Hamburger Beiträge zur Angewandten Mathematik

Transcription

1 Hamburger Beiträge zur Angewandten Matematik Optimal L 2 velocity error estimates for a modified pressure-robust Crouzeix-Raviart Stokes element A. Linke, C. Merdon und W. Wollner Tis preprint is also available as WIAS-Preprint No 2140 Nr July 2015

2

3 Optimal L 2 velocity error estimate for a modified pressure-robust Crouzeix Raviart Stokes element A. Linke C. Merdon W. Wollner July 21, 2015 Abstract Recently, a novel approac for te robust discretization of te incompressible Stokes equations was proposed tat sligtly modifies te nonconforming Crouzeix Raviart element suc tat its velocity error becomes pressure-independent. Te modification results in an O() consistency error tat allows straigtforward proofs for te optimal convergence of te discrete energy norm of te velocity and of te L 2 norm of te pressure. However, toug te optimal convergence of te velocity in te L 2 norm was observed numerically, it appeared to be nontrivial to prove. In tis contribution, tis gap is closed. Moreover, te dependence of te energy error estimates on te discrete inf-sup constant is traced in detail, wic sows tat classical error estimates are extremely pessimistic on domains wit large aspect ratios. Numerical experiments in 2D and 3D illustrate te teoretical findings. 1 Introduction For several decades, it was common belief in te numerical analysis community, tat in mixed discretizations for te Stokes equations in primal variables velocity u and pressure p, and wit data f L 2 (), g L 2 0 () and ν > 0 on a domain R d (d = 2, 3), ν u + p = f, x, u = g, x, u = 0, x, (1) a dependence of te discrete velocity on te continuous pressure was more or less practically unavoidable. In oter words, it became standard in finite Weierstrass Institute, Morenstr. 39, Berlin. alexander.linke@wiasberlin.de Universität Hamburg, Department of Matematics, Bundesstr. 55, Hamburg. winnifried.wollner@uni-amburg.de. 1

4 element analysis to prove for new mixed discretisations of te incompressible Stokes equations te following kind of finite element error estimate for te discrete velocity u u 1, C 1 β inf w X u w 1, + 1 ν inf p q L 2, (2) q Q wit some generic constant C 1, since tis estimate allows to conclude tat te discrete velocity converges wit an asymptotically optimal convergence order. However, te estimate (2) is not really optimal wit respect to two different aspects, i.e., qualitatively, it does not give te best possible teoretical result, wic one can ope for: i) Te first point concerns te appearance of te inverse of te discrete inf-sup constant β in te error estimate. Te inf-sup constant is well-known to degenerate for domains wit a large aspect ratio [16, 15, 38, 11], e.g., for practically relevant cannellike domains. Terefore, velocity error estimates containing te constant 1/β are extremely pessimistic. Indeed, suc estimates can be improved, if an appropriate, locally defined Fortin operator for a mixed finite element is known, as demonstrated in tis contribution. Furter, we will derive several explicit a-priori error estimates, were all involved constants (suc as C 1 in (3) below) only depend on te angles in te underlying finite element mes, but not on te inf-sup constant or te value of ν. ii) Te second point concerns te appearance of te pressure-dependent error contribution 1 ν inf q Q p q L 2. Toug mixed finite elements witout a pressure-dependent error contribution are rater classical [36, 35, 31, 19], tey were not really investigated by numerical analysts for many years. For te pressure-robust Crouzeix Raviart finite element metod, we prove in tis contribution te error estimate u u 1, C 1 T D 2 u L 2, (3) were T is te messize function. We remark tat te appearance of te pressure-dependent error contribution 1 ν inf q Q p q L 2 in (2) sows tat classical mixed metods do not fulfill a fundamental invariance property of te continuous Stokes equations (1) exactly: canging te rigt and side by f f + φ canges te Stokes solution by (u, p) (u, p + φ), i.e., gradient fields in te momentum balance are absorbed completely by te pressure gradient. A renewed interest [39, 40, 8, 13, 26, 21, 20, 37] in pressure-robust mixed metods for te Stokes equations tat allow for pressure-independent velocity error estimates was incited by te seminal work of S. Zang [39], wo constructed in 2005 te first pressure-robust 3D Stokes element. Te lack of robustness of classical mixed metods, wose velocity error is indeed pressure-dependent, was demonstrated in recent years for several flow problems, were te pressure is muc more complicated tan te velocity [24, 10, 17, 26]. 2

5 Recently, te observation was made tat te appearance of te pressuredependent error contribution 1 ν inf q Q p q L 2 is only due to te fact tat certain discrete velocity test functions in classical mixed metods are not divergence-free in te sense of H(div) [26, 25]. Tis problem also influences te approximation by adaptive finite element metods for stationary [27] and for nonstationary problems as noted in [4, 5], and special care needs to be taken in te transfer of solutions between different meses at different points in time to preserve te discrete divergence-free condition. Employing lowest-order H(div)-conforming Raviart Tomas elements in certain novel velocity reconstructions [26, 25], it was sown tat te nonconforming Crouzeix Raviart element [14] can be sligtly modified in its discretisation of te rigt-and side suc tat its velocity error becomes pressure-independent. Tese velocity reconstructions introduce an O() consistency error, wic allows for straigtforward proofs of te optimal convergence of te discrete velocity in its energy norm and of te L 2 -norm of te pressure [26]. However, te optimal convergence of te discrete velocity in te L 2 -norm seemed to be difficult to prove, altoug it was observed in numerical experiments [6]. Tis gap will be closed in tis contribution, using an Aubin Nitsce type duality argument and a certain iger regularity of te rigt and side. Te rest of te paper is outlined as follows. Section 2 intrudces continuous and discrete setting and all necessary notation. Section 3 recalls and refines known a priori error estimates for te energy norm and te L 2 norm of te pressure and eventually presents te proof for te optimal convergence of te L 2 velocity error. Section 4 concludes te paper wit tree numerical examples. 2 Continuous and Discrete Setting Tis section explains te continuous and te discrete setting for te model problem under consideration and employs te standard Sobolev spaces V := H0 1 () d := {v H 1 () d : v = 0 along }, Q := L 2 0() := {q L 2 () : qdx = 0}, 2.1 Continuous Setting H(div, ) := {v L 2 () d : v L 2 ()}. Te weak solution (u, p) V Q of te continuous steady incompressible Stokes problem wit rigt-and side f L 2 () d and g Q satisfies ω a(u, v) + b(v, p) = l(v), b(u, q) = χ(q) 3 for all (v, q) V Q (4)

6 wit a, b and l defined by a : V V R, a(u, v) := ν u : vdx, b : V Q R, b(u, q) := q udx, l : V R, l(v) := f vdx, χ : Q R, χ(q) := g qdx Wit te subset of functions tat satisfy te divergence constraint V g := {v V : v = g}, (5) te saddle point problem (4) transforms into a problem for te velocity alone, i.e., u V g suc tat 2.2 Notation a(u, v) = l(v) for all v V 0. (6) In te following, T denotes a sape-regular family of triangulations of te domain into triangles for d = 2 or tetraedra for d = 3, for simplicity, we assume te domain to be polygonal or polyedral respectively, so tat no special treatment of te boundary is needed. For any element T T, mid(t ) denotes te barycenter of T. Te set of all simplex faces, i.e., edges of triangles for d = 2 and faces of tetraedra for d = 3, is denoted by F. Te subset F() denotes te set of interior faces, wile F( ) denotes te set of boundary faces along. For any F F, mid(f ) denotes te barycenter of F and n F abbreviates a face unit normal vector. Te orientation of tese normal vectors for te interior faces F F() are arbitrary, but fixed. Te normal vector n F for boundary faces F F( ) points outwards of te domain. For every simplex T T, F(T ) denotes te set of faces of tis simplex and n T denotes te outer unit normal of te simplex T T. Te piecewise constant function T denotes te local mes size, i.e., T T := diam(t ) for all T T. Moreover, we let = T L. Te function space of P k (T ) contains piecewise polynomials of order k wit respect to T. For a piecewise Sobolev function v H 1 (T ) d and some face F F(), te notion [v n F ] denotes te jump of te normal flux over F, wile {{v n F }} denotes te average value of te normal flux over F. Te space of Crouzeix Raviart velocity trial functions is given by CR(T ) := { v P 1 (T ) d : [v ](mid(f )) = 0 for all F F() & v (mid(f )) = 0 for all F F( ) }. 4

7 Te pressure trial function space reads { Q(T ) := q P 0 (T ) : } q dx = 0. Te space of lowest order Raviart Tomas finite element functions reads { RT(T ) := v H(div, ) : T T a T R d, b T R, } v T (x) = a T + b T x. Any Raviart Tomas function is uniquely defined by its constant face normal fluxes v n F P 0 (F ) for all F F [7]. Te discrete setting employs te broken gradient and te broken divergence in te sense tat : V CR(T ) L 2 () d d ( ) : V CR(T ) L 2 () ( v ) T := (v T ), ( v ) T := (v T ) for all T T. Te discrete gradient norm for te space V CR(T ) reads ( 1/2 v 1, := v : v dx) = v L 2. (7) 2.3 Interpolation operators Te usual Crouzeix Raviart interpolation operator π CR : V CR(T ) is defined by (π CR v)(mid(f )) = 1 vds for all F F. F F Te Raviart Tomas interpolation operator π RT : V CR(T ) RT(T ) is defined by n F (π RT 1 v)(mid(f )) = F F v n F ds for all F F. Note tat, due to continuity in te face barycenters, tis is well-defined also for v CR(T ). Moreover, it olds te identity π RT πcr v = π RT v for any v V. For any γ Q and v V γ, it immediately follows - π RT v = π 0 γ and - π CR v = π 0 γ by Gauss teorem. Here, π 0 denotes te L 2 projector onto 5

8 P 0 (T ). Furtermore, tere are te well-known stability and approximation properties, elementwise on all T T, (π CR v) L 2 (T ) v L 2 (T ) for all v H 1 (T ), (8) (v π CR v) L 2 (T ) C I T D 2 v L 2 (T ) for all v H 2 (T ) d, (9) v π RT v L 2 (T ) C F T v L 2 (T ) for all v H 1 (T ), (10) were te generic constants C I and C F depend only on te sape of te simplices in te triangulation T, but not on teir size [7, 1, 9]. 2.4 Te finite element sceme wit divergence-conforming reconstruction Te discrete weak formulation of te model problem employs a (u, v) := ν u : v dx, b (u, q ) := q u dx, l (v ) := f v dx. Wit tis, te discrete Stokes problem seeks (u, p ) CR(T ) Q(T ) suc tat a (u, v ) + b (v, p ) = l (π RT v ), b (u, q ) = χ(q ) for all (v, q ) CR(T ) Q(T ). (11) In comparison to te classical Crouzeix Raviart nonconforming finite element metod [14], te introduction of π RT in te rigt-and side constitutes a variational crime tat maps discretely divergence-free test functions to divergence-free functions in H(div, ) wit certain benefits as discussed below. Like te continuous incompressible Stokes and Navier-Stokes equations, also te discretization (11) can be formulated as an problem [34, 18] witin te space of discretely constrained functions V g, := {v CR(T ) : v = π 0 g}. (12) Ten, u V g, is uniquely defined by a (u, v ) = l (π RT v ) for all v V 0,. (13) Remark 1. Te pair CR(T ) Q(T ) satisfies te discrete inf-sup condition 0 < β := inf sup q v dx. (14) q Q(T )\{0} v 1, q L 2 v CR(T ) \{0} Te inf-sup constant β for te Crouzeix Raviart element is independent of te mes [14]. 6

9 3 A Priori Error Estimates Tis section presents a priori finite element error estimates for te modified Crouzeix Raviart discretization of te incompressible Stokes equations (11). Te analysis is based on te estimates of te consistency error in [1], wic apply te Raviart Tomas interpolation to te best advantage and avoid te use of a trace inequality. However, some sligt canges due to te divergenceconforming reconstruction deliver fundamentally improved results, since te sceme (11) allows for an error estimate of te discrete velocity tat is independent of te pressure. Te proof involves te interpolation error estimates from above and te elementwise Poincaré constant C P : = sup { v π 0 v L 2/ T v L 2 : v V CR(T )} sup max sup { v π 0 v L T T 2 (T )/ T v L 2 (T ) : v H 1 (T ) }. >0 Lemma 1. For all v CR(T ), it olds i) b (v, q) = b(π RT v, q) for all q L 2 (), ii) b(π RT v, q) = q (π RT v )dx for all q H 1 (). Proof. Te divergence teorem and te definition of π RT T (v π RT v )dx = F F(T ) F sows (v π RT v ) n T ds = 0. Since (v π RT v ) is elementwise constant, te above implies te relation v = (π RT v ) wic proves te first identity. An integration by parts, wic is allowed since π RT v H(div, ), sows te second identity and concludes te proof. Lemma 2. For all v V H 2 () d, w V CR(T ), it olds v : w + v π RT wdx (2C F + C P ) T D 2 v L 2 w 1,. Proof. Let Π RT denote te rowwise Raviart Tomas interpolator and Π 0 te L 2 projection onto P 0 (T ) d. Since te normal fluxes (Π RT v)n F are continuous for all F F and constant on te boundary faces F F( ) and w is zero at least at te centers of any F F( ), it olds T T T ( Π RT v n ) wds = 0. 7

10 An elementwise integration by parts and te commutation property of te divergence wit te Raviart Tomas interpolation (Π RT v) = Π 0( v) sow Π RT v w + Π 0 ( v) wdx = 0. Tis and elementary calculations reveal v : w + v π RT wdx ( = v Π RT v ) : wdx + + v (πrt w w) dx. ( v Π 0 ( v)) wdx For te first integral, a Caucy-Scwarz inequality and te rowwise version of (10) yield ( v Π RT v ) : wdx C F T D 2 v L 2 w 1,. For te second integral, te L 2 ortogonality of v Π 0 ( v) and w Π 0 w w.r.t. P 0 (T ) d and elementwise Poincaré inequalities sow ( v Π 0 ( v)) wdx = ( v Π 0 ( v)) (w Π 0 w)dx v L 2 w Π 0 w L 2 C P T v L 2 w 1,. Anoter Caucy-Scwarz inequality and (10) bound te tird integral by (πrt v w w) dx C F T v L 2 w 1,. (15) Te combination of te last tree estimates concludes te proof. Te estimate of te consistency error is a corollary to Lemma 2. Lemma 3 (Pressure-independent consistency error estimate). Given te solution (u, p) H 2 () d H 1 () of te continuous Stokes equations (4), it olds sup 0 w V CR(T ), π 0 ( w )=0 a (u, w ) l (π RT w ) ν(2c F + C P ) T D 2 u w L 2. 1, 8

11 Proof. For all 0 w V CR(T ) wit π 0 ( w ) = 0 it olds p π RT w dx = 0. Tis and (4) and sow 1 ν a (u, w ) l (π RT w ) = 1 ν ν u : w f π RT w dx = 1 ν ν u : w + (ν u p) π RT w dx (16) = u : w + u π RT w dx. Lemma 2 concludes te proof. Remark 2. Note tat Lemma 3 does not old in a pressure-independent way for te standard Crouzeix Raviart finite element metod, since in (16) p and w for w V 0 + V 0, are not ortogonal in te L 2 scalar product. Te estimate of te pressure-independent consistency error leads to te following optimal a priori estimates. Teorem 1. For te solution (u, p) H 2 () d H 1 () of te continuous Stokes equations (4) and te discrete solution (u, p ) of (11), it olds i) u u 1, (2C I + 2C F + C P ) T D 2 u L 2, ii) iii) π 0 p p L 2 (2C I + 4C F + 2C P )β 1 ν T D 2 u L 2, p p 2 L C 2 2 P T p 2 L + (2C 2 I + 4C F + 2C P ) 2 β 2 ν2 T D 2 u 2 L. 2 Proof of i). Formulation (13) and w := u v V 0, for an arbitrary v V g, yield ν w 2 1, = a (w, w ) = a (u v, w ) = a (u v, w ) + a (u, w ) a (u, w ) = a (u v, w ) + l (π RT w ) a (u, w ) ν u v 1, w 1, + a (u, w ) l (π RT w ). Te triangle inequality for u u 1, = (u v ) w 1, produces Strang s second lemma in te form u u 1, 2 inf u v 1, + 1 a (u, w ) l (π RT sup w ). v V g, ν w V 0, w 1, Since π CR u V g,, te first error term can be bounded wit (9) by inf u v 1, u π CR u 1, C I T D 2 u L 2. v V g, Te second error term is estimated wit Lemma 3. 9

12 Proof of ii). Due to te discrete inf-sup stability (14), we can estimate te second term by π 0 p p L 2 1 β b (v, π 0 p p ) sup. v CR(T ) v 1, Since v is constant and π 0 p p is ortogonal on constants, te term in te numerator of tis expression equals b (v, π 0 p p ) = b (v, p p ). Elementary calculations, te application of Lemma 1 i) and ii), and f = ν u + p sow b (v, p p ) = b (v, p) + a (u, v ) l (π RT v ) = b(π RT v, p) + a (u, v ) f π RT v dx = p π RT v dx + a (u, v ) + (ν u p) π RT v dx = a (u u, v ) + ν { u : v + u π RT v } dx. Te first term is estimated by i) wit a (u u, v ) (2C I + 2C F + C P ) T D 2 u L 2 v 1, Lemma 2 yields te concluding argument ν { u : v + u π RT v } dx (2CF + C P )ν T D 2 u L 2 v 1,. Proof of iii). For te pressure estimate, te Pytagoras teorem sows p p 2 L 2 = p π 0 p 2 L 2 + π 0 p p 2 L 2. Elementwise Poincaré inequalities wit constant C P bound te first term by p π 0 p L 2 C P T p L 2. Te combination wit ii) concludes te proof. Remark 3. Te constants C I, C F and C P in Teorem 1 are independent of te inf-sup-constant β. Estimates for mixed metods tat depend on β are dramatically pessimistic for cannel domains wit large aspect ratio [38, 15, 16, 11]. 10

13 Remark 4. Guaranteed upper bounds for all involved constants C I, C F and C P in Teorem 1 are known. Te Fortin interpolation constant is bounded by C F for rectangular triangles, for details see te maximum angle estimate from [9, Teorem 5.1]. In 2D, [23] sows C P = 1/j 1,1 were j 1,1 = is te first positive root of te first Bessel function J 1. In 3D, te constant C P = 1/π is valid for every convex domain [29, 3]. Moreover, te constant C I is in fact also bounded by C P, since te Crouzeix-Raviart interpolation operator π CR as te property T (v π CR v ) dx = 0 for all T T and so allows for a Poincaré type inequality in (9). Remark 5. Te modified Crouzeix Raviart metod (11) or equivalently (13) is usually muc more accurate tan te standard Crouzeix Raviart metod, see [26]. However, te standard Crouzeix Raviart metod performs better in tose (very special) situations, wenever te continuous pressure p vanises. In order to get an estimate, ow te modified Crouzeix Raviart metod beaves in tis worst case, let û denote te solution of te standard Crouzeix Raviart finite element metod and let u denote te solution of te modified Crouzeix Raviart finite element solution from (11). Ten, by (10) it olds ν û u 2 1, = a (û u, û u ) = l (û u ) l (π RT (û u )) C F T f L 2 û u 1,. Hence, te difference between te two solutions is at most û u 1, C F ν T f L 2. Tis estimate reflects te considerations above in te following way: te worst case for te classical Crouzeix Raviart finite element metod is for f = p wic means u = 0. Here, te modified Crouzeix Raviart metod delivers te exact velocity solution u = 0, wile û deteriorates in general wit O(1/ν) for ν 0. On te oter and, te worst case for te modified Crouzeix Raviart finite element metod appens for f = ν u wic means p = 0. Ten, it olds wic is independent of 1/ν. û u 1, C F T u L 2 Corollary 1 (Invariance property). Te modified Crouzeix Raviart finite element metod satisfies a continuous invariance property in te sense tat for all f L 2 () d and all φ H 1 ()/R it olds f f + φ = (u, p) (u, p + φ), f f + φ = (u, p ) (u, p + π 0 φ). 11

14 Proof. Tis is a direct consequence of Teorem 1. Teorem 1 i) sows, tat te discrete solution u for te rigt-and side φ and for g = 0 is zero (because te exact solution is zero) and Teorem 1 ii) sows π 0 φ p L 2 = 0. Lemma 4. Given a rigt-and side r L 2 () d, let u r V 0 denote te solution of a(u r, v) = (r, v) for all v V 0, and let u r, V 0, denote te solution of a (u r,, v ) = (r, π RT v ) for all v V 0, Ten, for te solutions u from (4) and u from (11), it olds { u u L 2 sup ν u u 1, u r u r, 1, r L 2 () d, r L 2 =1 + a (u u, u r ) ( r, π RT (u u ) ) + a (u, u r u r, ) ( f, π RT (u r u r, ) ) + ( r, (u u ) π RT (u u ) ) + ( f, u r π RT u ) } r. Proof. Te proof is based on te duality argument u u L 2 = sup (r, u u ) / r L 2. r L 2 () d \{0} Elementary algebra yields (r, u u ) = a (u, u r, ) a (u, u r ) + (r, u u ) + ( f, u r π RT u ) r, = a (u u, u r, ) a (u, u r u r, ) + (r, u u ) + ( f, u r π RT u r, = a (u u, u r u r, ) a (u u, u r ) + ( r, π RT (u u ) ) a (u, u r u r, ) + ( f, π RT (u r u r, ) ) + ( r, (u u ) π RT (u u ) ) + ( f, u r π RT u r). Triangle and Caucy-Scwarz inequalities conclude te proof. Teorem 2. Assuming tat is convex, simply connected, and tat for te solution of te continuous Stokes equations (4) olds (u, p) H 2 () d H 1 (), u H 2 (), we obtain for te discrete solution (u, p ) of te ) 12

15 sceme (11) te following L 2 error estimate of optimal order for te discrete velocity u u L 2 C 2 ( u H 2 + u H 2), (17) wit a constant C depending on te sape regularity of te triangulation. Proof. Since we assume tat te domain is convex. we obtain by classical regularity results for te incompressible Stokes equations tat u r H 2 () d for all r L 2 () d and tat te following a-priori estimates ν u r H 2 C r L 2, ν u r L 2 C r L 2 (18) old. Ten, we apply te abstract error estimate from Lemma 4, and ave to estimate te corresponding five different terms. First, we obtain ν u u 1, u r u r, 1, ν (C u H 2) (C u r H 2) C 2 u H 2 r L 2 using Teorem 1 and (18). Te second term a (u u, u r ) ( r, π RT (u u ) ) νc u r H 2 u u 1, C 2 u H 2 r L 2 can be estimated by te consistency error for te adjoint problem from Lemma 3, Teorem 1 and (18). By Lemma 3, we obtain analogously a (u, u r u r, ) ( f, π RT (u r u r, ) ) νc u H 2 u r u r, 1, νc 2 u H 2 u r H 2 C 2 u H 2 r L 2, using te estimate of te consistency error for te original problem. For te fourt term, we obtain by Teorem 1 and (10) ( r, (u u ) π RT (u u ) ) C u u 1, r L 2 C 2 u H 2 r L 2. Bounding te fift term goes beyond standard arguments. We introduce te L 2 -interpolation Π 0 into elementwise constants and obtain ( f, u r π RT u ) r ν ( u Π 0 u, u r π RT u ) r + ( ν Π0 u, u r π RT u ) (19) r since ( p, u r π RT u r) = 0. For te first summand, standard error estimates for te L 2 -projection and π RT give te desired bound ν ( u Π 0 u, u r π RT u r) c 2 u L 2 r L 2 13

16 To estimate te second term on te rigt of (19), we notice tat u r = 0 and ence, utilizing te exactness of te de Ram complex on a simply connected domain, tere is a function σ r suc tat σ r = u r. Furter, since u r H 1 () d it olds σ r H 2 () if d = 2 and σ r H 2 () 3 if d = 3. In bot cases, it olds σ r H 2 c r L 2, see, e.g., [22, Lemma 2.6]. Furter, tere is a finite element space Ṽ and a corresponding interpolation operator I : H 2 Ṽ suc tat u r π RT u r = (σ r I σ r ). Since te space Ṽ takes a different form for different dimensions d = 2, 3, we proceed by cases: 2d In tis case Ṽ consists of piecewise linear polynomials and I is te standard nodal interpolation, see, e.g., [2, Table 5.1] or [30] for te original definition of te element. Ten, we can estimate te remaining term as follows utilizing Green s formula ( Π 0 u, u r π RT u ) r = T T ( Π0 u, u r π RT u ) r = (Π 0 u, (σ r I σ r ) T T T ( Π 0 u, σ r I σ r ) T T T + (Π 0 u, n (σ r I σ r )) T T T T (20) Given tat Π 0 u is elementwise constant, te volume term vanises. For te boundary term, we calculate (Π 0 u, n (σ r I σ r ) T Π 0 u L ( T ) σ r I σ r L 1 ( T ). Standard interpolation estimates for linear polynomials imply σ r I σ r L 1 ( T ) c 2 σ r H 2 (T ). For te sake of completeness, we derive tis estimate step by step. Te trace identity for any function v H 1 (T ) and triangle T = conv{e, p} wit edge E and opposite node P reads E vds = E T T vdx + E v (x P )dx. 2 T T Setting v := σ r I σ r in tis identity yields σ r I σ r ds E E T 1 ( ) σ r I σ r L 1 (T ) + 1/2 x P L 2 (T ) (σ r I σ r ) L 2 (T ) E T 1/2 ( ) σ r I σ r L 2 (T ) + T /2 (σ r I σ r ) L 2 (T ) c 2 T σ r H 2 (T ) 14

17 for some constant c tat depends only on te sape of T. For te term Π 0 u L ( T ), we observe, tat Π 0 u L ( T ) = 1 T Combining tis, we can bound (20) as follows ( Π 0 u, u r π RT u r) T u dx u L (T ) c u H 2 (T ). (Π 0 u, n (σ r I σ r ) T T T c 2 T T u H 2 (T ) σ r H 2 (T ) c 2 u H 2 σ r H 2 c 2 u H 2 r L 2. Here, te elementwise L norm of u was estimated by u H 2 (T ) to avoid a dependence on te number of elements. 3d In 3d, Ṽ is te space of Nedelec elements wit te corresponding interpolation, see, e.g., [2, Table 5.2] or [28] for te original definition of te elements. Unfortunately, Ṽ does contain all constants, but not all linear polynomials. Hence, te argument from te 2d case needs to be modified, since te interpolation of σ r can at most provide one power of. To tis end, we utilize te representation (20) ( Π 0 u, u r π RT u r) = (Π 0 u, n (σ r I σ r ) T T T and proceed wit a different splitting (Π 0 u, n (σ r I σ r )) T Π 0 u L 1 ( T ) σ r I σ r L ( T ). A straigt forward calculation utilizing te sape regularity, i.e., T c 2, gives Π 0 u L 1 ( T ) = Π 0 u ds T Π 0 u L ( T ) ds c 2 u L (T ) c 2 u H 2 (T ). For te interpolation error estimate, we employ a Bramble-Hilbert type argument. For tis, we note tat by assumption any elements T can be obtained by an affine linear transformation A T : T T from some reference element T. For any function f on T, we denote by f its pullback onto T, i.e., f( x) = f(a T x) for any x T. By definition of te nodal-variables of te T 15

18 Nedelec element it is Îf = Î f were Î is te local interpolation operator on te reference element. Utilizing L stability of Î, we conclude using tat σ r is continuous σ r I σ r L ( T ) σ r Î σ r L ( T ) c σ r L ( T ) c σ r L (T ) c σ r H 2 (T ) wit a constant c depending on te sape regularity of te element only. Analogous to te 2d case, te assertion follows. Remark 6. Again, te significance of Teorem 2 lies in te fact tat te velocity error u u L 2 is independent of te pressure. Te additional regularity assumption, needed for te proof of te optimal O( 2 ) error estimate is a consequence of te variational crime commited in te definition of (11), were piecewise linear discretely divergence-free functions are mapped onto divergence-free piecewise constant Raviart Tomas functions. It is wellknown from te classical teory of variational crimes tat iger regularity assumptions tan usual are necessary for proving optimal error estimates, wen te rigt-and side is projected onto a polynomial space of less tan optimal order as it appens e.g. wit quadrature rules [12]. 4 Numerical Experiments Tis section deals wit tree numerical experiments to validate and confirm te teory. Te first example demonstrates te benefits of te modified metod. Te next two examples focus on te convergence rate of te L 2 velocity error under low regularity. Here, we set in eac case p = 0, since in tis case te classical Crouzeix Raviart element performs best and we want to compare wit tose results witout being distracted by pressure effects in te non-modified standard metod. Please note, tat p = 0 is te (quite unrealistic) worst-case for te modified Crouzeix Raviart element. 4.1 First Example Te first bencmark example studies te Stokes problem wit te exact solution u = rotξ P 7 () 2 V for te stream function ξ = x 2 (1 x) 2 y 2 (1 y) 2 and te pressure p = x 3 + y 3 1/2 on te unit square = (0, 1) 2. For given viscosity ν, te volume force equals f := ν u + p. 16

19 ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e e e e e e e Table 1: Convergence istory and convergence order for all error norms for te standard metod in te example of Section 4.1 for ν = 1. ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e e e e e e e Table 2: Convergence istory and convergence order for all error norms for te modified metod in te example of Section 4.1 for ν = 1. Tables 1-2 sow te error norms and teir convergence orders for ν = 1. Te error for te standard metod is significantly larger tan for te modified metod due to te influence of te pressure. Table 3 compares te results on a fixed mes for different ν towards zero. Te velocity errors of te standard metod get polluted more and more as indicated by te a priori error estimate, wile te modified metod is robust and sows no canges in te velocity error. 4.2 Second Example Te second example considers te stream function w(x, y) = r 2 log log r wit r(x, y) := x 2 + y 2 wit te exact solution u := rot(w(x, y)) and rigt- u u L 2 u u 1, p p L 2 ν (standard) (modified) (standard) (modified) (standard) (modified) 1e e e e e e e-02 1e e e e e e e-02 1e e e e e e e-02 1e e e e e e e-02 1e e e e e e e-02 Table 3: Comparison of error norms for bot metods in te example of Section 4.1 for different ν and a fixed mes wit ndof =

20 ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e Table 4: Convergence istory and convergence order for all error norms for te standard metod in te example of Section 4.2. ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e Table 5: Convergence istory and convergence order for all error norms for te modified metod in te example of Section 4.2. and side f := u on te domain := ( 3/7, 4/7) 2. Te exact solution satisfies u H 2 () but not u L (). Te unstructured meses for te computations were generated wit Triangle [32] and te unsymmetric bounds of te domain ensure tat te singular point (0, 0) is not a node of te meses. Tables 4 and 5 sow tat all error norms under consideration converge wit optimal speed for bot metods. Te results indicate, tat te required regularity u H 2 () in te statement of Teorem 2 can potentially be relaxed. 4.3 Tird Example Te tird example concerns te 3d velocity u(x, y, z) := 13 5y 3z 5x + 2z r 1/2+1/100 wit r 2 := x 2 + y 2 + z 2 5 3x 2y and te rigt-and side f := u on te unit cube := ( 0.5, 1) 3. Te exact solution satisfies u H 2 () but not u L (). Te unstructured meses for te computations were generated wit TetGen [33] and te unsymmetric bounds of te domain ensure tat te singular point (0, 0) is not a node of te meses. Tables 6 and 7 suggest tat tere is no reduction of te optimal convergence order in case u H 2 () but u / L (). 18

21 ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e e e e Table 6: Convergence istory and convergence order for all error norms for te standard metod in te example of Section 4.3. ndof u u L 2 order u u 1, order p p L 2 order e e e e e e e e e e e e e e e Table 7: Convergence istory and convergence order for all error norms for te modified metod in te example of Section 4.3. References [1] G. Acosta and R. G. Durán. Te maximum angle condition for mixed and nonconforming elements: application to te Stokes equations. SIAM J. Numer. Anal., 37(1):18 36 (electronic), [2] D. N. Arnold, R. S. Falk, and R. Winter. Finite element exterior calculus, omological tecniques, and applications. Acta Numer., 15:1 155, [3] M. Bebendorf. A note on te Poincaré inequality for convex domains. Z. Anal. Anwendungen, 22(4): , [4] M. Besier and W. Wollner. On te pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meses. Internat. J. Numer. Metods Fluids, 69(6): , [5] M. Braack, J. Lang, and N. Tascenberger. Stabilized finite elements for transient flow problems on varying spatial meses. Comput. Metods Appl. Mec. Engrg., 253: , [6] C. Brennecke, A. Linke, C. Merdon, and J. Scöberl. Optimal and pressure-independent L 2 velocity error estimates for a modified Crouzeix Raviart Stokes element wit BDM reconstructions. J. Comput. Mat., 33(2): ,

22 [7] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Elements, volume 15 of Springer Series in Computational Matematics. Springer, [8] E. Burman and A. Linke. Stabilized finite element scemes for incompressible flow using Scott-Vogelius elements. Appl. Numer. Mat., 58(11): , [9] C. Carstensen, J. Gedicke, and D. Rim. Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Tomas finite element metods. J. Comput. Mat., 30(4): , [10] M. A. Case, V. J. Ervin, A. Linke, and L. G. Rebolz. A connection between Scott-Vogelius and grad-div stabilized Taylor-Hood FE approximations of te Navier-Stokes equations. SIAM J. Numer. Anal., 49(4): , [11] E. V. Cizonkov and M. A. Olsanskii. On te domain geometry dependence of te LBB condition. M2AN Mat. Model. Numer. Anal., 34(5): , [12] P. G. Ciarlet. Basic error estimates for elliptic problems. In Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pages Nort-Holland, Amsterdam, [13] B. Cockburn, G. Kanscat, and D. Scotzau. A locally conservative LDG metod for te incompressible Navier-Stokes equations. Mat. Comp., 74(251): (electronic), [14] M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element metods for solving te stationary Stokes equations. I. Rev. Française Automat. Informat. Recerce Opérationnelle Sér. Rouge, 7(R-3):33 75, [15] M. Dobrowolski. On te LBB constant on stretced domains. Mat. Nacr., 254/255:64 67, [16] M. Dobrowolski. On te LBB condition in te numerical analysis of te Stokes equations. Appl. Numer. Mat., 54(3 4): , [17] K. J. Galvin, A. Linke, L. G. Rebolz, and N. E. Wilson. Stabilizing poor mass conservation in incompressible flow problems wit large irrotational forcing and application to termal convection. Comput. Metods Appl. Mec. Engrg., 237/240: , [18] V. Girault and P.-A. Raviart. Finite Element Metods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Matematics. Springer-Verlag, Berlin,

23 [19] M. D. Gunzburger. Te inf-sup condition in mixed finite element metods wit application to te Stokes system. In Collected lectures on te preservation of stability under discretization, pages SIAM, Fort Collins, [20] J. Guzmán and M. Neilan. Conforming and divergence-free Stokes elements in tree dimensions. IMA J. Numer. Anal., 34(4): , [21] J. Guzmán and M. Neilan. Conforming and divergence-free Stokes elements on general triangular meses. Mat. Comp., 83(285):15 36, [22] R. Hiptmair. Finite elements in computational electromagnetism. Acta Numer., 11: , [23] R. S. Laugesen and B. A. Siudeja. Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. J. Differential Equations, 249(1): , [24] A. Linke. Collision in a cross-saped domain a steady 2d navier stokes example demonstrating te importance of mass conservation in CFD. Computer Metods in Applied Mecanics and Engineering, 198(41 44): , [25] A. Linke. A divergence-free velocity reconstruction for incompressible flows. C. R. Mat. Acad. Sci. Paris, 350(17-18): , [26] A. Linke. On te role of te Helmoltz decomposition in mixed metods for incompressible flows and a new variational crime. Comput. Metods Appl. Mec. Engrg., 268: , [27] A. Linke and C. Merdon. Guaranteed energy error estimators for a modified robust crouzeix raviart stokes element. Journal of Scientific Computing, 64(2): , [28] J.-C. Nédélec. Mixed finite elements in R 3. Numer. Mat., 35(3): , [29] L. E. Payne and H. F. Weinberger. An optimal Poincaré inequality for convex domains. Arcive for Rational Mecanics and Analysis, 5(1): , [30] P.-A. Raviart and J. M. Tomas. A mixed finite element metod for 2nd order elliptic problems. In Matematical aspects of finite element metods (Proc. Conf., Consiglio Naz. delle Ricerce (C.N.R.), Rome, 1975), volume 606 of Lecture Notes in Mat., pages Springer, Berlin,

24 [31] L. R. Scott and M. Vogelius. Norm estimates for a maximal rigt inverse of te divergence operator in spaces of piecewise polynomials. Matematical Modelling and Numerical Analysis, 19(1): , [32] J. R. Sewcuk. Triangle: Engineering a 2D Quality Mes Generator and Delaunay Triangulator. In Ming C. Lin and Dines Manoca, editors, Applied Computational Geometry: Towards Geometric Engineering, volume 1148 of Lecture Notes in Computer Science, pages Springer-Verlag, From te First ACM Worksop on Applied Computational Geometry. [33] H. Si. Tetgen, a delaunay-based quality tetraedral mes generator. ACM Trans. Mat. Softw., 41(2):11:1 11:36, February [34] R. Temam. Navier-Stokes equations. Elsevier, Nort-Holland, [35] M. Vogelius. An analysis of te p-version of te finite element metod for nearly incompressible materials. uniformly valid, optimal error estimates. Numer. Mat., 41:39 53, [36] M. Vogelius. A rigt-inverse for te divergence operator in spaces of piecewise polynomials. Application to te p-version of te finite element metod. Numer. Mat., 41:19 37, [37] J. Wang, Y. Wang, and X. Ye. A robust numerical metod for Stokes equations based on divergence-free H(div) finite element metods. SIAM J. Sci. Comput., 31(4): , [38] M. Wolmut and M. Dobrowolski. Numerical analysis of Stokes equations wit improved LBB dependency. Electron. Trans. Numer. Anal., 32: , [39] S. Zang. A new family of stable mixed finite elements for te 3d Stokes equations. Mat. Comp., 74(250): , [40] S. Zang. A family of Q k+1,k Q k,k+1 divergence-free finite elements on rectangular grids. SIAM J. Numer. Anal., 47(3): ,