGFEM FOR INCOMPRESSIBLE FLOW. Math. Models Methods Appl. Sci., Vol. 13, pp , 2003

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1 Mathematical Models and Methods in Applied Sciences c World Scientiic Publishing Company STABILIZED HP-DGFEM FOR INCOMPRESSIBLE FLOW D. SCHÖTZAU1 C. SCHWAB 2 A. TOSELLI 2 1 Department o Mathematics, University o Basel, Rheinsprung 21, CH-4051 Basel, Switzerland 2 Seminar or Applied Mathematics, ETH Zürich, ETH Zentrum, CH-8092 Zürich, Switzerland Math. Models Methods Appl. Sci., Vol. 13, pp , 2003 We consider stabilized mixed hp-discontinuous Galerkin methods or the discretization o the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that Q k Q k and Q k Q k 1 elements satisy a generalized in-sup condition on geometric edge and boundary layer meshes that are reined anisotropically and non quasi-uniormly towards aces, edges, and corners. The discrete in-sup constant is proven to be independent o the aspect ratios o the anisotropic elements and to decrease as k 1/2 with the approximation order. We also show that the generalized in-sup condition leads to a global stability result in a suitable energy norm. Keywords: hp-fem, discontinuous Galerkin methods, Stokes problem, anisotropic reinement 1. Introduction Over the last ew years, several discontinuous Galerkin (DG) methods or incompressible low problems and or certain saddle-point problems with incompressibility constraints have been proposed in the literature. Here we only mention the piecewise solenoidal discontinuous Galerkin methods 6,20, the local discontinuous Galerkin (LDG) methods 11,10, and the interior penalty methods 18,29,17. The methods above all rely on discrete velocity spaces consisting o piecewise polynomial unctions with no continuity constraints between the elements in the underlying triangulation. Interelemental communication is achieved through so-called numerical luxes, as in the original discontinuous Galerkin methods or non-linear hyperbolic systems. 12,9,13 The main motivations or using DG methods in luid low problems lie in their robustness in convection-dominated regimes, their conservation properties, and their great lexibility in the mesh-design. Based on completely discontinuous inite element spaces, DG methods easily handle elements o various types and shapes, non-matching grids and even local spaces o dierent orders; they are thereore ideal or hp-adaptivity. Even i transport phenomena may be dominant in incompressible low problems, mixed DG methods still require suitable velocity-pressure pairs in order to ensure stability and convergence o the underlying Stokes discretization. It was shown recently that discontinuous P k P k 1 and Q k Q k 1 pairs are in-sup stable with 1

2 2 Stabilized hp-dgfem or incompressible low respect to the mesh-size, as opposed to their conorming counterparts. 18,29 These elements are optimal rom an approximation point o view. A slightly dierent approach was proposed or the LDG methods. 11,10 There, the introduction o a pressure stabilization term was proven to also render the convenient equal-order P k P k and Q k Q k elements stable, uniormly in the mesh-size. The study o mixed hp-discontinuous Galerkin methods was recently initiated and it was shown that several discontinuous velocity-pressure pairs possess better stability properties than their conorming versions. 29 In particular, the numerical results reported in the experiments o Re. 29 or two-dimensional uniorm meshes show that discontinuous Q k Q k 1 elements are also uniormly stable with respect to the polynomial degree k. For this pair, the best available bound o the in-sup constant in terms o k was then shown to decrease as k 1, on shape-regular tensorproduct meshes in two and three dimensions, possibly with hanging nodes. 24 This bound ensures the same p-version convergence rate or the velocity and the pressure as that o conorming Q k Q k 2 elements in three dimensions. However, the latter elements are mismatched with respect to h-approximation. In laminar regimes, solutions o incompressible low problems in polyhedral domains have corner and edge singularities. In addition, strong boundary layers may arise at aces, edges, and corners. In the hp-version o the inite element method, these solution components can be approximated at exponential rate o convergence provided that the meshes are geometrically and anisotropically graded towards aces, edges, and corners. 2,5,21,27,28 These anisotropically reined meshes raise serious stability issues in mixed approximations as the in-sup constants might in general be very sensitive to the aspect ratios o the elements. It was recently shown or two- and three-dimensional conorming approximations employing Q k Q k 2 elements that, on corner, edge, and boundary-layer tensor-product meshes, the in-sup constant or the Stokes problem is in act independent o the aspect ratios o the anisotropic elements in the meshes. 22,23,1,30. Recently, discontinuous Q k Q k 1 elements were studied on geometric edge meshes designed to resolve corner and edge singularities in the absence o boundary layers. 25 By suitably deining the discontinuity stabilization parameters in the DG bilinear orms on anisotropic elements, it was proven that this velocity-pressure pair is divergence stable, with an in-sup constant that is independent o the aspect ratios o the anisotropic elements and that decreases as k 3/2 in the approximation order. In this paper, we analyze stabilized hp-discontinuous Galerkin methods on geometric meshes in three dimensions. We show that the introduction o the pressure stabilization term originally proposed or the LDG discretization 11 leads to a generalized in-sup constant or Q k Q k 1 and Q k Q k elements that decreases only as k 1/2 in the polynomial degree, and is independent o possibly large aspect ratios o the mesh. As opposed to the work o Re. 25 that only considers geometric edge meshes, the results here also hold or geometric boundary layer meshes that are additionally geometrically reined towards the aces. As a consequence o the generalized in-sup condition, we obtain a global stability result in a suitable energy norm and derive p-version error bounds that are better than those available in the recent work o Re. 24, by o hal an order o k in the velocity and a ull order o k in the pressure, respectively. We emphasize that, in our analysis, we use a similar uniying setting as that proposed in the analysis o Re. 24. Thus, although we only consider the so-called interior penalty discontinuous Galerkin method, our results hold true verbatim or the analogues o the methods analyzed there, and, in particular, extend the LDG methods 11,10 to the hp-context. The outline o the paper is as ollows. In Section 2, we introduce stabilized mixed hp-dgfem or the Stokes problem. Two classes o geometric meshes are deined in Section 3. Continuity and coercivity properties o the DG orms on these meshes are established in Section 4. Our main result is the generalized in-sup condition

3 Stabilized hp-dgfem or incompressible low 3 that we present and prove in Section 5. A global stability result or the proposed DG discretizations is then derived in Section 6, together with hp-error bounds on shape-regular elements. 2. Stabilized mixed hp-dgfem or the Stokes problem In this section, we introduce stabilized mixed hp-discontinuous Galerkin methods using the pressure stabilization orm that was originally proposed or the LDG discretization The Stokes problem Let Ω be a bounded polyhedron in R 3, and let n be the outward normal unit vector to its boundary Ω. Given a source term L 2 (Ω) 3 and a Dirichlet datum g H 1/2 ( Ω) 3 with g n ds = 0, the Stokes problem consists in inding a Ω velocity ield u and a pressure p such that ν u + p = in Ω, u = 0 in Ω, (2.1) u = g Thanks to the continuous in-sup condition 7,16 in 0 q L 2 (Ω)/R sup 0 v H0 1(Ω)3 on Ω. Ω q v dx v 1 q 0 C Ω > 0, (2.2) with a constant C Ω depending only on Ω, the Stokes problem (2.1) has a unique solution (u, p) with u V := H 1 (Ω) 3, p Q := L 2 0 (Ω) = L2 (Ω)/R. Here, we denote by s,d and s,d the norm and seminorm o the Sobolev space H s (D), s 0, on a domain D in R d, d = 1, 2, 3. The same notation is used to denote norms or vector ields. In case D = Ω, we drop the subscript Meshes and trace operators Throughout, we consider triangulations T on Ω that consist o aine hexahedral elements {K}. More precisely, each element K T is obtained rom the reerence cube Q = ( 1, 1) 3 by an aine mapping. In general, we allow or irregular meshes, i.e., meshes with hanging nodes (see, e.g., Sect o Re. 26), but suppose that the intersection between neighboring elements is either a common vertex, a common edge, a common ace, or an entire ace o one o the two elements. An interior ace o T is the (non-empty) two-dimensional interior o K + K, where K + and K are two adjacent elements o T. Similarly, a boundary ace o T is the (non-empty) two-dimensional interior o K Ω which consists o entire aces o K. We denote by F I the union o all interior aces o T, by F B the union o all boundary aces, and set F = F I F B. For an element K T, we denote its diameter by h K and the radius o the largest circle that can be inscribed into K by ρ K. A mesh T is called shape-regular i h K cρ K, K T, (2.3) or a shape-regularity constant c > 0 that is independent o the elements. As will be discussed below, our meshes are not necessarily shape-regular.

4 4 Stabilized hp-dgfem or incompressible low We next deine some trace operators. Let F I be an interior ace shared by two elements K + and K and v, q, and τ be vector-, scalar- and matrix-valued unctions, respectively, that are smooth inside each element K ±. We denote by v ±, q ± and τ ± the traces o v, q and τ on rom the interior o K ± and deine the mean values { } and normal jumps [ ] at x as {v } := (v + + v )/2, [v] := v + n K + + v n K, {q } := (q + + q )/2, [[q ]] := q + n K + + q n K, {τ } := (τ + + τ )/2, [[τ ]] := τ + n K + + τ n K. Here, we denote by n K the outward normal unit vector to the boundary K o an element K. We also deine the matrix-valued jump o the velocity v given by [v] := v + n K + + v n K, where, or two vectors a and b, [a b] ij = a i b j. On a boundary ace F B given by = K Ω, we set {v } := v, {q } := q, {τ } := τ, as well as [v] := v n, [v] := v n, [[q ]] := qn and [[τ ]] := τ n Finite element spaces Given a mesh T on Ω and an approximation order k 0, we introduce the inite element spaces Vh k(t ) and Qk h (T ): V k h (T ) := { v L2 (Ω) 3 : v K Q k (K) 3, K T }, Q k h(t ) := { q L 2 0(Ω) : q K Q k (K), K T }, (2.4) where Q k (K) is the space o polynomials o maximum degree k in each variable on element K Mixed discontinuous Galerkin approximations We approximate the velocities and pressures in the spaces V h and Q h given by V h := Vh k (T ), Q h := Q l h (T ), (2.5) with k 1 and l = k or l = k 1. We reer to these velocity-pressure pairs as (discontinuous) equal-order Q k Q k elements and mixed-order Q k Q k 1 elements, respectively. We consider the ollowing stabilized mixed DG methods: ind (u h, p h ) V h Q h such that { Ah (u h, v) + B h (v, p h ) = F h (v) (2.6) B h (u h, q) + C h (p h, q) = G h (q) or all (v, q) V h Q h. The orms A h, B h, and C h are given by A h (u, v) = ν h u : h v dx ( {ν h v } : [u] + {ν h u } : [v] ) ds Ω F +ν δ [u] : [v] ds, F B h (v, q) = q h v dx + {q }[v] ds, C h (p, q) = ν 1 F I γ [[p ]] [[q ]] ds. Ω E

5 Stabilized hp-dgfem or incompressible low 5 Here, h is the discrete gradient, taken elementwise. The unctions δ L (F) and γ L (F I ) are the so-called discontinuity and pressure stabilization unctions, respectively, or which we will make a precise choice below. Finally, the corresponding right-hand sides F h and G h are F h (v) = v dx (g n) : {ν h v } ds + ν δ g v ds, Ω F B F B G h (q) = q g n ds. F B Remark 2.1 It ollows rom the stability results below that problem (2.6) has a unique solution (u h, p h ) V h Q h. Remark 2.2 The orm A h (, ) discretizing the Laplacian is the so-called interior penalty (IP) orm. Several other choices are possible or A h (, ). All the results o this paper also hold verbatim or the orms considered in the work o Re. 24. The orm B h (, ) is related to the incompressibility constraint; it is used in several related mixed DG approaches. 11,18,29,24,17 Finally, C h (, ) is the pressure stabilization orm that was originally introduced in the local discontinuous Galerkin methods. 11, Perturbed mixed ormulation For the purpose o the analysis, we introduce perturbed orms Ãh and B h, ollowing the ideas o Re. 4 and o Re. 24. To this end, we deine the space V(h) := V + V h, and introduce the liting operators L : V(h) Σ h and M : V(h) Q h by L(v) : τ dx = [v] : {τ } ds, τ Σ h, Ω F M(v)q dx = [v] {q } ds, q Q h, Ω E where we use the auxiliary space Σ h = { τ L 2 (Ω) 3 3 : τ K Q k (K) 3 3, K T }. We then introduce the ollowing perturbed orms on V(h) V(h) and V(h) Q: Ã h (u, v) = ν [ h u : h v L(u) : h v L(v) : h u ] dx + ν δ [u] : [v] ds, Ω F B h (v, q) = q [ h v M(v)] dx. Ω (2.7) We have Ãh = A h on V h V h, and B h = B h on V h Q h, respectively. Thus, we may rewrite the method (2.6) as: ind (u h, p h ) V h Q h such that { Ãh (u h, v) + B h (v, p h ) = F h (v) B (2.8) h (u h, q) + C h (p h, q) = G h (q) or all (v, q) V h Q h. 3. Geometric edge and boundary layer meshes In this section, we deine two classes o geometric meshes, namely geometric edges meshes that are employed in the presence o corner and edge singularities (as,

6 6 Stabilized hp-dgfem or incompressible low e.g., in Stokes low or nearly incompressible elasticity), and geometric boundary layer meshes that are used when, in addition to corner/edge singularities, boundary layers are present as well. Both meshes are characterized by a geometric grading actor σ (0, 1) and the number o layers n, the thinnest layer having width proportional to σ n. 2,5,21,27,28, Geometric edge meshes A geometric edge mesh T n,σ edge is constructed by considering an initial shaperegular macro-triangulation T m = {M} o Ω, with no hanging nodes, possibly consisting o just one element. The macro-elements M in the interior o Ω are then reined isotropically and regularly (not discussed urther) while the macroelements M on the boundary o Ω are reined geometrically and anisotropically towards edges and corners. This geometric reinement is obtained by ainely mapping corresponding reerence triangulations (reerred to as patches) on Q onto the macro-elements M using the elemental maps F M : Q M. This process is illustrated in Figure 1. For edge meshes, the ollowing patches on Q = I 3, I = ( 1, 1), are used or the geometric reinement towards the boundary o Ω: Edge patches: An edge patch Te edge on Q is given by T edge e := {K xy I K xy T xy }, where T xy is an irregular corner mesh, geometrically reined towards a vertex o Ŝ = ( 1, 1) 2 with grading actor σ and n reinement levels; see Figure 1 (level 2, let). Corner patches: In order to build a corner patch Tc edge on Q, we irst consider an initial, irregular, corner mesh T c,m, geometrically reined towards a vertex o Q, with grading actor σ and n reinement levels; see the mesh in bold lines in Figure 1 (level 2, right). The elements o this mesh are then irregularly reined towards the three edges adjacent to the vertex in order to obtain the mesh Tc edge ; see also Figure 3. For simplicity, we always assume that the only hanging nodes in geometric edge are those in the closure o edge and corner patches. meshes T n,σ edge Level 1 Level 2 Figure 1: Hierarchic structure o a geometric edge mesh T n,σ edge. The macro-elements M on the boundary o Ω (level 1) are urther reined as edge and corner patches (level 2). The geometric grading actor is here σ = 0.5.

7 Stabilized hp-dgfem or incompressible low Geometric boundary layer meshes As or edge meshes, the construction o a geometric boundary layer mesh T n,σ bl starts rom an initial shape-regular macro-triangulation T m = {M} o Ω, with no hanging nodes, possibly consisting o just one element. The macro-elements M on the boundary o Ω are now also reined geometrically towards aces; see Figure 2. More precisely, the ollowing ace, edge, and corner patches on Q = I 3 are used: Face patches: A ace patch T bl on Q is given by an anisotropic triangulation o the orm T bl := {K x I I K x T x }, where T x is a mesh o I, geometrically reined towards one o the vertices, say x = 1, with grading actor σ (0, 1) and total number o layers n; see Figure 2 (level 2, let). Edge patches: An edge patch Te bl on Q is given by T bl e := {K xy I K xy T xy }, where T xy is a triangulation o Ŝ = I2 obtained by irst considering an irregular corner mesh T xy as in a patch Te edge o an edge mesh, geometrically reined towards a vertex o Ŝ, say (x, y) = (1, 1), with grading actor σ and n reinement levels (see Figure 1 below, level 2, let). The elements o the mesh T xy are then anisotropically reined towards the two edges x = 1 and y = 1, in order to obtain a regular mesh T xy. We reer to Figure 2 (level 2, center) or an example. Corner patches: In order to build a corner patch Tc bl on Q, we irst consider the same initial, irregular corner mesh T c,m, geometrically reined towards a vertex o Q, with grading actor σ and n reinement levels; see the mesh in bold lines in Figure 2 (level 2, right). The elements o T c,m are then anisotropically reined towards the three aces x = 1, y = 1, and z = 1 in order to obtain a regular mesh Tc bl ; see also Figure 3. For simplicity, we always assume that the three types o patches above are combined in such a way that geometric boundary layer meshes T n,σ bl do not contain hanging nodes. Level 1 Level 2 Figure 2: Hierarchic structure o a geometric boundary layer mesh T n,σ bl. The macro-elements M on the boundary o Ω (level 1) are urther reined as ace, edge and corner patches (level 2). The geometric grading actor is here σ = 0.5.

8 8 Stabilized hp-dgfem or incompressible low Remark 3.1 We note that the underlying mesh T c,m is the same or the corner patches Tc edge and Tc bl in edge and boundary layer meshes, respectively. However, Tc edge is irregular and contains hanging nodes. Figure 3 shows the dierence between corner patches or boundary layer and edge meshes. Figure 3: Geometrically reined corner patches Tc bl and Tc edge or boundary layer (let) and edge (right) meshes. The geometric grading actor is σ = 0.5. The geometric edge and boundary layer meshes deined above satisy the ollowing property; see Sect. 3 o Re. 25. Property 3.2 Let T be a geometric edge mesh T n,σ edge or a geometric boundary layer mesh T n,σ bl, with a grading actor σ (0, 1) and n levels o reinement. Then, any K T can be written as K = F K (K xyz ), where K xyz is o the orm K xyz = I x I y I z = (x 1, x 2 ) (y 1, y 2 ) (z 1, z 2 ), and F K is an aine mapping, the Jacobian o which satisies det(j K ) C, det(j 1 K ) C, DF K C, DF 1 K C, with constants only depending on the angles o K but not on its dimensions. We note that the constants in Property 3.2 only depend on the shape-regularity constant in (2.3) o the underlying macro-element mesh T m. The dimensions o K xyz on the other hand may depend on the geometric grading actor and the number o reinements. For an element K o a geometric edge mesh, we deine, according to Property 3.2, h K x = h x = x 2 x 1, h K y = h x = y 2 y 1, h K z = h x = z 2 z Stabilization on geometric meshes In this section, we deine the discontinuity and pressure stabilization unctions δ L (F) and γ L (F I ) on geometric meshes. To this end, let be an entire ace o an element K o a geometric mesh T on Ω. According to Property 3.2, K can be obtained by a stretched parallelepiped K xyz by an aine mapping F K that only changes the angles. Suppose that the ace is the image o, e.g., the ace {x = x 1 }. We set h = h x. For a ace perpendicular to the y- or z-direction, we choose h = h y or h = h z, respectively.

9 Stabilized hp-dgfem or incompressible low 9 Let then K and K be two elements with entire aces and that share an interior ace = in F I. We have ch h c 1 h, (3.1) with a constant c > 0 that only depends on the geometric grading actor σ and the constant in (2.3) or the underlying macro-mesh T m. We deine the unction h L (F) by { min{h h(x) :=, h } x F I, (3.2) h x F B. We then set and δ(x) = δ 0 h 1 k 2, x F, (3.3) γ(x) = γ 0 min{h, h } max{1, l} 1, x F I, (3.4) with δ 0 > 0 and γ 0 > 0 independent o h and k. Remark 3.3 For isotropically reined and shape-regular meshes, the deinitions in (3.2) and (3.3) are equivalent to the usual deinition o δ. 24 Similarly, the deinition o γ in (3.4) generalizes the deinition o Re. 11 to the hp-version context on geometric meshes. 4. Continuity and coercivity on geometric meshes On geometric meshes, the continuity o à h and B h as well as the coercivity o A h can be established as in Sect. 4 o Re. 25. To this end, we equip V(h) = V + V h with the broken norm v 2 h := v 2 1,K + δ [v] 2 ds, v V(h). K T F We have the ollowing result. Theorem 4.1 Let T be a geometric edge mesh T n,σ edge or a geometric boundary layer, with a grading actor σ (0, 1) and n levels o reinement. Let the stabilization unctions δ be deined as in (3.2) and (3.3). Then, the orms Ãh and B h in (2.7) are continuous: mesh T n,σ bl Ãh(v, w) να 1 v h w h v, w V(h), B h (v, q) α 2 v h q 0 u V(h), q Q, with continuity constants α 1 and α 2 that depend on δ 0 and the constants in Property 3.2, but are independent o ν, k, n, and the aspect ratio o T. Furthermore, there exists a constant δ min > 0 that depends on the constants in Property 3.2, but is independent o ν, k, n, and the aspect ratio o T, such that, or any δ 0 δ min, A h (v, v) νβ v 2 h v V h, or a coercivity constant β > 0 depending on δ 0 and the constants in Property 3.2, but independent o ν, k, n, and the aspect ratio o T.

10 10 Stabilized hp-dgfem or incompressible low Remark 4.2 The results in Theorem 4.1 are based on anisotropic stability estimates or the liting operators L and M that can be ound in Sect. 4 o Re. 25. These operators are identical or all the DG orms considered in the ramework o Re. 24 and, thus, the results in Theorem 4.1 also hold or all the mixed DG methods considered there. We note that the restriction on δ 0 is typical or the interior penalty orm A h and can be avoided i A h were chosen to be, e.g., the local discontinuous Galerkin orm, the nonsymmetric interior penalty orm, or the second Bassi-Rebay orm. 24 Next, we address the continuity o F h and G h. Theorem 4.3 Let T be a geometric edge mesh T n,σ edge or a geometric boundary layer mesh T n,σ bl, with a grading actor σ (0, 1) and n levels o reinement. Let the stabilization unctions δ be deined as in (3.2) and (3.3). Then we have F h (v) C [ 0 + ν δ 1 2 g 0, Ω ] v h v V h, G h (q) C δ 1 2 g 0, Ω q 0 q Q h, with continuity constants that depend on δ 0, Ω, and the constants in Property 3.2, but are independent o ν, k, n, and the aspect ratio o T. Proo: We irst note that we have the Poincaré inequality v 0 C v 1,h v V(h), (4.1) with a constant depending on δ 0, Ω, and the constants in Property 3.2. The bound (4.1) ollows by proceeding as in the original proo in Lemma 2.1 o Re. 3, taking into account elliptic regularity theory or polyhedral domains and by using the anisotropic trace inequality ϕ 0, Ch 1 ϕ 3/2+ε,K, ε > 0, or an element K T and an entire ace o K, with a constant depending on the constants in Property 3.2. Let now v V h. From (4.1), we obtain Ω v dx C 0 v h. Further, applying the discrete trace inequality rom Lemma 3.3 o Re. 25 as in the proo o Theorem 4.1 o Re. 25, (g n) : {ν h v } ds Cν δ 1 2 g 0, Ω v h, E B with a constant depending on δ 0, and the constants in Property 3.2. Finally, the Cauchy-Schwarz inequality yields ν E B δg v ds ν δ 1 2 g 0, Ω v h. This proves the assertion or F h. Similarly, or q Q h, ( G h (q) q g n ds δ 1 2 g 0, Ω δ 1 q 2 ds ) 1 2. E B E B Using again the techniques in Lemma 3.3 and Theorem 4.1 o Re. 25, we have E B δ 1 q 2 ds C q 2 0, with a constant depending on δ 0, and the constants in Property 3.2. This completes the proo.

11 Stabilized hp-dgfem or incompressible low 11 Remark 4.4 The same continuity properties hold or all the unctionals F h and G h in the mixed DG methods analyzed in the setting o Re Generalized in-sup condition on geometric meshes Our main result establishes a generalized in-sup condition on geometric meshes. To this end, we introduce the ollowing seminorm on Q h q 2 F I := γ [[q ]] 2 ds, F I with γ the pressure stabilization unction deined in (3.4). We have the ollowing result. Theorem 5.5 Let T be a geometric edge mesh T n,σ edge or a geometric boundary layer mesh T n,σ bl, with a grading actor σ (0, 1) and n levels o reinement. Let the stabilization unctions δ and γ be deined according to (3.2), (3.3), and (3.4). Then, there exists a constant C > 0 that depends on Ω, δ 0, γ 0, and the constants in Property 3.2 and (3.1), but is independent o ν, k, l, n, and the aspect ratio o T, such that, or any n and k 1, l = k or l = k 1, B h (v, q) sup Ck ( 1 q FI ) 2 q 0 1, q Qh \ {0}. 0 v V h v h q 0 Remark 5.6 For h-version DG approximations on shape-regular meshes, the generalized in-sup condition in Theorem 5.5 was established recently or the LDG discretizations, in a orm that also involves the auxiliary stresses present in the LDG approach. 11,10 Similar in-sup conditions also play an important role in the analysis o conorming stabilized mixed methods. 15,14 The proo o Theorem 5.5 is carried out in the rest o this section. We begin by collecting several properties o L 2 -projections and by deriving bounds or averages and jumps over aces o geometric meshes. We then complete the proo o Theorem L 2 -projections I x I x For an interval I x = (x 1, x 2 ), let Π x : L 2 (I x ) Q k (I x ) denote the onedimensional L 2 -projection onto the space Q k (I x ) o polynomials o degree at most k on I x ; given v L 2 (I x ), this projection is deined by imposing Π x v ϕ dx = v ϕ dx, ϕ Q k (I x ). The L 2 -projection is stable: Π x v 0,Ix v 0,Ix, v L 2 (I x ). (5.2) Moreover, applying similar techniques as in Theorem 2.2 o Re. 8, we have, or k 1, Π x v 1,Ix Ck 1 2 v 1,Ix, v H 1 (I x ), (5.3) with a constant C > 0 independent o k, I x, and v. We next recall the ollowing approximation result rom Lemma 3.5 o Re. 19.

12 12 Stabilized hp-dgfem or incompressible low Lemma 5.7 Let I x = (x 1, x 2 ), h x = x 2 x 1 and v H 1 (I x ). Then, there holds v(x 1 ) Π x v(x 1 ) 2 + v(x 1 ) Π x v(x 2 ) 2 C h x k 1 v 2 0,I x, or k 1 and with a constant C > 0 independent o h x, k, and v. We will also make use o an approximation result rom Lemma 3.9 o Re. 19 or the two-dimensional L 2 -projection Π x Π y ; here, the subscripts indicate the variables the projections Π x and Π y act on. Lemma 5.8 Let I x = (x 1, x 2 ), I y = (y 1, y 2 ), h x = x 2 x 1 and h y = y 2 y 1. Assume that there exists a constant c > 0 such that ch x h y c 1 h x. Then, or v H 1 (I x I y ) and k 1, we have v Π x Π y v 2 0, (I x I y) C h xk 1 v 2 1,I x I y, K xyz K xyz with a constant C > 0 depending on c, but independent o h x, h y, k, and v. For an axiparallel element K xyz = (x 1, x 2 ) (y 1, y 2 ) (z 1, z 2 ), the L 2 -projection Π Kxyz : L 2 (K xyz ) Q k (K xyz ) is the product operator Π Kxyz = Π x Π y Π z o one-dimensional L 2 -projections. For v L 2 (K xyz ), it satisies Π Kxyz v ϕ dx = v ϕ dx, ϕ Q k (K xyz ). For an element K o a geometric edge or boundary layer mesh T, the L 2 -projection Π K : L 2 (K) Q k (K) is deined by Π K v ϕ dx = v ϕ dx, ϕ Q k (K). K K Thanks to Property 3.2, we have K = F K (K xyz ) or an axiparallel element K xyz = (x 1, x 2 ) (y 1, y 2 ) (z 1, z 2 ). For v L 2 (K), we thereore have Π K v F K = Π Kxyz [ v FK ], on Kxyz. (5.4) We have the ollowing stability result. Lemma 5.9 Let T be a geometric edge or boundary layer mesh. Let K T and v H 1 (K). Then we have or k 1 Π K v 1,K Ck 1 2 v 1,K, with a constant C > 0 depending on the bounds in Property 3.2, but independent o k, K, and v. Proo: Let K = K xyz according to Property 3.2. The bounds (5.2) and (5.3) imply that x Π Kxyz v 0,Kxyz Ck 1 2 x v 0,Kxyz, y Π Kxyz v 0,Kxyz Ck 1 2 y v 0,Kxyz, z Π Kxyz v 0,Kxyz Ck 1 2 z v 0,Kxyz, or any v H 1 (K xyz ), with a constant C > 0 independent o k, K xyz, and v. A scaling argument and the bounds in Property 3.2 prove the assertion or a general element K.

13 Stabilized hp-dgfem or incompressible low 13 Finally, the L 2 -projection Π : L 2 (Ω) {v L 2 (Ω) : v K Q k (K), K T } is deined elementwise by Πv K = Π K v K, K T. For vector ields, we use bold-ace notation (such as Π Kxyz, Π K, and Π) to denote the L 2 -projections that are applied componentwise Auxiliary results In this section, we derive bounds or the averages and jumps over aces. We start by considering interior aces Interior aces Let K and K be two elements o a geometric mesh with entire aces and that share an interior ace in F I. We may assume that is an entire ace o K, that is, =. By Property 3.2, we have K = F K (K xyz ) and K = F K (K xyz ) with, e.g., K xyz = (x 1, x 2 ) (y 1, y 2 ) (z 1, z 2 ), K xyz = (x 2, x 3 ) (y 1, y 3 ) (z 1, z 2 ), and y 2 y 3. The ace is then given by = F ( yz ), with yz = {x 2 } (y 1, y 2 ) (z 1, z 2 ), and F (y, z) = F K (x 2, y, z) = F K (x 2, y, z) or y 1 y y 2, z 1 z z 2. Similarly, we have = F ( yz ). For a unction v H 1 (K K ) 3, we deine v xyz = v K F K and v xyz = v K F K. We set h x = x 2 x 1, h x = x 3 x 2, h y = y 2 y 1, h y = y 3 y 1, and h z = z 2 z 1, and may assume that ch x h x c 1 h x, (5.5) according to (3.1). In the case where the elements K and K match regularly (i.e., = ) the ratios o the mesh-sizes h x, h y and h z can be arbitrary. However, when K and K match irregularly (i.e., ), it is essential to observe that, by deinition o geometric meshes, we also have ch y h y c 1 h y, ch y h x c 1 h y, (5.6) with a constant c > 0 depending solely on the bounds in Property 3.2. The situation when K and K match irregularly is shown in Figure 4. We point out that the above coniguration covers all interior aces in geometric edge and boundary layer meshes. We irst show the ollowing result. Lemma 5.10 Let K, K T share a ace F I. Then, or q Q h and v H 1 (K K ) 3, ( [[q ]] {v Πv } ds C ) 1 ( ) 1 γ [[q ]] 2 2 ds v 2 1,K + v 2 2 1,K, with a constant C > 0 that depends on γ 0 and the bounds in Property 3.2 and (3.1). Proo: We begin by noting that [[q ]] {v Πv } ds = 1 2 S K S K, where S K = S K = [[q ]] (v K Π K v K ) ds, [[q ]] (v K Π K v K ) ds.

14 14 Stabilized hp-dgfem or incompressible low z 2 K x x 2 x3 1 xyz yz K xyz y 1 y 2 y z 3 1 Figure 4: The axiparallel elements K xyz and K xyz match irregularly. The ace yz is given by yz = {x 2 } (y 1, y 2 ) (z 1, z 2 ). Step 1: We start by bounding the term S K. Setting q yz = [[q ]] F, we obtain S K = = = = [[q ]] (v K Π K v K ) ds q yz (v xyz Π Kxyz v xyz ) det(df ) dy dz yz yz yz q yz (v xyz Π x Π y Π z v xyz ) det(df ) dy dz q yz (Π y Π z v xyz Π x Π y Π z v xyz ) det(df ) dy dz. Here, we have used identity (5.4), the actorization Π Kxyz = Π x Π y Π z into onedimensional L 2 -projections, and the act that each component o q yz is a polynomial o degree l = k or l = k 1 in y- and z-direction. The Cauchy-Schwarz inequality, the deinition o h in (3.1), the deinition o γ in (3.4), and (5.5) yield ( ) 1 S K T K h x max{1, l} 1 q yz 2 2 det(df ) dy dz yz ( C T K with the term T K given by T 2 K := max{1, l}h 1 x yz ) 1 γ [[q ]] 2 2 ds, Π y Π z v xyz Π x Π y Π z v xyz 2 det(df ) dy dz. From the stability o the one-dimensional projections Π y and Π z in (5.2) (taking into account that det(df ) is constant), the approximation result in Lemma 5.7,

15 Stabilized hp-dgfem or incompressible low 15 and the bounds in Property 3.2, we obtain TK 2 max{1, l}h 1 x det(df ) yz v xyz Π x v xyz 2 dy dz C max{1, l}k 1 det(df ) x v xyz 2 0,K xyz C det(df ) DF K det(df 1 K ) v 2 1,K C v 2 1,K. Combining the above estimates shows that S K C v 1,K ( ) 1 γ [[q ]] 2 ds 2, (5.7) with a constant C depending on γ 0 and the bounds in Property 3.2 and (3.1). Step 2: Let us now consider the term S K. We note that there is an entire ace o K, such that =. I =, then S K can be bounded as S K in Step 1. Thus, we only need to consider the case where is an irregular ace o K, i.e., is a proper subset o as in Figure 4. As in the proo o Step 1, since q yz is a polynomial in z-direction, we have: S K = [[q ]] (v K Π K v K ) ds = = yz yz q yz (v xyz Π K xyz v xyz) det(df ) dy dz q yz (Π zv xyzv xyz Π x Π y Π zv xyz) det(df ) dy dz. Here, we denote by Π x, Π y, and Π z the one-dimensional L 2 -projections on K xyz. We obtain ( ) 1 S K C T K γ [[q ]] 2 2 ds, with the term T K given by TK 2 := max{1, l}h 1 x Π z v xyz Π x Π y Π z v xyz 2 det(df ) dy dz. yz From the stability (5.2) o Π z in z-direction, (5.5), (5.6) and Lemma 5.8, we obtain TK 2 max{1, l}h 1 x det(df ) v xyz Π x Π yv xyz 2 dy dz C v xyz 2 1,K xyz v 2 1,K. Combining the bounds above gives S K C v 1,K ( yz ) 1 γ [[q ]] 2 ds 2, (5.8) with a constant C > 0 depending on γ 0 and the bounds in Property 3.2 and (3.1). Combining (5.7) and (5.8) concludes the proo. Next, we estimate the jump o the L 2 -projection over the ace.

16 16 Stabilized hp-dgfem or incompressible low Lemma 5.11 Let K, K T share a ace F I. Suppose that =, with and entire aces o K and K, respectively. Then, or v H 1 (K K ) 3, [Πv] 2 ds C min{h, h }k 1[ v 2 1,K + v 2 1,K ], with a constant C > 0 that depends only on the bounds in Property 3.2 and (3.1). Proo: Equality (5.4) ensures that [Πv] 2 ds = yz Π K v K Π K v K 2 ds Π Kxyz v xyz Π K xyz v xyz 2 det(df ) dy dz. We consider two cases separately. Case 1: Let K and K match regularly, i.e., =. Since v xyz and v xyz coincide on the ace yz, we have Π y Π z v xyz = Π y Π z v xyz on yz. We thus obtain rom the triangle inequality [Πv] 2 ds C det(df ) [ T K + T K ], with T K = T K = yz Π y Π z v xyz Π x Π y Π z v xyz 2 dy dz, Π y Π zv xyz Π x Π y Π zv xyz 2 dy dz. yz Using the stability (5.2) o the projections Π y and Π z in y- and z-directions, as well as the approximation result in Lemma 5.7, we obtain T K Ch x k 1 x v xyz 2 0,K xyz Ch k 1 v 2 1,K. An analogous bound or T K and (5.5) prove the assertion in this case. Case 2: Assume that K and K are non-matching ( ). We then have that Π z v xyz = Π z v xyz on yz. Thus, [Πv] 2 ds C det(df ) [ T K + T K ], with T K = T K = Π z v xyz Π x Π y Π z v xyz 2 dy dz, yz yz Π z v xyz Π x Π y Π z v xyz 2 dy dz.

17 Stabilized hp-dgfem or incompressible low 17 Since the underlying elements are shape-regular in x- and y-directions thanks to (5.5) and (5.6), we can invoke the stability (5.2) o Π z and the approximation result in Lemma 5.8. This gives v xyz Π x Π yv xyz 2 dy dz T K yz yz v xyz Π x Π y v xyz 2 dy dz Ch x k 1 v xyz 2 1,K xyz Ch k 1 v 2 1,K. An analogous bound or T K and (5.5) prove the assertion in this case Boundary aces We conclude by stating an analogous result to Lemma 5.11 or boundary aces that can be proved with exactly the same techniques. Let K be an element on the boundary and an entire ace o K in F B. Lemma 5.12 For v H0 1 (K) 3, we have [Πv] 2 ds C h k 1 v 2 1,K, with a constant C > 0 depending on the bounds in Property Proo o Theorem 5.5 Fix q Q h. From the continuous in-sup condition (2.2), there exists a ield w H0 1(Ω)3 such that q w dx = q 2 0, w 1 C 1 Ω q 0, (5.9) Ω where C Ω > 0 is the continuous in-sup constant. We then set v = Πw, with Π the L 2 -projection deined previously. Using [w] = 0 on F, (5.9), integration by parts, and the properties o the L 2 -projection, we ind B h (v, q) = B h (w, q) + B h (Πw w, q) = q h q (Πw w) dx Ω = q [[q ]] {w Πw } ds. Applying Lemma 5.10 gives [[q ]] {w Πw } ds F I F I F I ( C K T F I [ [[q ]] {w Πw } ds w 2 1,K C w 1 q FI. ) 1 2 ( [q ]] {Πw w } ds F I ) 1 γ [[q ]] 2 2 ds

18 18 Stabilized hp-dgfem or incompressible low Combining the above estimates with (5.9) yields B h (v, q) C q 2 0 ( 1 q F I q 0 ), (5.10) with a constant C > 0 depending on C Ω, γ 0, and the bounds in Property 3.2 and (3.1). We have rom Lemma 5.9, Lemma 5.11 and Lemma 5.12, together with the deinition o the discontinuity stabilization unction δ, v 2 h = Πw 2 1,K + δ [Πw] 2 ds K T F Ck K T w 2 1,K + Ck K T w 2 1,K Ck w 2 1. Thus, invoking (5.9), v h Ck 1 2 q 0. (5.11) Combining (5.10) and (5.11) concludes the proo o Theorem Global stability and a-priori error estimates In this section, we show how the stability results in the previous sections can be used to obtain a global stability result and to derive a-priori error estimates. The technique we use is closely related to that used in the analysis o conorming stabilized mixed methods. 15, Global stability Let W h be the product space W h = V h Q h, endowed with the norm In W h we deine the orms (v, q) 2 DG = ν v 2 h + ν 1 k 1 q ν 1 q 2 F I. A h (u, p; v, q) = Ãh(u, v) + B h (v, p) B h (u, q) + C h (p, q), L h (v, q) = F h (v) + G h (q), and reormulate (2.8) equivalently as: ind (u h, p h ) W h such that A h (u h, p h ; v, q) = L h (v, q) (6.1) or all (v, q) W h. The ollowing stability result holds. Theorem 6.1 Let T be a geometric edge mesh T n,σ edge or a geometric boundary layer mesh T n,σ bl, with a grading actor σ (0, 1) and n levels o reinement. Let the stabilization unctions δ and γ be deined according to (3.2), (3.3), and (3.4). Then, there exists a constant C > 0 that depends on Ω, δ 0, γ 0, and the constants in Property 3.2 and (3.1), but is independent o ν, k, l, n, and the aspect ratio o o T, such that, or any n and k 1, l = k or l = k 1, in (0,0) (u,p) W h A h (u, p; v, q) sup C. (0,0) (v,q) W h (u, p) DG (v, q) DG

19 Stabilized hp-dgfem or incompressible low 19 Proo: Fix (0, 0) (u, p) V h Q h. Thanks to the coercivity o A h in Theorem 4.1 and the deinition o C h, we have A h (u, p; u, p) νβ u 2 h + ν 1 p 2 F I. (6.2) Furthermore, Theorem 5.5 guarantees the existence o a velocity w V h satisying B h (w, p) C p 2 0 C p F I p 0, w h Ck 1 2 p 0. (6.3) From the deinition o A h, the continuity properties in Theorem 4.1, weighted Cauchy-Schwarz inequalities, and (6.3), we obtain A h (u, p; w, 0) = A h (u, w) + B h (w, p) Cε 1 ν u 2 h Cε 1 1 ν w 2 h + C p 2 0 Cε 1 2 p 2 0 Cε 2 p 2 F I C(1 ε 1 2 νε 1 1 k) p 2 0 Cε 1ν u 2 h Cε 2 p 2 F I, (6.4) with parameters ε 1, ε 2 > 0 at our disposal. We next set (v, q) = (u, p) + ε 3 (w, 0), with ε 3 > 0. Then, combining (6.2) and (6.4), yields A h (u, p; v, q) Cν(1 ε 1 ε 3 ) u 2 h +C(ν 1 ε 3 ε 2 ) p 2 F I +Cε 3 (1 ε 1 2 νε 1 1 k) p 2 0. It is now easy to see that one can select ε 1 o order O(kν), ε 2 o order O(k), and ε 3 o order O(ν 1 k 1 ), respectively, in such a way that A h (u, p; v, q) Cν u 2 h + Cν 1 p 2 F I + Cν 1 k 1 p 2 0 = C (u, p) 2 DG. (6.5) Using the act that ε 3 is o order O(ν 1 k 1 ) and (6.3) give (v, q) 2 DG Cν u 2 h + Cνε 2 3 w 2 h + ν 1 k 1 p ν 1 p 2 F I Cν u 2 h + Cν 1 k 1 p ν 1 k 1 p ν 1 p 2 F I C (u, p) 2 DG. (6.6) Combining (6.5) and (6.6) completes the proo A-priori error estimates In order to derive a-priori error estimates, we let (u, p) be the exact solution o the Stokes system (2.1) and assume that p H 1 (Ω int ) in a domain Ω int Ω containing all the interior aces in F I. Thus, [[p ]] = 0 on F I. We deine Q(h) := Q h + H 1 (Ω int ) and W(h) := V(h) Q(h), equipped with the norm (v, q) DG. From the continuity properties in Theorem 4.1, Theorem 4.3 and the Cauchy- Schwarz inequality, it can be seen that and A h (u, p; v, q) Ck 1 2 (u, p) DG (v, q) DG, (u, p), (v, q) W(h), (6.7) L h (v, q) C [ ν νk δ 1 2 g 2 0, Ω ] 1 2 (v, q) DG, (v, q) W h, (6.8) with constants as in Theorem 4.1 and Theorem 4.3, respectively. Taking into account (6.7), the global in-sup condition in Theorem 6.1, and the non-consistency o the orms Ãh and B h, we obtain straightorwardly the ollowing a-priori bound.

20 20 Stabilized hp-dgfem or incompressible low Corollary 6.2 Let (u, p) be the exact solution o the Stokes system (2.1), with p H 1 (Ω int ), and let (u h, p h ) be its discontinuous Galerkin approximation (2.6) on a geometric edge mesh T = T n,σ n,σ edge or a geometric boundary layer mesh T = Tbl, with a grading actor σ (0, 1) and n levels o reinement. Let the stabilization unctions δ and γ be deined as in (3.2), (3.3) and (3.4), respectively. Then, (u u h, p p h ) DG Ck 1 2 in (u v, p q) DG + C R h (u, p), (v,q) W h with a constant C > 0 that depends on Ω, δ 0, γ 0, and the constants in Property 3.2 and (3.1), but is independent o ν, k, l, n, and the aspect ratio o the anisotropic elements in T. Here, R h (u, p) is the residual R h (u, p) = A h (u, p; w, s) L h (w, s) sup. (w,s) W h (w, s) DG Let us make precise the abstract error bound above or a smooth solution (u, p) H s+1 (Ω) 3 H s (Ω), s 1, on isotropically reined meshes with mesh-size h with possible hanging nodes and or mixed-order elements where l = k 1. In this case, the residual R h (u, p) can be bounded (see Proposition 8.1 o Re. 24) by F R h (u, p) sup {ν u T (ν u) } : [w] ds + {p T (p) }[w] ds F, (w,s) W h (w, s) DG where T and T are the L 2 -projections onto Σ h and Q h, respectively. The Cauchy- Schwarz inequality and standard hp-approximation properties then give Furthermore, ] R h (u, p) C [ν hmin{s,k} 12 u s+1 + ν 12 q s. k s+ 1 2 ] in (u v, p q) DG C [ν hmin{s,k} 12 u s+1 + ν 12 q s (v,q) W h k s 1 2 and thus ] (u u h, p p h ) DG C [ν hmin{s,k} 12 u s+1 k s 1 + ν 12 q s. (6.9) This estimate is optimal in the mesh-size h and suboptimal in k by one power o k in the velocity and by a power k 3/2 in the pressure, respectively. Similarly to Sect. 8 o Re. 24, we obtain a slightly better result on conorming meshes, that is, ] (u u h, p p h ) DG C [ν hmin{s,k} 12 u s+1 + ν 12 q s. (6.10) k s 1 2 We point out that the a-priori error bounds (6.9) and (6.10) hold verbatim or equal-order elements.

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