Exponential convergence in a Galerkin least squares hp‐FEM for Stokes flow

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(1)IMA Journal of Numerical Analysis (2001) 21, 53–80. Exponential convergence in a Galerkin least squares hp-FEM for Stokes flow † ¨ D OMINIK S CH OTZAU School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455, USA. AND. C HRISTOPH S CHWAB‡ Seminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland [Received 6 June 1999 and in revised form 14 March 2000] A stabilized hp-finite element method (FEM) of Galerkin least squares (GLS) type is analysed for the Stokes equations in polygonal domains. Contrary to the standard Galerkin FEM, the GLSFEM admits the implementationally attractive equal-order interpolation in the velocity and the pressure. In conjunction with geometrically refined meshes and linearly increasing approximation orders it is shown that the hp-GLSFEM leads to exponential rates of convergence for solutions exhibiting singularities near corners. To obtain this result a novel hp-interpolation result is proved that allows the approximation of pressure functions in certain weighted Sobolev spaces in a conforming way and at exponential rates of convergence on geometric meshes.. 1. Introduction The Stokes equations describe the motion of incompressible fluids at moderate values of the Reynolds number and are identical in form to the equations of isotropic incompressible elasticity. Despite their linear structure, they give the right setting for studying stability aspects encountered when discretizing the incompressibility constraint in fluid mechanics problems. Therefore, these equations have become an important model problem in CFD, and the ability to solve Stokes-type problems accurately and efficiently is an inalterable requirement in many engineering applications. The classical mixed Galerkin finite element method (FEM) for the Stokes equations leads to a saddle point problem for which stability is only guaranteed if the velocity and pressure FE spaces satisfy the ‘Babuˇska–Brezzi’ condition (see, e.g., Girault & Raviart (1986) and Brezzi & Fortin (1991)). Typically, this stability condition imposes the use of different polynomial orders for the velocity and the pressure spaces (for low-order pairs see Girault & Raviart (1986), Stenberg (1990), Brezzi & Fortin (1991) and the references therein; for high-order and spectral elements see Bernardi & Maday (1992), Stenberg & Suri (1996), Bernardi & Maday (1999), Schwab & Suri (1999) and the references therein). † Email: schoetza@math.umn.edu ‡ Email: schwab@sam.math.ethz.ch. c The Institute of Mathematics and its Applications 2001 .

(2) 54. ¨ D . SCH OTZAU AND C . SCHWAB. Many attractive velocity–pressure combinations, such as equal-order spaces, are unstable which results in wildly oscillating pressures in computations. In Hughes et al. (1986) a Galerkin least squares (GLS) approach was introduced which allows the circumventing of the Babuˇska–Brezzi condition and the use of equalorder spaces. In this method the standard Galerkin form is perturbed by adding weighted residuals of the differential equation which results in enhanced stability properties of the scheme. Franca et al. (1993) and Franca & Stenberg (1991) showed optimal h-version convergence rates for this GLS approach. The idea of stabilizing FEMs by appropriately chosen (and mesh-dependent) weights is by now widely used in the FE community, and has been applied successfully in a variety of problems in fluid flow, elasticity and continuum mechanics. We mention here only Hughes et al. (1986), Brezzi & Douglas (1988), Douglas & Wang (1989), Tobiska & Verfürth (1996) and the references therein. We refer also to the survey article by Franca et al. (1993). However, all these works are concerned with the h-version of the FEM where convergence is achieved by decreasing the mesh width h at a fixed (typically) low approximation order. Recently, there have been several attempts to extend stabilized methods to high-order and spectral elements (see, e.g., Boillat & Stenberg (1994), Gervasio & Saleri (1998), Nabh (1998), Houston et al. (2000) and the references therein). In Boillat & Stenberg (1994) a complete hp-error analysis of the GLSFEM for Stokes flow has been given. Error bounds are derived for the pure p-version where convergence is obtained by increasing the polynomial approximation order on a fixed (quasi-uniform mesh). However, these estimates give only algebraic rates of convergence and are restricted to smooth solutions, which is unrealistic in domains with corners where corner singularities are present (see, e.g., Orlt (1998)). In the present work we deal with these corner singularities and extend the GLSFEM discussed in Franca & Stenberg (1991), Franca et al. (1993) and Boillat & Stenberg (1994) to hp-FEM on geometric meshes. To do so, we consider Stokes flow in non-smooth domains where solutions exhibit singularities at re-entrant corners. The solution regularity is described in terms of countably normed, weighted spaces from Babuˇska & Guo (1988a, 1989), Guo & Babuˇska (1993) and Guo & Schwab (2000). To our best knowledge, we present the first error analysis for the GLSFEM valid for such singular solutions. The reduced regularity near corners and the continuous pressure approximations impose several technical difficulties and require a careful treatment of the elements near vertices of the domain. To numerically resolve the corner singularities we employ geometrically refined meshes combined with linearly increasing approximation orders and show that this hpGLSFEM approach leads to exponential rates of convergence. This result indicates that the performance of the hp-GLSFEM is not downgraded by singular solution components despite the appearance of stabilization terms of higher order in the variational formulations. To prove our main results, a novel hp-interpolation is constructed that approximates pressure functions in certain weighted Sobolev spaces in a conforming way and at exponential rates of convergence on geometric meshes. This result, proved along the lines of Guo & Babuˇska (1986a,b), Babuˇska & Guo (1988b) and Guo (1988) (see also Schwab (1998)), is important from an implementational point of view: in the hp-GLSFEM all field variables can be interpolated by the same H 1 -conforming FE space without losing the.

(3) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 55. exponential convergence property. We remark that our approximation theorem immediately implies corresponding results for the classical Galerkin FEM. The theoretical results of this paper have been announced in Schötzau et al. (1998) and confirmed numerically in Gerdes & Schötzau (1999). Here, we give their detailed proof. The outline of the paper is as follows. In Section 2 we review the Stokes problem and discuss the regularity of solutions near corners. The GLS discretization is presented in Section 3. We analyse the stability properties of the method and derive error bounds valid for solutions exhibiting corner singularities. In Section 4 we prove exponential rates of convergence in the hp-version of the GLSFEM on geometric meshes. In Section 4.4 we show that the same result holds true for the standard Galerkin FEM. The proof of these convergence properties is based on a novel hp-approximation result, the detailed derivation of which is presented in Section 5. Subsequently, standard notations and conventions are followed. We denote by C, C1 , C2 , . . . generic constants, not necessarily identical at different places, but always independent of the polynomial degrees and the mesh widths. 2. The Stokes problem 2.1. The Stokes equations. Let Ω ⊂ R2 be a polygonal and bounded domain with boundary Γ = ∂Ω . The Stokes problem is to find a velocity field u and a pressure p such that −∆ u + ∇ p = f in Ω , ∇ · u = 0 in Ω , u = 0 on Γ .. (2.1) (2.2) (2.3). Here, the right-hand side f ∈ L 2 (Ω )2 is an exterior body force per unit mass. Introducing the spaces H01 (Ω )2 = { u ∈ H 1 (Ω )2 : u|Γ = 0 in the sense of trace}, L 20 (Ω ) = { p ∈ L 2 (Ω ) : ( p, 1) = 0}, with (·, ·) denoting the inner product in L 2 (Ω ), L 2 (Ω )2 or L 2 (Ω )2×2 , the usual mixed formulation of (2.1)–(2.3) is as follows. Find ( u , p) ∈ H01 (Ω )2 × L 20 (Ω ) such that B0 ( u , p; v, q) = F0 ( v , q). ∀( v , q) ∈ H01 (Ω )2 × L 20 (Ω ),. (2.4). with B0 ( u , p; v, q) = (∇ u, ∇ v) − (∇ · v, p) − (∇ · u, q), F0 ( v , q) = ( f, v).. (2.5) (2.6). The system (2.4) has a unique solution ( u , p) in H01 (Ω )2 × L 20 (Ω ) (see, e.g., Brezzi & Fortin (1991) or Girault & Raviart (1986))..

(4) ¨ D . SCH OTZAU AND C . SCHWAB. 56 2.2. Regularity near corners. We assume that the right-hand side f in (2.1) is analytic in Ω . Then it follows from elliptic M A where {A } M regularity theory that the exact solution ( u , p) is analytic in Ω \ ∪i=1 i i i=1 denotes the vertices of Ω . Near a vertex Ai there arise corner singularities which are solution components that are, in polar coordinates (r, ϕ) at Ai , essentially of the form u(r, ϕ) = r λ g(r, ϕ),. p(r, ϕ) = r λ−1 h(r, ϕ). (2.7). for λ 0 and analytic functions g, h. The value of the exponent λ depends only on the angle of the corner at Ai . Decompositions of the solutions into smooth parts and corner singularities have been given, e.g., in Orlt (1998). For closely related elasticity and potential problems Babuˇska and Guo described elliptic regularity in terms of weighted Sobolev spaces (see Babuˇska & Guo (1988a,b, 1989), Guo & Babuˇska (1993), Schwab (1998) and the references therein). In particular, they showed that under the analyticity assumption on the data the solutions belong to the countably normed spaces Bβl (Ω ). These are defined as follows. We denote by A1 , . . . , A M the vertices of the domain Ω . To each Ai we assign a weight βi 0 and store these numbers in the M-tuple β = (β1 , . . . , β M ). We define β ± k := (β1 ± k, . . . , β M ± k) and use the shorthand notation C1 > β > C2 to mean C1 > βi > C2 for i = 1, . . . , M. For ri∗ (x) = min{1, |x − Ai |} we define the weight function Φβ (x) := M ∗ β i i=1 ri (x) and, for integers m l 0, we introduce the semi-norms |u|2. Hβm,l (Ω ). :=. m k=l. |D k u|Φβ+k−l 2L 2 (Ω ) .. . m,l α 2 Here, |D k u|2 = |α|=k |D u| . For m l 0 integers, we denote by Hβ (Ω ) the completion of the set of all infinitely often differentiable functions with respect to the norm. u2. := u2H l−1 (Ω ) + |u|2. u2. :=. Hβm,l (Ω ). Hβm,0 (Ω ). Hβm,l (Ω ). m k=0. ,. l 1,. |D k u|Φβ+k 2L 2 (Ω ) .. D EFINITION 1 Fix an M-tuple β = (β1 , . . . , β M ) and l 0. The countably normed space Bβl (Ω ) consists of all functions u for which u ∈ Hβm,l (Ω ) for m l and |D k u|Φβ+k−l L 2 (Ω ) Cd (k−l) (k − l)!,. k = l, l + 1, . . .. for some constants C > 0, d 1 independent of k. We will also need the weighted spaces Hβs,l (Ω ) for non-integer s = k + θ , 0 < θ < 1, defined by the K-method of interpolation as Hβk+θ,l (Ω ) = (Hβk,l (Ω ), Hβk+1,l (Ω ))θ,∞ . If u ∈ Bβl (Ω ), there holds for any k l u H k+θ,l (Ω ) Cd k+θ −l Γ (k + θ − l + 1). β.

(5) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 57. The spaces Bβl (Ω ) describe the singular behaviour as in (2.7) near corners and the analytic one in the interior. For the Stokes problem (2.1)–(2.3) the corresponding regularity statement is then (see Guo & Schwab (2000)) as follows. There exist numbers 0 β i < 1 depending on the angles at Ai , i = 1, . . . , M, such that for f ∈ Bβ0 (Ω )2 , β < β < 1, there holds u ∈ Bβ2 (Ω )2 ,. p ∈ Bβ1 (Ω ).. (2.8). We remark that in general Bβ2 (Ω )2 ⊂ H 2 (Ω )2 and Bβ1 (Ω ) ⊂ H 1 (Ω ). Throughout, we assume that f ∈ L 2 (Ω )2 ∩ B 0 (Ω )2 for β < β < 1. β. R EMARK 2 Our analysis below is restricted to affine meshes consisting of triangles and parallelograms. To obtain corresponding exponential convergence results on curvilinear elements we presume that a slightly stronger regularity requirement has to be imposed on the solutions, in accordance with the results of Babuˇska & Guo (1988b). 3. GLS discretization 3.1. Finite element spaces. In order to discretize (2.1)–(2.3) we start with an affine and regularly-shaped mesh T that partitions Ω into triangular and quadrilateral elements {K } subject to the following standard assumptions. (i) Ω = ∪ K ∈T K . (ii) Each element K is the image of the reference element Kˆ under an affine element mapping FK , K = FK ( Kˆ ). Kˆ is either the unit square Qˆ = (0, 1)2 or the unit triangle Tˆ = {(x, y) : 0 < x < 1, 0 < y < 1 − x}. (iii) The mesh is regular, i.e. the intersection K ∩ K of two elements K and K is either empty, a single vertex or an entire side. (iv) There exist constants C1 , C2 such that C1 h K ρ K C2 h K with h K denoting the diameter of K and ρ K the diameter of the largest circle that can be inscribed into K . If (iii) is not satisfied, the mesh is called irregular and contains hanging nodes. The mesh width h of T is h = max K ∈T {h K }. T is quasi-uniform if there are constants κ1 , κ2 > 0 such that κ1 h h K κ2 ρ K for all K ∈ T . The space of polynomials of degree k on an interval I = (a, b) is denoted by P k (I ). In two dimensions we introduce the reference polynomial spaces Qˆ k = span{xˆ1κ1 xˆ2κ2 : 0 κ1 , κ2 k}, Pˆ k = span{xˆ1κ1 xˆ2κ2 : 0 κ1 , κ2 , κ1 + κ2 k}. For a (triangular or quadrilateral) element K ∈ T we set Qk (K ) = {q : K → R : q ◦ FK = p| Kˆ , p ∈ Qˆ k }, P k (K ) = {q : K → R : q ◦ FK = p| Kˆ , p ∈ Pˆ k }, and define further. S (K ) = k. Qk (K ) P k (K ). if K is a quadrilateral, if K is a triangle..

(6) ¨ D . SCH OTZAU AND C . SCHWAB. 58. We assign to each element K ∈ T a polynomial degree k K . These degrees are stored in the vector k = {k K : K ∈ T } and we set |k| := max{k K : K ∈ T }. The hp-FE spaces are then S k,l (T ) = {u ∈ H l (Ω ) : u| K ∈ S k K (K ) : K ∈ T },. l = 0, 1.. If the polynomial degree k K = k throughout the mesh, we simply write S k,l (T ). 3.2. GLSFEM with equal-order spaces. Let k be a polynomial degree vector on T . In the GLS method we approximate the velocities and the pressure by equal-order FE spaces VN ⊂ H 1 (Ω )2 and M N ⊂ L 2 (Ω ), respectively. These spaces are given by VN := S k,1 (T )2 ,. M N := S k,l (T ),. l = 0, 1.. (3.1). The index l indicates whether the pressure p ∈ M N is approximated by L 2 - or H 1 conforming FE functions. Less formally, these spaces are often called ‘S k × S k elements’. Throughout, we set VN ,0 := VN ∩ H01 (Ω )2 and M N ,0 := M N ∩ L 20 (Ω ). The spaces in (3.1) satisfy the following inverse estimates (see, e.g., Bernardi & Maday (1992) or Schwab (1998)). L EMMA 3 There exists a constant Cinv > 0 only depending on the shape regularity constants of the mesh such that for all u ∈ VN and p ∈ M N Cinv. h2 K 4 K ∈T k K. D 2 u2L 2 (K ) ∇ u2L 2 (Ω ) ,. Cinv. h2 K 4 K ∈T k K. ∇ p2L 2 (K ) p2L 2 (Ω ) .. We are now ready to define the GLS method. D EFINITION 4 Let VN and M N be the equal-order spaces in (3.1) with k K 2,. ∀K ∈ T .. (3.2). Let α be a user-specified stabilization parameter in the range 0 < α < min{Cinv , |k|2 }. with Cinv defined in Lemma 3.. (3.3). The GLS discretization of (2.1)–(2.3) is: find a discrete velocity field u N ∈ VN ,0 and a discrete pressure p ∈ M N ,0 such that u N , p N ; v, q) = Fα ( v , q) Bα (. ∀( v , q) ∈ VN ,0 × M N ,0 ,. where the perturbed form Bα and the perturbed functional Fα are given by Bα ( u , p; v, q) := (∇ u, ∇ v) − (∇ · v, p) − (∇ · u, q) h2 K (−∆ u + ∇ p, −∆ v + ∇q) L 2 (K ) , −α 4 k K K ∈T Fα ( v , q) := ( f, v) − α. h2 K. 4 K ∈T k K. ( f, −∆ v + ∇q) L 2 (K ) .. (3.4) (3.5).

(7) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 59. R EMARK 5 The assumptions (3.2) and (3.3) guarantee the stability of the method, as will be shown in Proposition 8 below. For linear (or bi-linear) approximations where k K = 1, this stability property holds true as well, provided that M N ⊂ H 1 (Ω ) or that additional terms containing jumps over inter-element boundaries are added to the forms Bα and Fα (see Franca & Stenberg (1991)). R EMARK 6 For α = 0 the GLS method in Definition 4 coincides with the standard Galerkin approach which is considered in Section 4.4 below. u , p) ∈ H01 (Ω )2 × L 20 (Ω ) be the exact solution of (2.1)–(2.3). Since the R EMARK 7 Let ( right-hand side f belongs to L 2 (Ω )2 , we have (−∆ u + ∇ p)| K = f| K ∈ L 2 (K )2 ,. K ∈T.. (3.6). This implies that the GLSFEM is fully consistent without further regularity assumptions on the exact solution, i.e. we have Bα ( u , p; v, q) = Fα ( v , q). ∀( v , q) ∈ VN ,0 × M N ,0 .. (3.7). Consequently, there holds the orthogonality property Bα ( u − u N , p − p N ; v, q) = 0. ∀( v , q) ∈ VN ,0 × M N ,0 .. (3.8). 3.3 Stability The following stability proposition is proved in Boillat & Stenberg (1994) for a constant polynomial degree distribution where k K = k for all K ∈ T . The extension to variable degree distributions k is straightforward. Nevertheless, we present the proof for the sake of completeness. P ROPOSITION 8 We assume (3.2) and (3.3) and set γ (N )2 := α|k|−2 . Then for every v , q) ∈ VN ,0 × M N ,0 such that ( u , p) ∈ VN ,0 × M N ,0 there exists ( Bα ( u , p; v, q) C( u 2H 1 (Ω ) + γ (N )2 p2L 2 (Ω ) ), v 2H 1 (Ω ) + γ (N )2 q2L 2 (Ω ) C( u 2H 1 (Ω ) + γ (N )2 p2L 2 (Ω ) )· The constant C > 0 is independent of the approximation orders k K and the mesh widths hK . R EMARK 9 Although this seems to be the best stability estimate for the GLSFEM known at the moment, it is suboptimal with respect to the approximation order k since γ (N ) appears in the estimate. Hence, error bounds for the pure p-version GLSFEM fall two powers of k short from being quasi-optimal, at least for the pressure approximation; see Boillat & Stenberg (1994). However, the inf-sup condition in Proposition 8 is sufficient to establish exponential rates of convergence in the hp-GLSFEM. In addition, it can be seen that the stability estimate in Proposition 8 deteriorates as α → 0 (or as α → Cinv ), in agreement with the fact that Galerkin FEM with equal-order elements are unstable (see Section 4.4 below). The sensitivity of the hp-GLSFEM with respect to the stabilization parameter α is numerically investigated in Gerdes & Schötzau (1999)..

(8) ¨ D . SCH OTZAU AND C . SCHWAB. 60. Proof. We fix ( u , p) ∈ VN ,0 × M N ,0 . We now proceed in several steps as follows. S TEP 1 We have Bα ( u , p; u, − p) C u 2H 1 (Ω ) + α. h2 K 4 K ∈T k K. ∇ p2L 2 (K ) .. (3.9). To see (3.9), we use Lemma 3 and get Bα ( u , p; u, − p) = ∇ u2L 2 (Ω ) − α. h2 K 4 K ∈T k K. ∆ u 2L 2 (K ) + α. h2 K 4 K ∈T k K. ∇ p2L 2 (K ). h2 α K ∇ u2L 2 (Ω ) + α 1− ∇ p2L 2 (K ) . 4 Cinv k K ∈T K . The assertion (3.9) follows with (3.3) and Poincaré’s inequality. S TEP 2 There exists w ∈ VN ,0 such that w H 1 (Ω ) C p L 2 (Ω ) , Bα ( u , p; −w, 0) −C u 2H 1 (Ω ) + C p2L 2 (Ω ) − C. h2 K ∈T. h 2K K + α k 2K k 4K. (3.10). . ∇ p2L 2 (K ) .. To prove (3.10), we apply Lemma 4.2 in Boillat & Stenberg (1994) observing (3.2): there exists a w ∈ VN ,0 such that w H 1 (Ω ) C p L 2 (Ω ) , (∇ · w, p) . p2L 2 (Ω ). − C p L 2 (Ω ). . 1. h 2K. ∇ p2L 2 (K ) 2 k K K ∈T. 2. .. Thus, using the properties of w, the Cauchy–Schwarz inequality, Lemma 3 and (3.3), we arrive at Bα ( u , p; −w, 0) = Bα ( u , 0; −w, 0) + Bα (0, p; −w, 0) h2 K −C u H 1 (Ω ) w H 1 (Ω ) + (∇ · w, p) − α (∇ p, ∆w) L 2 (K ) 4 k K ∈T K 2 1 2 hK 2 −C u H 1 (Ω ) p L 2 (Ω ) + C p2L 2 (Ω ) − C p L 2 (Ω ) ∇ p L 2 (K ) 2 K ∈T k K 1 2 h2 2 K −C p L 2 (Ω ) α ∇ p . 2 L (K ) 4 K ∈T k K Above, each of the negative terms is estimated with ab 2ε a 2 + weights ε > 0. This yields (3.10).. b2 2ε. with suitably chosen.

(9) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 61. S TEP 3 We now prove the assertion. We define ( v , q) = ( u − δ w, − p) ∈ VN ,0 × M N ,0 where δ > 0 is still at our disposal. Using Step 1 and Step 2 we obtain Bα ( u , p; v, q) (C1 − δC2 ) u 2H 1 (Ω ) + δC3 p2L 2 (Ω ) 2 h2 h h2 α 4K − C4 δ 2K + α 4K ∇ p2L 2 (K ) . + k k k K K K K ∈T. Selecting δ = min. C1 1 α 2C2 , C4 , 3C4 |k|2. . α , |k|2. we get. B( u , p; v, q) C u 2H 1 (Ω ) + Cα|k|−2 p2L 2 (Ω ) .. (3.11). Furthermore, v 2H 1 (Ω ) + α|k|−2 q2L 2 (Ω ) C( u 2H 1 (Ω ) + δ 2 p2L 2 (Ω ) + α|k|−2 p2L 2 (Ω ) ). (3.12) Since δ 2 γ (N )2 , the assertion follows from (3.11) and (3.12).. . R EMARK 10 The GLS discretization in Definition 4 has a unique solution due to Proposition 8. 3.4. Error analysis. To capture singular solution behaviour near the corners we partition the mesh T into two disjoint sets T0 and T1 where T0 consists of all the elements that abut at a corner of the domain, i.e. T0 := {K ∈ T : K ∩ Ai = ∅ for some corner Ai of Ω },. T1 := T \ T0 .. (3.13). Throughout, let γ (N ) be the stability constant from Proposition 8, that is, γ (N )2 := α|k|−2 .. (3.14). L EMMA 11 We assume (2.8), (3.2) and (3.3). Let u I ∈ VN and p I ∈ M N be arbitrary approximations of u and p, correspondingly. Then we have Bα ( u − u I , p − p I ; v, q)2 Cγ (N )−2 ( v 2H 1 (Ω ) + γ (N )2 q2L 2 (Ω ) )(E 12 + E 22 + E 32 ) for all ( v , q) ∈ VN × M N , with E 12 = u − u I 2H 1 (Ω ) + p − p I 2L 2 (Ω ) ,. (3.15). E 22 =. (∆( u − u I )2L 2 (K ) + ∇( p − p I )2L 2 (K ) ),. (3.16). f + ∆ u I − ∇ p I 2L 2 (K ) ·. (3.17). E 32 =. h2 K 4 k K ∈T 1 K h2 K 4 K ∈T 0 k K.

(10) ¨ D . SCH OTZAU AND C . SCHWAB. 62. R EMARK 12 Due to (2.8), u and p are analytic in the domain covered by T1 . Therefore, the term E 2 containing second-order derivatives of u and first-order derivatives of p is well defined. Proof. With the inequality of Cauchy–Schwarz we have |Bα ( u − u I , p − p I ; v, q)| √ ( u − u I ) H 1 (Ω ) v H 1 (Ω ) + 2 v H 1 (Ω ) p − p I L 2 (Ω ) √ + 2 u − u I H 1 (Ω ) q L 2 (Ω ) hK hK +α − ∆( u − u I ) + ∇( p − p I ) L 2 (K ) 2 − ∆ v + ∇q L 2 (K ) 2 kK K ∈T 1 k K hK hK +α f + ∆ u I − ∇ p I L 2 (K ) 2 − ∆ v + ∇q L 2 (K ) , 2 kK K ∈T k K 0. where we used the differential equation in the elements of T0 (see (3.6)). We apply again Cauchy–Schwarz and the inverse inequalities in Lemma 3 for the terms −∆ v +∇q L 2 (K ) . Because of (3.3), this results in |Bα ( u − u I , p − p I ; v, q)|2 C( v 2H 1 (Ω ) + q2L 2 (Ω ) )(E 12 + E 22 + E 32 ). Due to (3.3), we have v 2H 1 (Ω ) + q2L 2 (Ω ) γ (N )−2 ( v 2H 1 (Ω ) + γ (N )2 q2L 2 (Ω ) ), which completes the proof. P ROPOSITION 13 We assume (2.8), (3.2) and (3.3). Let ( u N , p N ) ∈ VN ,0 × M N ,0 be the GLSFEM solution. Then we have, for any ( u I , p I ) ∈ VN ,0 × M N , 1. u − u N H 1 (Ω ) + γ (N ) p − p N L 2 (Ω ) Cγ (N )−1 (E 12 + E 22 + E 32 ) 2. (3.18). with E 1 , E 2 , E 3 defined in (3.15)–(3.17), respectively. R EMARK 14 If ( u , p) ∈ H s (Ω )2 × H s−1 (Ω ) for some s 2, we can choose T0 = ∅ and have E 3 = 0. Proposition 13 then gives optimal convergence rates in the h-version GLSFEM (cf. Franca & Stenberg (1991)). However, in the p-version GLSFEM, the rates from Proposition 13 are suboptimal (cf. Remark 9): in the approximation of the velocity one power of k is lost, whereas the pressure approximation falls two powers of k short of being optimal. In Boillat & Stenberg (1994), it is discussed how this p-version result can be improved in certain situations. We remark that in the presence of re-entrant corners the assumption ( u , p) ∈ H 2 (Ω )2 × H 1 (Ω ) does not hold true anymore and (2.8) is the realistic regularity assumption in that case. However, this requires a careful treatment of the elements in T0 near the corners. Proof. Let ( u I , p I ) ∈ VN ,0 × M N . We denote by Π the L 2 -projection of L 2 (Ω ) onto 2 L 0 (Ω ) given by Π p = p − |Ω1 | Ω p dx. We decompose p I into p I = p1,I + p2,I where. p2,I = |Ω1 | Ω p I dx and p1,I = Π p I . The inf-sup condition in Proposition 8 for ( uN − u, p N − p1,I ) implies the existence of ( v , q) ∈ VN ,0 × M N ,0 such that 1. ( v 2H 1 (Ω ) + γ (N )2 q2L 2 (Ω ) ) 2 C, 1. u N − u I , p N − p1,I ; v, q). ( u N − u I 2H 1 (Ω ) + γ (N )2 p N − p1,I 2L 2 (Ω ) ) 2 Bα (.

(11) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 63. We have Bα ( u N − u I , p N − p1,I ; v, q) = Bα ( u − u I , p − p1,I ; v, q) due to the Galerkin orthogonality (3.8). Thus, Lemma 11 yields 1. u N − u I H 1 (Ω ) + γ (N ) p N − p1,I L 2 (Ω ) Cγ (N )−1 (E 12 + E 22 + E 3 ) 2 with p1,I entering in the terms E 1 , E 2 and E 3 . Hence, the triangle inequality gives the desired bound (3.18) for ( u I , p1,I ). Since p − p1,I L 2 (Ω ) = Π ( p − p I ) L 2 (Ω ) p − p I L 2 (Ω ) ,. ∇ p1,I = ∇ p I , . p1,I can be replaced by p I .. u , p) satisfy (2.8) with βmax = max{βi } ∈ (0, 1). Let T0 consist C OROLLARY 15 Let ( only of quadrilateral elements and we assume (3.2) and (3.3). Then we have for any ( u I , p I ) ∈ VN ,0 × M N the a priori estimate 1. u − u N H 1 (Ω ) + p − p N L 2 (Ω ) Cγ (N )−2 (E 12 + E 22 + E32 ) 2 with E 1 , E 2 defined in (3.15), (3.16) and 2−2β E32 := h K max ( f2L 2 (K ) + u I 2. Hβ2,2 (K ). K ∈T0. + p I 2. Hβ1,1 (K ). (3.19). ).. Proof. Let K ∈ T be a quadrilateral element and let r denote the distance to one of its vertices. Then we claim for β 0 the weighted inverse inequality 4β. π2L 2 (K ). C(β). kK. 2β. hK.

(12) r 2β π 2 dx,. ∀π ∈ S k K (K ).. (3.20). K. To prove (3.20), we may assume that K = (−1, 1)2 and consider the vertex A = (−1, 1); the general case follows by a scaling argument. We recall from Schwab (1998) the onedimensional result that for all β 0 there holds

(13). 1. −1.

(14) q 2 (x) dx C(β)k 2β. 1 −1. q 2 (x)(1 − x 2 )β dx,. ∀q ∈ P k (−1, 1).. A tensor product argument yields easily:

(15)

(16) 4β π 2 dx dy Ck K π 2 (1 − x 2 )β (1 − y 2 )β dx dy K K

(17)

(18) 4β Ck K π 2 (1 − x 2 )2β dx dy + π 2 (1 − y 2 )2β dx dy . (3.21) K. K. The distance from a point (x, y) to A is given by r 2 = (1 + x)2 + (1 − y)2 . We have (1 − x 2 )2β C(1 + x)2β Cr 2β and the same estimate holds true for (1 − y 2 )2β . Referring to (3.21) finishes the proof of (3.20). Corollary 15 is now a direct consequence of Proposition 13 and (3.20). .

(19) ¨ D . SCH OTZAU AND C . SCHWAB. 64. xˆ2. xˆ2 ∆n,σ. 1. 1 K 34. 33. K 14. 13. K 24. 32 12 11 22. 23. 0. 1. xˆ1. 0. 1. xˆ1. n,σ and ∆n,σ with n = 3 and σ = 0·5. F IG . 1. The geometric meshes ∆. 4. Exponential rates of convergence 4.1. Geometric meshes. In order to resolve singular solution behaviour near corners we introduce meshes that are M . We define first the basic geometric geometrically refined towards the vertices {Ai }i=1 meshes on Qˆ = (0, 1)2 . ˆ the (irregular) geometric mesh D EFINITION 16 We fix n ∈ N0 and σ ∈ (0, 1). On Q, ˜ n,σ with n + 1 layers and grading factor σ is created recursively as follows. If n = 0, ∆ ˜ 0,σ = { Q}. ˜ n,σ for n 0, ∆ ˜ n+1,σ is generated by subdividing that square K ˆ Given ∆ ∆ with 0 ∈ K into four smaller rectangles by dividing its sides in a σ : (1 − σ ) ratio. With ˜ n,σ a (regular) geometric mesh ∆n,σ can be associated by removing the hanging each ∆ nodes by additional triangles. n,σ Examples of basic geometric meshes are shown in Fig. 1. The elements {K i j } of ∆ are numbered as indicated there. The elements K 1 j , K 2 j and K 3 j constitute layer j. D EFINITION 17 A geometric mesh Tn,σ in the polygon Ω ⊂ R2 is obtained by mapping the basic geometric meshes ∆n,σ from Qˆ affinely to a vicinity of each convex corner of Ω . At re-entrant corners three suitably scaled copies of ∆n,σ are used (as shown in Fig. 2). The remainder of Ω is subdivided with a fixed affine quasi-uniform and regular partition. In Fig. 2 this local geometric refinement is illustrated. For ease of exposition, we consider only mesh patches that are identically refined with a fixed σ and n, although different grading factors and numbers of layers may be used for the partition of each corner patch. D EFINITION 18 A polynomial degree distribution k on a geometric mesh Tn,σ is called linear with slope µ > 0 if the elemental polynomial degrees are layerwise constant in the geometric patches and given by k j := max(2, "µj#) in layer j, j = 1, . . . , n + 1. In the interior of the domain the elemental polynomial degree is set constant to max(2, "µ(n + 1)#)..

(20) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 65. A3 A4 A2 A1. A5 = A0 F IG . 2. Local geometric refinement near vertices of Ω .. 4.2. An hp-approximation result. On geometric meshes Tn,σ a function in Bβl (Ω ) with l > 1 can be approximated in H l−1 (Ω ) at an exponential rate of convergence (see Guo (1988)). The same result holds true for l = 1, i.e. a pressure function in Bβ1 (Ω ) can be approximated exponentially by a L 2 -conforming FE function in S k,0 (Tn,σ ). For certain values of β this has been proved in Maischak & Stephan (1997) in the context of boundary element methods. However, we are mainly interested in continuous pressure functions where M N = S k,1 (Tn,σ ). Therefore, we prove an H 1 -conforming interpolation results that allows us to approximate Bβ1 (Ω )functions at exponential rates of convergence in L 2 (Ω ). To do so, we modify the arguments given in Guo & Babuˇska (1986a), Guo & Babuˇska (1986b), Guo (1988) and Schwab (1998), and in the elements abutting on a solution singularity we use a weighted Poincaré inequality. Our main hp-approximation result is as follows. T HEOREM 19 Let l = 1, 2 and f ∈ Bβl (Ω ) for some β ∈ (0, 1). Let Tn,σ be a geometric mesh on Ω . Then there exists a µ0 > 0 such that for linearly increasing polynomial degree vectors k with slope µ µ0 there is an approximation Ψ l ∈ S k,1 (Tn,σ ) that satisfies f − Ψ l 2H l−1 ( Q) ˆ +. . h 2K. K ∈(Tn,σ )1. k 4K. | f − Ψ l |2H l (K ) C exp(−bN 1/3 ). with C and b independent of N = dim(S k,1 (Tn,σ )). Here, according to (3.13), we write (Tn,σ )1 for the elements away from the corners. On the elements K A = ∪{K ∈ Tn,σ : K ∩ A = ∅} near a vertex A the approximation Ψ 2 | K A is given by the (piecewise) bilinear nodal. 1 1 interpolation and Ψ | K A = |K A | K A f dx. We have Ψ l H l,l (K β. A). C f H l,l (K ) . β. A.

(21) ¨ D . SCH OTZAU AND C . SCHWAB. 66. Additionally, if f ∈ Bβ2 (Ω ) ∩ H01 (Ω ), the interpolation Ψ 2 can be chosen to satisfy the zero boundary conditions. The proof of Theorem 19 is rather lengthy and technical. We therefore postpone it to Section 5. 4.3. Convergence results. The next result establishes exponential rates of convergence for the hp-GLSFEM, irrespective of whether the pressures are approximated by H 1 - or L 2 -conforming spaces. T HEOREM 20 We assume that the exact solution ( u , p) of the Stokes equations satisfies (2.8) for some β ∈ (0, 1). We discretize these equations with the GLSFEM using the equal-order spaces in (3.1) with l = 0 or l = 1 on a geometric mesh Tn,σ assuming (3.2) and (3.3). Let ( u N , p N ) ∈ VN ,0 × M N ,0 be the discrete solution. Then there exists µ0 = µ0 (σ, β) > 0 such that for linear degree vectors k with slope µ µ0 there holds the error estimate 1. u − u N H 1 (Ω ) + p − p N L 2 (Ω ) C exp(−bN 3 ) with constants C, b > 0 independent of N = dim(VN ) ≈ dim(M N ) (but depending on µ, σ , β and on the stabilization parameter α). 2 = Proof. We assume first that M N = S k,1 (Tn,σ ). We insert the approximations u I = Ψ 2 2 1 (Ψ , Ψ ) and p I = Ψ from Theorem 19 into the abstract error bound (3.19). Using the interpolation properties yields u − u N 2H 1 (Ω ) + p − p N 2L 2 (Ω ) C exp(−bN 1/3 ) +C. M i=1. diam(K Ai )2−2βmax ( f2L 2 (K. Ai ). + u 2. Hβ2,2 (K Ai ). + p2. Hβ1,1 (K Ai ). ).. M are the vertices of Ω and K Here, {Ai }i=1 Ai := {K ∈ Tn,σ : K ∩ Ai = ∅}. The diameters of the sets K Ai are exponentially small, i.e.. diam(K Ai )2−2βmax Cσ 2n(1−βmax ) C exp(−bN 1/3 ). This proves the assertion if M N = S k,1 (Tn,σ ). If now M N = S k,0 (Tn,σ ), we have S k,0 (Tn,σ ) ⊃ S k,1 (Tn,σ ) and the approximation constructed in Theorem 19 can be employed to bound (3.19) in this case as well. R EMARK 21 If the polynomial degree is chosen to be constant throughout the mesh, i.e. VN = S k,1 (Tn,σ )2 and M N = S k,l (Tn,σ ), l = 0, 1, exponential convergence is still obtained by choosing k proportional to the number n of layers. This is due to the fact that for k ∼ n the approximation with linearly increasing polynomial approximation order constructed in the proof of Theorem 19 can still be used to bound the error estimate (3.19). R EMARK 22 The techniques applied in Section 5 to prove Theorem 19 allow us also to establish exponential rates of convergence for geometric meshes with hanging nodes where the local mesh refinement towards corners is obtained by mapping the irregular ˜ n,σ . Of course, other geometric refinement strategies are imaginable. mesh patches ∆.

(22) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 4.4. 67. A remark on Galerkin hp-FEM. For α = 0 the GLSFEM in Definition 4 results in the Galerkin scheme. Find ( u N , p N ) ∈ VN ,0 × M N ,0 such that B0 ( u N , p N ; v, q) = F0 (v, q). ∀( v , q) ∈ VN ,0 × M N ,0. (4.1). (with B0 and F0 defined in (2.5) and (2.6)). However, for the equal-order spaces in (3.1) this approach is highly unstable (see, e.g., Girault & Raviart (1986), Brezzi & Fortin (1991) and Bernardi & Maday (1999)), since the Babuˇska–Brezzi stability condition is not satisfied. To obtain stable discretizations different polynomial orders have to be chosen for the velocities and the pressure spaces: Schwab & Suri (1999) and Stenberg & Suri (1996) show that ‘S k × S k−2 ’ elements are stable on geometric meshes with an inf-sup constant γ (N ) depending algebraically on the approximation order k. Up to γ (N ), these elements lead to quasi-optimal error estimates and hence the approximation in Theorem 19 yields exponential convergence for this Galerkin approach as well. C OROLLARY 23 We assume that the exact solution ( u , p) of the Stokes equations satisfies (2.8) for some β ∈ (0, 1). We discretize these equations with the Galerkin scheme (4.1) using ‘S k × S k−2 ’ elements, i.e. VN = S k,1 (Tn,σ )2 ,. M N = S k−2,l (Tn,σ ),. l = 0, 1,. on a geometric mesh Tn,σ assuming (3.2). Let ( u N , p N ) ∈ VN ,0 × M N ,0 be the discrete solution. Then there exists µ0 = µ0 (σ, β) > 0 such that for linear degree vectors k with slope µ µ0 there holds the error estimate 1. u − u N H 1 (Ω ) + p − p N L 2 (Ω ) C exp(−bN 3 ) with constants C, b > 0 independent of N = dim(VN ) ≈ dim(M N ) (but depending on µ, σ , β). If the polynomial degree is constant throughout the mesh, i.e. VN = S k,1 (Tn,σ )2 and M N = S k−2,l (Tn,σ ), l = 0, 1, exponential convergence is still obtained by choosing k proportional to the number n of layers. 5. Approximation of functions in Blβ (Ω ) for l = 1, 2 This section is devoted to the proof of Theorem 19. Section 5.1 collects some auxiliary facts about the spaces Bβl (Ω ). In Section 5.2, we show that functions in Bβ1 (Ω ) can be approximated continuously at exponential rates of convergence on the basic geometric meshes. Finally, the proof of Theorem 19 is given in Section 5.3. 5.1. Auxiliary results. In this subsection the spaces Hβm,l (Ω ) are equipped with the weight Φβ (x) = r β for some β ∈ [0, 1) with (r, θ ) denoting polar coordinates at the origin. We assume the elements {K } to be affine and regularly shaped..

(23) ¨ D . SCH OTZAU AND C . SCHWAB. 68. We remark that Hβ2,2 (K ) → C(K ) so that point evaluation is allowed for Hβ2,2 functions. For the linear/bilinear interpolation, the following proposition holds. P ROPOSITION 24 Let K be an element with vertex A = 0. Let f ∈ Hβ2,2 (K ) for some β ∈ [0, 1). Then the linear/bilinear interpolation I K f of f at the vertices of K satisfies f − I K f 2 2,2 C| f |2 2,2 and Hβ (K ). Hβ (K ). 2(2−β). f − I K f 2L 2 (K ) Ch K. | f |2. Hβ2,2 (K ). ,. 2(1−β). f − I K f 2H 1 (K ) Ch K. | f |2. Hβ2,2 (K ). .. Proof. The proof can be found in Guo & Babuˇska (1986a) (see also Schwab (1998, Lemma 4.25)). In the following we construct an interpolation in Hβ1,1 (K ) satisfying completely analogous estimates. To this end, we follow the methods of Guo & Babuˇska (1986a,b) (see also Schwab (1998)), i.e. we establish a compactness property and then apply a Bramble– Hilbert type argument. We proceed in three propositions. P ROPOSITION 25 Let Kˆ be the reference element and let f ∈ Hβ1,1 ( Kˆ ) for some β ∈ [0, 1). Then we have r β f ∈ H 1 ( Kˆ ) and r β f 1 ˆ C f 1,1 ˆ . H (K ). Hβ ( K ). Proof. If β = 0, this is trivial. Thus, let β > 0 and assume first Kˆ = Tˆ . We set g := r β f . Then g L 2 (Tˆ ) C f H 1,1 (Tˆ ) and it remains to bound D 1 g. Since β. |D 1 g|2 = (βr β−1 f + r β fr )2 + it is sufficient to show that r β−1 f 2 2. 1 β 2 (r f θ ) C{r 2(β−1) f 2 + r 2β |D 1 f |2 }, r2. L (Tˆ ). C f 2. Hβ1,1 (Tˆ ). . From the inequality of Hardy. (see Babuˇska et al. (1979, Lemma 4.3)) the existence of a constant m follows such that

(24)

(25) r 2(β−1) | f − m|2 dx C r 2β |D 1 f |2 dx, (5.1) ˆ ˆ T T

(26) |m|2 C f 2L 2 (Tˆ ) + r 2β |D 1 f |2 dx · (5.2) Tˆ. Let S = {(r, θ ) : 0 < r < R, 0 < θ < θ0 } be a sector containing the triangle Tˆ . We get with (5.1) and (5.2)

(27)

(28)

(29) r 2(β−1) f 2 dx C r 2(β−1) | f − m|2 dx + C r 2(β−1) |m|2 dx Tˆ Tˆ Tˆ

(30)

(31) 2β 1 2 2 C r |D f | dx + C|m| r 2(β−1) dx.. Tˆ. S. The last integral S r 2(β−1) dx is bounded for β > 0, which proves the assertion for the reference triangle. ˆ we split Qˆ into Tˆ and a remainder T1 . The assertion To obtain the same result for Q, follows now directly from the triangle proof and the fact that dist(0, T1 ) > 0. .

(32) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 69. P ROPOSITION 26 Let Kˆ be the reference element. Then Hβ1,1 ( Kˆ ) is compactly embedded into L 2 ( Kˆ ) for every β ∈ [0, 1). Proof. Note that for β = 0 this is Rellich’s theorem. Let { f j } be a bounded sequence in Hβ1,1 ( Kˆ ). We define g j := r β f j . By Proposition 25, {g j } is bounded in H 1 ( Kˆ ). We choose s > 1 such that 2βs < 2 (this is possible since β ∈ [0, 1)) and let 1 s < ∞ be such that 1s + s1 = 1. By the theorem of Rellich, H 1 ( Kˆ ) is compactly embedded into L r ( Kˆ ) for r ∈ [1, ∞). So there exists a subsequence (again denoted by {g j }) which converges to a function g in L 2s ( Kˆ ). We define f := r −β g. With the inequality of Hölder we get

(33).

(34) f dx = 2. Kˆ. Kˆ. r.

(35). −2β 2. g dx . Kˆ. r. −2βs. 1s

(36) dx. Kˆ. g. 2s . 1 s. dx. < ∞,. since by construction Kˆ r −2βs dx < ∞. Hence, f ∈ L 2 ( Kˆ ). Analogously, we conclude that f j → f in L 2 ( Kˆ ), which finishes the proof. P ROPOSITION 27 Let K be an element with vertex A = 0. Let f ∈ Hβ1,1 (K ) for some. β ∈ [0, 1) and f = |K1 | K f dx. Then there holds 2(1−β). f − f 2L 2 (K ) Ch K. | f |2. Hβ1,1 (K ). ,. f − f 2. Hβ1,1 (K ). C| f |2. Hβ1,1 (K ). .. Proof. We remark that for β = 0 this is Poincaré’s inequality. First, let K be the reference element. We claim that there exists a constant C > 0 such that f H 1,1 (K ) C| f | H 1,1 (K ) + C| f |, β. β. f ∈ Hβ1,1 (K ).. (5.3). We prove (5.3) by contradiction: if (5.3) is not valid, there exists a sequence { f j } in Hβ1,1 (K ) such that f j H 1,1 (K ) = 1 and β. | f j | H 1,1 (K ) + | f j | → 0. for j → ∞.. β. (5.4). By Proposition 26, Hβ1,1 (K ) is compactly embedded into L 2 (K ). We can choose a subsequence (again denoted by { f j }) which is a Cauchy sequence in L 2 (K ). With (5.4) it follows that { f j } is also a Cauchy sequence in Hβ1,1 (K ). Hβ1,1 (K ) is complete and there exists f ∈ Hβ1,1 (K ) with f j → f in Hβ1,1 (K ). Taking into account (5.4) we get | f | H 1,1 (K ) | f − f j | H 1,1 (K ) + | f j | H 1,1 (K ) → 0, β. that is | f |2. Hβ1,1 (K ). =. β. K. β. j → ∞,. r 2β |D 1 f |2 dx = 0. Therefore, |D 1 f | = 0 and f is constant on K .. Since | f j | → 0 ( j → ∞), it follows that f = 0 and f = 0. But this is a contradiction to 1 = lim j→∞ f j H 1,1 (K ) = f H 1,1 (K ) . We conclude that (5.3) must be valid. β. β. Applying (5.3) to f − f yields the assertion on the reference element. A scaling argument finishes the proof. .

(37) ¨ D . SCH OTZAU AND C . SCHWAB. 70. R EMARK 28 Proposition 27 is a weighted Poincaré inequality. The same result has been obtained in Maischak & Stephan (1997) for values of β in the range β ∈ 12 , 1 . R EMARK 29 Let T be a regularly-shaped and affine mesh and K A = ∪{K ∈ T : K ∩ A = ∅} be the union of elements abutting on the vertex A = 0. Setting f = |K1A | K A f dx we get with the same techniques f − f L 2 (K A ) Cdiam(K A )2(1−β) f 2. Hβ1,1 (K A ). ,. f − f H 1,1 (K β. C f 2. A). Hβ1,1 (K A ). .. We need the following versions of weighted trace theorems. L EMMA 30 Let K be an element with vertex A = 0 and β ∈ [0, 1). Let γ be a side of K such that 0 ∈ γ . We have:. 1−2β (i) if f ∈ Hβ1,1 (K ) and f = |K1 | K f dx = 0, then f 2L 2 (γ ) Ch K | f |2 1,1 ; (ii) if f ∈. Hβ2,2 (K ). Hβ (K ). and f = 0 at the vertices of K , then 3−2β. f 2L 2 (γ ) Ch K. | f |2. Hβ2,2 (K ). ,. 1−2β. f 2H 1 (γ ) Ch K. | f |2. Hβ2,2 (K ). .. If β = 0, γ can be an arbitrary side of K . Proof. The proof of (ii) can be found in Schwab (1998, Lemma 4.55). (i) follows analogously, using Proposition 27. The next results are concerned with the extension of given polynomial trace functions into a domain. L EMMA 31 The following extensions are possible: (i) Let K be a triangle or a quadrilateral and let γ be a side of K . Let v be a polynomial in P k (γ ) which vanishes at the endpoints of γ . Then there exists V ∈ S k (K ) such that V ≡ v on γ , V ≡ 0 on the other sides of K , V 2L 2 (K ) Ch K v2L 2 (γ ) and. V 2H 1 (K ) Ch K |v|2H 1 (γ ) . (ii) Let K be a quadrilateral element and let γ be a side of K . Let v be a polynomial in P k (γ ). Then there exists V ∈ Qk (K ) such that V ≡ v on γ , V ≡ 0 on the side opposite to γ and V 2L 2 (K ) Ch K v2L 2 (γ ) . Further, if γ1 is a side adjacent to γ , then V L ∞ (γ1 ) V L ∞ (γ ) . (iii) Let K be a quadrilateral element, γ 1 = [A, B] and γ 2 = [B, C] two adjacent sides of K with common endpoint B. Let v1 and v2 be polynomials in P k (γ1 ) and P k (γ2 ), respectively. Let v1 (A) = v2 (C) = 0 and v1 (B) = v2 (B). Then there exists a polynomial V in Qk (K ) such that V ≡ v1 on γ1 , V ≡ v2 on γ2 , V ≡ 0 on the other sides of K and V 2L 2 (K ) Ch K k 2 (v1 2L 2 (γ ) + v2 2L 2 (γ ) ). 1. 2. Proof. (i) and (ii) are easy modifications of Lemma 4.55 in Schwab (1998). We verify (iii) on the reference square as follows. Let γ1 = {(0, y) : 0 < y < 1} and γ2 = {(x, 0) : 0 < x < 1}. We define V (x, y) = v1 (y)(1 − x) + [v2 (x) − v2 (0)(1 − x)](1 − y). In addition, V (0, y) = v1 (y) and V (x, 0) = v2 (x). Further,

(38) 1

(39) 1

(40) 1

(41) 1 V (x, y)2 dx dy C v12 (y) dy + C v22 (x) dx + Cv22 (0). 0. 0. 0. 0.

(42) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 71. The last term can be bounded using the one-dimensional inverse inequality, i.e. v2 2L ∞ (γ2 ) Ck 2 v2 2L 2 (γ ) , which yields the desired result. 2. 5.2 Approximation on the basic geometric meshes ˆ In this subsection we address the continuous approximation of a function f in Bβl ( Q), l = 1, 2, with weight Φβ (x) = |x|β for some β ∈ (0, 1) on a basic geometric mesh ∆n,σ n,σ . Note that if K ∈ ∆n,σ or K ∈ ∆ n,σ belongs to layer with underlying irregular mesh ∆ j, we have √ h K = 2σ n , j = 1, (5.5) C1 σ n+2− j h K C2 σ n+2− j , 2 j n + 1. The distance to the origin is bounded by dist(0, K ) Cσ n+2− j ,. 2 j n + 1.. (5.6). n,σ be layerwise constant, i.e. k = Let the polynomial degree vector k on ∆n,σ and ∆ n+1 {k K i j = k K l =: k j } j=1 for some k j 2. ij. ˆ for β ∈ (0, 1) and {k j } be a layerwise constant L EMMA 32 Let l = 1, 2, f ∈ Bβl ( Q) n,σ away from the origin there polynomial degree distribution. On the elements {K i j } of ∆ is a polynomial ϕ K i j of degree k j in each variable such that f = ϕ K i j at the vertices of K i j and 2s j Γ (k j − s j + 1) ρ 2 2(n+2− j)(l−m−β) | f − ϕ K i j | H m (K i j ) Cσ f 2 s j +3,l Γ (k j + s j + 3 − 2m) 2 Hβ (K i j ) (5.7) for any 1 i 3, 2 j n + 1, 0 m 2 and s j ∈ [1, k j ]. Above, ρ = max(1, 1−σ σ ). Proof. For l = 2 this is proved in Guo & Babuˇska (1986a) (see also Schwab (1998, Section 4.5.3)). The proof for l = 1 is analogous. We now define the auxiliary space Q k,1 (∆n,σ ) := { f K ∈ ∆n,σ }.. Qk K (K ),. ∈ H 1 (Ω ) :. f |K ∈. ˆ with weight Φβ (x) = |x|β for β ∈ (0, 1). T HEOREM 33 Let l = 1, 2 and f ∈ Bβl ( Q). Then for every layerwise constant polynomial degree distribution {k j 2}n+1 j=1 on ∆n,σ with k j+1 k j there exists an approximation Ψ l ∈ Q k,1 (∆n,σ ) such that f − Ψ l 2H l−1 ( Q) ˆ + +Ck24. 3 n+1 j=2 i=1. σ. . h 2K. K ∈(∆n,σ )1. k 4K. | f − Ψ l |2H l (K ) Ck24 σ 2n(1−β) f 2 l,l. − s j + 1) ρ 2s j f 2 s j +3,l Γ (k j + s j − 1) 2 Hβ (K i j ). 2(n+2− j)(1−β) Γ (k j. Hβ (K 11 ). (5.8).

(43) ¨ D . SCH OTZAU AND C . SCHWAB. 72. where ρ = max(1, 1−σ σ ), γ 0 and s j ∈ [1, k j ]. On the first element K 11 , the approximation Ψ 2 | K 11 is given by the bilinear interpolation in Proposition 24 and. ˆ H 1 ( Q), ˆ the approximation Ψ 2 can Ψ 1 | K 11 = |K111 | K 11 f dx. Additionally, if f ∈ Bβ2 ( Q)∩ 0 be chosen to satisfy the zero boundary conditions. (In (5.8), we use the notation in (3.13) with respect to the vertex A = 0.) The estimate (5.8) for l = 1 is the main novelty in the result of Theorem 33. For l = 2 estimates of this type can be found, e.g., in Guo & Babuˇska (1986a,b), Guo (1988) and Schwab (1998, Section 4.5.3). We present the proof for l = 1 in full detail and sketch the case l = 2 for the convenience of the reader. Proof. We prove the cases l = 1 and l = 2 separately. 1. Approximation of the pressure. We establish (5.8) for a pressure p in Bβ1 (Ω ). n,σ . On the A first approximation of p is given by Lemma 32 on the irregular mesh ∆ elements {K i j } away from the origin there exist polynomials ϕ K i j of degree k j on K i j such that p = ϕ K i j at the vertices of K i j and such that (5.7) holds with l = 1. Observing (5.5) yields p − ϕ K i j 2L 2 (K. + h 2K i j | p − ϕ K i j |2H 1 (K ) + h 4K i j | p − ϕ K i j |2H 2 (K ) ij ij 2s j Γ (k j − s j + 1) ρ Cσ 2(n+2− j)(1−β) p H s j +3,1 (K ) . ij Γ (k j + s j − 1) 2 ij). In the smallest element K 11 at the origin we choose ϕ K 11 := Proposition 27 we get p − ϕ K 11 2L 2 (K. 2(1−β). 11 ). Ch K 11. p2. Hβ1,1 (K 11 ). 1 |K 11 |. = Cσ 2n(1−β) p2. K 11. Hβ1,1 (K 11 ). (5.9). p dx. From. .. (5.10). n,σ by splitting the elements K i j ∈ The regular geometric mesh ∆n,σ is obtained from ∆ n,σ for i = 2, 3 and 3 j n + 1 into three triangular elements K 1 , K 2 and K 3 . We ∆ ij ij ij write ϕ K l = ϕ K i j | K l and remark that ϕ K l ∈ Qk j (K il j ). (In general, ϕ K l ∈ P k j (K il j ).) ij ij ij ij Note also that C1 h K i j h K l C2 h K i j . ij Obviously, the elementwise approximation {ϕ K : K ∈ ∆n,σ } is discontinuous over inter-element boundaries and, generally, ϕ K (Q) = p(Q) at an irregular node Q of n,σ . We remove the inter-element discontinuities and K considered as an element in ∆ construct in the following a continuous approximation {Ψ K : K ∈ ∆n,σ }. A SSERTION 34 There exists Ψ = {Ψ K } K ∈∆n,σ ∈ Q k,1 (∆n,σ ) with K ∈∆n,σ. p − Ψ K 2L 2 (K ) + . Ck24 h K 11. 2(1−β). p2. . h 2K. K ∈(∆n,σ )1. k 4K. Hβ1,1 (K 11 ). +. 2 . | p − Ψ K |2H 1 (K ). . n,σ )1 i=0 K ∈(∆.  2  h 2i K | p − ϕ K | H i (K ) .. (5.11).

(44) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 73. F IG . 3. The first two layers of ∆n,σ near the origin.. The estimate (5.8) is then a direct consequence of Assertion 34, (5.9) and (5.10). It remains to prove (5.11): we set Ψ K 11 = ϕ K 11 and construct in a first step the approximation Ψ on the first two layers of ∆n,σ near the origin using the notations in Fig. 3. Remember that ϕ K 32 (B) = ϕ K 22 (B) = ϕ K 12 (B) = p(B). We define w12 := (ϕ K 12 − ϕ K 11 )|γ2 w32 := (ϕ K 32 − ϕ K 11 )|γ1. polynomial of degree k2 on γ2 , polynomial of degree k2 on γ2 .. By Lemma 31(ii) there exist polynomials W12 and W32 of degree k2 in each variable such that W12 2L 2 (K. 12 ). Ch K 12 w12 2L 2 (γ ) ,. W32 2L 2 (K. 2. 32 ). Ch K 32 w32 2L 2 (γ ) , 1. W12 |γ2 = w12 , W32 |γ1 = w32 , W12 ≡ 0 on γ6 and W32 ≡ 0 on γ5 . By construction, it holds that W12 2L 2 (K. 12 ). Ch K 12 w12 2L 2 (γ ) Ch K 12 ϕ K 12 − ϕ K 11 2L 2 (γ. 2). 2. Ch K 12 p − ϕ K 12 2L 2 (γ ) + Ch K 12 p − ϕ K 11 2L 2 (γ ) . 2. 2. By Lemma 30(ii) with β = 0 we get p − ϕ K 12 2L 2 (γ ) Ch 3K 12 | p − ϕ K 12 |2H 2 (K 2. 12 ). .. Moreover, Lemma 30(i) gives 1−2β. p − ϕ K 11 2L 2 (γ ) h K 11 p − ϕ K 11 2 2. Hβ1,1 (K 11 ). 1−2β. h K 11 p H 1,1 (K β. 11 ). ..

(45) ¨ D . SCH OTZAU AND C . SCHWAB. 74. Combining the above estimates and observing (5.5) results in W12 2L 2 (K. 12 ). Ch 4K 12 | p − ϕ K 12 |2H 2 (K. 2−2β. 12 ). + h K 11 p2. Hβ1,1 (K 11 ). .. (5.12). The analogous estimate holds for W32 . We define now, on the elements K 12 and K 32 , Ψ K 12 := ϕ K 12 − W12 ,. Ψ K 32 := ϕ K 32 − W32 .. It holds that Ψ K 12 |γ2 = ϕ K 11 |γ2 and Ψ K 32 |γ1 = ϕ K 11 |γ1 . Hence, Ψ K 11 , Ψ K 12 and Ψ K 32 are continuous across γ1 and γ2 and ϕ K 11 (B) = Ψ K 32 (B) = Ψ K 12 (B). The triangle inequality yields p − Ψ K 12 2L 2 (K. C p − ϕ K 12 2L 2 (K. 12 ). 12 ). + CW12 2L 2 (K. 12 ). .. Further, with inverse inequalities, h 2K 12 k 4K 12. | p − Ψ K 12 |2H 1 (K. Ch 2K 12 | p − ϕ K 12 |2H 1 (K. 12 ). 12 ). + CW12 2L 2 (K. 12 ). .. From (5.12) we get thus p − Ψ K 12 2L 2 (K 2(1−β). Ch K 11. 12. + ). p2. h 2K 12 k 4K 12. Hβ1,1 (K 11 ). | p − Ψ K 12 |2H 1 (K. +C. 2 i=0. 12 ). 2 h 2i K 12 | p − ϕ K 12 | H i (K. 12 ). .. (5.13). The analogous estimate holds true for Ψ32 . Next, we construct Ψ K 22 . We define v3 := (Ψ K 32 − ϕ K 22 )|γ3 v4 := (Ψ K 12 − ϕ K 22 )|γ4. polynomial of degree k2 on γ3 , polynomial of degree k2 on γ4 .. We have by construction and the properties of {ϕ K i j }, W32 and W12 v3 (A) = 0,. v4 (C) = 0,. v3 (B) = v4 (B) = ϕ K 11 (B) − ϕ K 22 (B).. Lemma 31(iii) can be applied and yields the existence of a polynomial W22 ∈ Qk2 (K 22 ) with W22 |γ3 = v3 , W22 |γ4 = v4 , W22 ≡ 0 on γ7 , γ8 and W22 2L 2 (K. 22 ). Ck22 h K 22 v3 2L 2 (γ ) + Ck22 h K 22 v4 2L 2 (γ ) . 3. 4. (5.14). We define, on K 22 , Ψ K 22 := ϕ K 22 + W22 . We have Ψ K 22 |γ3 = Ψ K 32 |γ3 and Ψ K 22 |γ4 = Ψ K 12 |γ4 , such that by construction Ψ K 11 , Ψ K 12 , Ψ K 22 and Ψ K 32 are a continuous approximation of p in the first two layers..

(46) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 75. To estimate p − Ψ K 22 L 2 (K 22 ) , we bound W32 2L 2 (γ ) using Lemma 31(ii) and inverse 3 inequalities W32 2L 2 (γ ) h K 22 W32 2L ∞ (γ3 ) h K 22 W32 2L ∞ (γ1 ) 3. k22 W32 2L 2 (γ ) = Ck22 W32 2L 2 (γ ) = Ck22 w32 2L 2 (γ ) 1 1 1 h K 22. Ch K 22. Ck22 ϕ K 32 − ϕ K 11 2L 2 (γ . Ck22 ϕ K 32. −. p2L 2 (γ ) 1. 1). + Ck22 p − ϕ K 11 2L 2 (γ ) . 1. With Lemma 30(ii) (β = 0) and Lemma 30(i) we get W32 2L 2 (γ ) Ck22 h 3K 32 | p − ϕ K 32 |2H 2 (K 3. 1−2β. 32 ). + Ck22 h K 11 p2. Hβ1,1 (K 11 ). .. (5.15). Using again Lemma 30(ii) with β = 0 and (5.15) gives v3 2L 2 (γ ) = Ψ K 32 − ϕ K 22 2L 2 (γ ) = ϕ K 32 − W32 − ϕ K 22 2L 2 (γ 3. . 3 3) 2 2 C p − ϕ K 32 L 2 (γ ) + C p − ϕ K 22 L 2 (γ ) + W32 2L 2 (γ ) 3 3 3 3 2 3 2 Ch K 32 | p − ϕ K 32 | H 2 (K ) + Ch K 22 | p − ϕ K 22 | H 2 (K ) 32 22 2 3 2 2 1−2β +Ck2 h K 32 | p − ϕ K 32 | H 2 (K ) + Ck2 h K 11 p − ϕ K 11 2 1,1 . 32 H (K 11 ) β. We see with (5.5) that k22 h K 22 v3 2L 2 (γ ) 3 Ck24 h 4K 32 | p − ϕ K 32 |2H 2 (K. 32 ). + h 4K 22 | p − ϕ K 22 |2H 2 (K. 2−2β. + h K 11 p2. 22 ). Hβ1,1 (K 11 ). (5.16) .. The analogous estimate holds true for k22 h K 22 v4 2L 2 (γ ) . The desired bound for Ψ K 22 4 follows now by the triangle inequality, inverse estimates, (5.14) and (5.16): p − Ψ K 22 2L 2 (K. 22. + ). h 2K 22 k 4K 22. | p − Ψ K 22 |2H 1 (K. 22 ). C p − ϕ K 22 2L 2 (K ) + Ch 2K 22 | p − ϕ K 22 |2H 1 (K ) + CW22 2L 2 (K ) 22 22 22 Ck24 p − ϕ K 22 2L 2 (K ) + h 2K 22 | p − ϕ K 22 |2H 1 (K ) 22. +. 3 i=1. 22. h 4K i,2 | p − ϕ K i,2 |2H 2 (K. 2−2β. i,2. + h K 11 p − ϕ K 11 2 ). Hβ1,1 (K 11 ). .. Note that Ψ K 32 = ϕ K 32 on γ5 ,. Ψ K 22 = ϕ K 22 on γ7 , γ8 ,. Ψ K 12 = ϕ K 12 on γ6 .. (5.17).

(47) ¨ D . SCH OTZAU AND C . SCHWAB. 76. 1 K 2, j−1. 2 K 2, j. 1 K 2, j. K 1, j−1. 3 K 2, j. 3 K 2, j. Case 1 ( j > 3). Case 2 ( j > 2). Case 3 ( j > 2). 2 K 2, j 1 K 2, j. K 1, j. 2 K 2, j. Case 4 ( j > 2). Case 5 ( j > 2) F IG . 4. The five basic correction cases.. The construction of the continuous approximation Ψ in the elements away from the origin is analogous to Guo & Babuˇska (1986a). We present the details for the sake of completeness. Consider 3 j n + 1 and i = 2, 3. Let vil j be linear on K il j (l = 1, 2, 3) such that n,σ ), vil j (Q) = p(Q) − ϕ K i j (Q) = p(Q) − ϕ K l (Q) at the irregular node Q (in ∆ ij. vil j (P). =0. at the other nodes P of K il j .. We define ϕ˜ K l := ϕ K l + vil j . ϕ˜ K l is equal to p at all the vertices of K il j . Proposition 24 ij ij ij with β = 0 yields p − ϕ˜ K l 2L 2 (K l ) Ch 4K i j | p − ϕ K l |2H 2 (K l ) ,. (5.18). p − ϕ˜ K l 2H 1 (K l ) Ch 2K i j | p − ϕ K l |2H 2 (K l ) ,. (5.19). p − ϕ˜ K l 2H 2 (K l ) C| p − ϕ K l |2H 2 (K l ) .. (5.20). ij. ij. ij. ij. ij. ij. ij. ij. ij. ij. ij. ij. We have ϕKi j = p. at the vertices of K i j ,. i = 1, 3 j n + 1,. ϕ˜ K l = p. at the vertices of K il j ,. i = 1, 2, 3 j n + 1, l = 1, 2, 3.. ij. The jumps over a common edge of two neighbouring elements can be corrected without affecting the polynomials {ϕ K i j } or {ϕ˜ K l } on the remaining sides. By symmetry, we have ij to consider the five correction cases shown in Fig. 4 . We present the details only for cases 2 and 3; the remaining ones are completely analogous..

(48) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 77. C ASE 2 ϕ K 1, j−1 and ϕ˜ K 2 coincide with p at the vertices of K 1, j−1 and K 22 j , respectively. 2j We define w := (ϕ K 1, j−1 − ϕ˜ K 2 )|γ 2j. where γ = K 1, j−1 ∩ K 22 j . w is a polynomial of degree k j on γ . By Lemma 31(i) there is a polynomial W in P k j (K 22 j ) ⊆ Qk j (K 22 j ) such that W ≡ w on γ , W ≡ 0 on the other sides of K 22 j and W 2 2 2 Ch K 2 j w2L 2 (γ ) . We get, using Lemma 30(ii) with β = 0 L (K 2 j ). and (5.20), W 2L 2 (K 2 ) Ch K 2 j ϕ K 1, j−1 − ϕ˜ K 2 2L 2 (γ ) 2j. 2j. Ch K 2 j p . − ϕ K 1, j−1 2L 2 (γ ). Ch K 2 j h 3K 1, j−1 |ϕ K 1, j−1. −. + Ch K 2 j ϕ˜ K 2 − p2L 2 (γ ) 2j. p|2H 2 (K 1, j−1 ). +Ch K 2 j h 3K 2 j |ϕ˜ K 2 − p|2H 2 (K 2 ) . 2j. 2j. We conclude with (5.5) W 2L 2 (K 2 ) Ch 4K 1, j−1 |ϕ K 1, j−1 − p|2H 2 (K 2j. 1, j−1 ). + Ch 4K 2 j |ϕ K 2 j − p|2H 2 (K. 2j). .. (5.21). We define Ψ K 2 := ϕ˜ K 2 + W in K 22 j , 2j. Ψ K 1, j−1 := ϕ K 1, j−1 in K 1, j−1 .. 2j. Clearly, Ψ K 2 = Ψ K 1, j−1 on γ and we get from (5.18)–(5.20) and (5.21) with the triangle 2j inequality and inverse inequalities h2 2 p − Ψ K 2 L 2 (K 2 ) + 2j. 2j. K2 j. k4 2 K2 j. | p − Ψ K 2 j |2H 1 (K 2. 2j). C p − ϕ˜ K 2 2L 2 (K 2 ) + h 2K 2 | p − ϕ˜ K 2 | H 1 (K 2 ) + CW 2L 2 (K 2 2j. . Ch 4K 2 j | p. 2j. +Ch 4K 2 j |ϕ K 2 j − p|2H 2 (K. 2j. 2j. − ϕ K 2 |2H 2 (K 2 ) 2j 2j. + Ch 41, j−1 |ϕ K 1, j−1. 2j). 2j). 2j. −. p|2H 2 (K 1, j−1 ). .. (5.22). 3 C ASE 3 Here, ϕ˜ K 1 = ϕ˜ K = p at the endpoints of γ = K 21 j ∩ K 23 j . We define w := 2j 2j. (ϕ˜ K 1 − ϕ˜ K 3 )|γ . Since w = v21 j − v32 j is linear, w ≡ 0 on γ and the piecewise polynomial 2j. 2j. {ϕ˜ K l }l=1,3 is continuous across γ . With Ψ K 1 = ϕ˜ K 1 and Ψ K 3 = ϕ˜ K 3 it holds that (cf. 2j. 2j. 2j. 2j. 2j.

(49) ¨ D . SCH OTZAU AND C . SCHWAB. 78 (5.18)). p − Ψ K 1 2L 2 (K 2 ) + p − Ψ K 3 2L 2 (K 3 2j. +. h2 1 K2 j k4 1. | p − Ψ K 1 |2H 1 (K 1 ) + 2j. K2 j. 2j). 2j. 2j. 2j. Ch 4K i j | p − ϕ K 2 j |2H 2 (K. h2 3 K2 j k4 3. | p − Ψ K 3 |2H 1 (K 3 2j. K2 j. 2j). 2j). .. (5.23). Referring to (5.13), (5.17), (5.22) and (5.23) proves (5.11). ˆ In Guo & 2. Approximation of the velocity. We establish (5.8) for a function u ∈ Bβ2 ( Q). Babuˇska (1986a) an approximation Ψ ∈ Q k,1 (∆n,σ ) is constructed that satisfies u − Ψ 2H 1 ( Q) ˆ r.h.s. of (5.8).. (5.24). We refer also to Schwab (1998, Section 4.5.3). The proof of (5.24) is again based on Lemma 32, which provides a discontinuous interpolation on the irregular mesh n,σ . Considering the correction cases in Fig. 4 a C 0 -conforming approximation Ψ ∆ is constructed on ∆n,σ . On the first element K 11 near the vertex, Ψ is the bilinear ˆ Ψ can be constructed interpolation in Proposition 24. Moreover, if u ∈ H01 ( Q), in such a way that the zero boundary conditions are satisfied, the additional terms h 2K 2 K ∈(∆n,σ )1 4 |u − Ψ | H 2 (K ) in the left-hand side of (5.8) can be controlled with the same kK. techniques used in the approximation of the pressure. This finishes the proof of Theorem 33.. . An immediate consequence of Theorem 33 is (see Guo & Babuˇska (1986a), Guo (1988) and Schwab (1998, Section 4.5.3)) the following corollary. ˆ for some β ∈ (0, 1). Then there exists a C OROLLARY 35 Let l = 1, 2 and f ∈ Bβl ( Q) µ0 > 0 such that for linearly increasing polynomial degree vectors k with slope µ µ0 there is an approximation Ψ l ∈ S k,1 (∆n,σ ) that satisfies f − Ψ l 2H l−1 ( Q) ˆ +. . h 2K. K ∈(∆n,σ )1. k 4K. | f − Ψ l |2H l (K ) C exp(−bN 1/3 ). with C and b independent of N = dim(S k,1 (∆n,σ )). On the first element K 11 near the origin Ψ 2 | K 11 is given by the bilinear interpolation in Proposition 24 and we have Ψ 1 | K 11 =. 1 2 ˆ 1 ˆ 2 |K 11 | K 11 f dx. Additionally, if f ∈ Bβ ( Q)∩ H0 ( Q), the approximation Ψ can be chosen to satisfy the zero boundary conditions. 5.3 Approximation on polygons: proof of Theorem 19 P ROOF OF T HEOREM 19 Tn,σ is obtained by mapping affinely up to three geometric mesh patches ∆n,σ to a neighbourhood of each corner (cf. Fig. 2). Locally, we can construct Ψ in each of these parallelogram patches according to Corollary 35 (a generalization of.

(50) EXPONENTIAL CONVERGENCE IN A GLS HP - FEM FOR STOKES FLOW. 79. Theorem 33 or Corollary 35 to parallelograms can be established straightforwardly, cf. Guo & Babuˇska (1986a), Guo (1988) and Schwab (1998)). Near a re-entrant corner A the approximation Ψ is constructed over three geometric patches with the techniques presented in the proof of Theorem 33 (the analogue of Corollary 35 holds true in that case). Here, the continuous pressure approximation is in the first elements K A = ∪{K : K ∩ A = ∅}. near A chosen as Ψ = |K1A | K A f dx, f ∈ Bβ1 (Ω ) (see also Remark 29). In the interior Ωint of the domain the polynomial approximation order is set to k = "µ(n + 1)# on a fixed quasi-uniform partition Tq . Bβl -functions behave analytically away from the vertices and it follows thus from standard approximation theory that inf. Ψ ∈S k,1 (Tq ). f − Ψ 2H l−1 (Ω. int. + ). h2 K K ∈Tq. k 4K. | f − Ψ |2H l (K ). C exp(−bk).. Joining continuously together the local and the interior approximations as in the proof of Theorem 33 gives the desired approximation Ψ . Acknowledgements The authors are grateful to Prof. Rolf Stenberg for many helpful discussions and suggestions and Prof. Klaus Gerdes for the help in implementing the GLS method in Fortran 90; see Gerdes & Schötzau (1999). R EFERENCES BABU Sˇ KA , I. & G UO , B. 1988a The hp-version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25, 837–861. BABU Sˇ KA , I. & G UO , B. 1988b Regularity of the solution of elliptic problems with piecewise analytic data I. SIAM J. Math. Anal. 19, 172–203. BABU Sˇ KA , I. & G UO , B. 1989 Regularity of the solution of elliptic problems with piecewise analytic data II. SIAM J. Math. Anal. 20, 763–781. ¨ BABU Sˇ KA , I., K ELLOG , R., & P ITK ARANTA , J. 1979 Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 447–471. B ERNARDI , C. & M ADAY , Y. 1992 Approximations Spectrales De Problèmes Aux Limites Elliptiques. Paris: Springer. B ERNARDI , C. & M ADAY , Y. 1999 Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Methods Appl. Sci. 9, 395–414. B OILLAT , E. & S TENBERG , R. 1994 An hp error analysis of some Galerkin least squares methods for the elasticity equations. Technical Report 21-94. Helsinki University of Technology. B REZZI , F. & D OUGLAS J R ., J. 1988 Stabilized mixed methods for the Stokes problem. Numer. Math. 53, 225–236. B REZZI , F. & F ORTIN , M. 1991 Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15, New York: Springer. D OUGLAS J R , J. & WANG , J. 1989 An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52, 495–508. F RANCA , L. & S TENBERG , R. 1991 Error analysis of some Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28, 1680–1697..

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