Energy norm a-posteriori error estimation for divergence-free discontinuous Galerkin approximations of the Navier-Stokes equations

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1 INTRNATIONAL JOURNAL FOR NUMRICAL MTHODS IN FLUIDS Int. J. Numer. Meth. Fluids 19007; 1:1 [Version: 00/09/18 v1.01] nergy norm a-posteriori error estimation for divergence-free discontinuous Galerkin approximations of the Navier-Stokes equations Guido anschat 1 and Dominik Schötzau 1 Department of Mathematics, Texas A & M University, College Station, TX , USA Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z, Canada International Journal for Numerical Methods in Fluids, Volume 57, Pages , 008 SUMMARY We develop the energy norm a-posteriori error analysis of exactly divergence-free discontinuous RT k /Q k Galerkin methods for the incompressible Navier-Stokes equations with small data. We derive upper and local lower bounds for the velocity-pressure error measured in terms of the natural energy norm of the discretization. Numerical examples illustrate the performance of the error estimator within an adaptive refinement strategy. Copyright c John Wiley & Sons, Ltd. 1. Introduction In this paper we derive a residual-based energy norm a-posteriori error estimator for exactly divergence-free discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations ν u + (u )u + p = f in R, u = 0 in, (1.1) u = 0 on Γ =. Here, ν > 0 is the kinematic viscosity, u the velocity, p the pressure, and f L () an external body force. The domain is assumed to be a Lipschitz polygon in R. Throughout, Contract/grant sponsor: The first author was partially supported by the National Science Foundation (NSF) under grant DMS The second author was partially supported by the National Science and ngineering Research Council of Canada (NSRC). Copyright c John Wiley & Sons, Ltd.

2 we assume that ν f L () is sufficiently small. Then the Navier-Stokes system has a unique solution (u, p) H0 1 () L 0(), where H0 1 () is the usual vector-valued Sobolev space with zero boundary values and L 0 () is the space of square integrable functions with vanishing mean value. Moreover, we have the stability bound u L () C P ν 1 f L (), (1.) with C P > 0 denoting the Poincaré constant of. xactly divergence-free discontinuous Galerkin methods for the incompressible Navier- Stokes equations (1.1) were recently introduced in [14]. They are based on divergenceconforming finite element spaces for the approximation of the velocity, such as Brezzi-Douglas- Marini (BDM) or Raviart-Thomas (RT) spaces, and on matching discontinuous spaces for the approximation of the pressure. The H 1 -conformity of the velocity approximation is then enforced weakly using the discontinuous Galerkin approach. The resulting methods have been shown to be locally conservative, inf-sup stable, and optimally convergent. They were inspired by the use of BDM elements for the analysis of DG schemes in [18]. In the lowest-order case, they have been shown to be closely related to the well-known marker and cell (MAC) scheme; cf. []. The use of element pairs beyond BDM and RT is discussed in [31]. In this article, we will focus on RT elements but remark here that the analysis applies in exactly the same way to BDM elements. The exactly divergence-free methods in [14] originated from the ideas introduced in [13]. There, an element-by-element post-processing procedure was devised to render DG velocity approximations exactly divergence-free. The divergence-conforming velocity approximations in [14] can then be understood and analyzed in the setting of [13], with a post-processing operator that is equal to the identity operator. The work [13], in turn, builds upon the earlier papers [11, 1, 15], where discontinuous Galerkin methods for linear incompressible flow problems were proposed and analyzed. In this paper, we develop the energy-norm a-posteriori error estimation for the RT k /Q k method proposed in [14] for quadrilateral meshes. Here, the approximation of the velocity is based on Raviart-Thomas elements of order k while the pressure is discretized using tensor product polynomials of order k. We derive a residual-based error estimator that is both reliable and efficient for the error measured in a natural energy norm that includes the broken H 1 - norm for the error in the velocity and the L -norm for the error in the pressure. Our technique of proof relies on the approach introduced in [0] for the Stokes problem. Let us also mention that a-posteriori error analyses for DG methods applied to elliptic problems can be found, e.g., in [6, 7, 10, 19, 1, 3, 4] and the references therein. The outline of the paper is as follows. In Section, we describe the RT k /Q k methods from [14], and establish the stability properties that are crucial for our analysis. For simplicity, we restrict ourselves to the interior penalty approach for the discretization of the diffusive terms. In Section 3, we present our energy norm error estimator and state our main result. The proof of these results is carried out in Section 4. In Section 5 we present a series of numerical experiments that show the usefulness of our estimator in adaptive refinement strategies. Finally, we conclude our presentation in Section 6.

3 3. Divergence-free DG methods In this section, we recall the RT k /Q k DG method for the Navier-Stokes equations. We follow [13] and [14]. However, instead of the local discontinuous Galerkin (LDG) approach presented there, we employ the interior penalty (IP) method for the discretization of the Laplace operator..1. Discretization We assume that the domain can be partitioned into shape-regular rectangular meshes T h = {}. The diameter of an element is denoted by h, and the mesh size parameter h of T h is the piecewise constant function h with h = h. We will assume that each edge of a mesh cell is a boundary edge, an edge of a neighboring cell or consists of two equally long edges of refined neighboring cells. The latter corresponds to so called one-irregular meshes with one hanging node on refined edges. We further assume that the neighboring nodes of hanging nodes are not hanging themselves. The adaptively generated meshes in our numerical experiments satisfy these properties, see [4]. Remark.1. The inf-sup stability of discretizations with hanging nodes using Raviart- Thomas elements is in part still an open question. For quadrilaterals with one-irregular meshes, a stability proof exists only for the pair RT k /Q k defined in (.1) below with k ; see [8]. Nevertheless, we conjecture from our computational results that stability also holds for k = 1. For triangles with hanging nodes, there is no stability result for the divergence-free elements proposed in [14]; on the other hand, locally refined triangular meshes without hanging nodes can be obtained using bisection. The results below are all to be read in view of the restrictions cited in this remark. We will use the standard average and jump operators. To do define them, we denote by (T h ) the set of all edges in T h, by I (T h ) the set of all interior edges, and by B (T h ) the set of all boundary edges. The length of an edge is denoted by h. Let now + and be two adjacent elements of T h that share an interior edge = + I (T h ). Let ϕ be a any piecewise smooth function (matrix-, vector- or scalarvalued), and let us denote by ϕ ± the traces of ϕ on taken from within the interior of ±. We then define the average operator { } across the edge as {ϕ } = 1 (ϕ+ + ϕ ). Further, let σ, u, and p be a piecewise smooth matrix-valued, vector-valued and scalar-valued function, respectively. The jumps [ ] of these functions across are defined as [σn] = σ + n + + σ n, [u n] = u + n + + u n, [pn] = p + n + + p n, where n ± denote the outward unit normal vectors on the boundary ± of the elements ±. Analogously, for a boundary edge B (T h ), we set {ϕ } = ϕ, [σn] = σn, [u n] = u n, and [pn] = pn. Here, n is the outward unit normal on Γ. Finally, if the jump appears quadratically, we abbreviate [u] = u + u.

4 4 For an approximation order k 1, we then seek DG approximations to the Navier-Stokes equations in the finite element space V h Q h where V h = { v H(div; ) : v RT k (), T h, v n = 0 on Γ }, Q h = { q L 0() : q Q k (), T h }, (.1) with Q k () denoting the tensor product polynomials of degree k on, RT k () Q k () the Raviart-Thomas polynomials of order k on, see e.g., [9] and the references therein, and H(div; ) = { v L () : v L () }. A crucial property of the pair V h Q h is that, on the meshes considered, V h Q h ; (.) see the discussion in [14, Section..]. We consider the discontinuous Galerkin method: find (u h, p h ) V h Q h such that A h (u h, v) + O h (u h ; u h, v) + B h (v, p h ) = f v dx, B h (u h, q) = 0 (.3) for all (v, q) V h Q h. The forms A h, O h and B h are associated to the discretizations of the Laplacian, the convective term, and the incompressibility condition, respectively. The form B h is given by B h (v, q) = vq dx. From the second equation in (.3), it then follows that u h is orthogonal to all pressures in Q h. Hence, the inclusion (.) implies that u h is indeed exactly divergence-free. For the form A h, we choose the classical (symmetric) interior penalty discretization defined by A h (u, v) = ν u : v dx [v n] : {ν u } ds T h (T h ) (T h ) [u n] : {ν v } ds + (T h ) κ ν [u] [v ] ds. The parameter κ is the interior penalty parameter stabilizing the discontinuous Galerkin form. In order to ensure the coercivity of the form A h, it has to be chosen sufficiently large. For a boundary edge B (T h ) of a mesh cell, its lower limit can be determined by an inverse estimate and is of the form s/h, where h is the length of orthogonal to and s depends on the polynomial degree and the shape of the cell. In particular, for rectangular mesh cells, stability is obtained for κ > k(k + 1) h,

5 5 and we usually choose twice that value in our numerical tests. On interior edges, stability is obtained by taking 1/ of the mean value of this value from both cells, respectively. We will always assume that this parameter is chosen such that stability is guaranteed. We point that, for the discretization of the Laplace operator, several other discontinuous Galerkin methods can be chosen, for which our results hold true as well; see the discussions in [3, 8] and in [14, Table I]. For these methods, we have to at least require stability and consistency; for our purposes, we considered the LDG and (symmetric) interior penalty methods. Finally, to define the convective form O h, let w be a piecewise smooth and divergence-free flow field in the space J(T h ) = { v H(div; ) : v 0 in, v H 1 (), T h }. Clearly, the discrete velocity field u h belongs to this space. We then take O h to be the standard upwind form introduced in [6, 7]: O h (w; u, v) = (w )v u dx T h + T h [ w n {u } 1 ] w n (u e u) v ds. Here, u e denotes the exterior trace of u taken over the edge under consideration and set to zero on the boundary. Upon integration by parts, we also have O h (w; u, v) = (w )u v dx T h + T h [ 1 w n (u e u) 1 ] w n (u e u) v ds. This completes the definition of the DG methods for the incompressible Navier-Stokes equations. We point out that, since u h is exactly divergence-free, the resulting DG methods are locally conservative and energy-stable; cf. [13, 14]... Stability properties In this section, we recapitulate the main stability properties of the DG forms from [13, 14, 8]. First, we rewrite the interior penalty form A h in the form A h (u, v) = ν u : v dx ν u : L(v) dx ν v : L(u) dx + κ ν [u] [v ] ds. (T h ) where the lifting operator L : V h Σ h is given by L(v) : τ dx = [v n] : {τ } ds τ Σ h, (T h )

6 6 with Σ h = { τ L () : τ Q k+1 (), T h }. We can now easily extend the form A h to the space V (h) = H0 1 () + V h, using the same definition of the lifting operator. This space is endowed with the broken H 1 -norm u 1,h = u L () + κ ν [u] L (). T h (T h ) Then, the form A h satisfies the following continuity and coercivity properties: If the interior penalty parameter is chosen as above, then there are constants c a > 0 and α > 0, independent of the viscosity and the mesh size, such that A h (u, v) νc a u 1,h v 1,h, u, v V (h), (.4) A h (u, u) να u 1,h, u V h. (.5) The form O h is Lipschitz continuous: There is a constant c 0 > 0, independent of the viscosity and the mesh size, such that O h (w 1 ; u, v) O h (w ; u, v) c o w 1 w 1,h u 1,h v 1,h, (.6) for all w 1, w J(T h ), u V (h) and v V (h). This statement has been proven in [13, Proposition 4.] for v V h. Using similar arguments, it can be readily seen that it also holds for v H 1 0 (). Moreover, there holds O h (w; u, u) 0, (.7) for w J(T h ) and u V h or u H 1 0 (). Finally, the form B h is continuous and satisfies the discrete inf-sup condition: There exist constants c b > 0 and β > 0, independent of the viscosity and the mesh size, such that B h (u, p) c b u 1,h p L (), u V (h), p L (), (.8) B h (u, p) sup β p L u V h u (), p Q h. (.9) 1,h While the continuity of B h is obvious, the inf-sup condition follows from the results in [8]. It holds on regular meshes for any k. For k, it holds on the irregular meshes considered in this paper. As pointed out in Remark.1, the stability on irregular meshes for k = 0, 1 remains an open problem. We further refer the reader to [18] for the stability of BDM elements on conforming triangular meshes. With these stability properties at hand and by proceeding as in the analysis presented in [14, Theorem 3.1] for triangular elements, we readily obtain the following result: If ν f L () is sufficiently small, then the DG discretization (.3) has a unique solution (u h, p h ) V h Q h and we have the a-priori error estimate u u h 1,h + p p h L () C ( h k u H k+1 (T h ) + h k p Hk (T h )), with a constant C > 0 independent of the mesh size. Here, the. H k (T h ) denotes the cellwise H k -norm. For later use, we also note that the discrete velocity u h satisfies the bound u h 1,h c p ν 1 f L (), (.10)

7 7 where c p is the constant in the discrete Poicaré inequality: There is c p > 0 independent of the mesh size such that see [, Lemma.1] and [8]..3. Additional stability properties v L () c p v 1,h, v V h, (.11) In this section, we establish two additional stability results that are key in our a-posteriori error analysis. To that end, we introduce the global form A h (w)(u, p; v, q) = A h (u, v) + O h (w; u, v) + B h (v, p) B h (u, q), for any w J(T h ), (u, p) and (v, q) in V (h) L (). We then rewrite the DG methods in the form: Find (u h, p h ) V h Q h such that A h (u h )(u h, p h ; v, q) = f v dx (v, q) V h Q h. (.1) We further introduce the product norm and define (u, p) = ν u 1,h + ν 1 p L (), c p = max{c S, c p α 1 }. In the following, we consider small data and assume that Then the following continuity and stability properties hold. Lemma.1. Assume (.13) and let w J(T h ) satisfy c o c p ν f L () < 1. (.13) w 1,h c p ν 1 f L (). Then there is a continuity constant c A > 0, only depending c a and c b, such that for all u, v V (h) and p, q L (). A h (w)(u, p; v, q) c A (u, p) (v, q) Proof. The assertion follows from the continuity of A h and B h, the Cauchy-Schwarz inequality and the fact that O h (w; u, v) c o w 1,h u 1,h v 1,h c o c p ν f L ()ν 1 u 1,h ν 1 v 1,h ν 1 u 1,h ν 1 v 1,h, which is due to (.6), the bound on w and the smallness assumption (.13). Lemma.. Assume (.13) and let w J(T h ) satisfy w 1,h c p ν 1 f L (). Then there is constant γ > 0, only depending on, such that for any tuple (u, p) H0 1() L 0 () there is (v, q) H 0 1() L 0 () with (v, q) 1 and A h (w)(u, p; v, q) γ (u, p).

8 8 Proof. Fix (u, p) H0 1() L 0 (). We have A h (w)(u, p; u, p) = A h (u, u) + O h (w; u, u) ν u 1,h. (.14) Here, we have used (.7) and that, for u H 1 0 (), A h (u, u) = ν u L () = ν u 1,h. Further, due to the continuous inf-sup condition, see [9, 17], there is a field v H 1 0 () that satisfies B h ( v, p) C ν 1 p L (), ν 1 v 1,h ν 1 p L (). with C > 0 denoting the continuous inf-sup constant, which only depends on. Then, A h (w)(u, p; v, 0) = A h (u, v) + O h (w; u, v) + B h ( v, p) C ν 1 p L () A h(u, v) O h (w; u, v). The H 1 -conformity of the functions u, v and the bound for v yield A h (u, v) ν u 1,h v 1,h u 1,h p L (). Furthermore, employing the continuity of O h, the bounds for w and v and the smallness assumption (.13), we obtain O h (w; u, v) c o ν 1 w 1,h u 1,h p L () u 1,h p L (). Hence, the inequality ab ε a + 1 ε b now gives A h (w)(u, p; v, 0) C ν 1 p L () u 1,h p L () ( C 1 ) ν 1 p L ε () νε u 1,h, (.15) for any ε > 0. From (.14) and (.15), we conclude that, for δ > 0, Taking ε = C A h (w)(u, p; u + δ v, p) = A h (w)(u, p; u, p) + δa h (w)(u, p; v, 0) ( (1 εδ)ν u 1,h + δ C 1 ) ν 1 p L ε (). and δ = C 4, we obtain that A h (w)(u, p; u + δ v, p) min{ 1, C 8 } (u, p). (.16) On the other hand, using the triangle inequality and the bound for v, it can be readily seen that ( (u + δ v, p) 1 + C ) (u, p). (.17) 4 The assertion now readily follows from (.16) and (.17).

9 9 3. nergy norm a-posteriori error estimation In this section, we present a reliable and efficient energy norm error estimator for the exactly divergence-free DG approximations of the Navier-Stokes equation with small data rror indicators We begin by introducing the error indicators. To that end, let (u h, p h ) be the discontinuous Galerkin approximation of (.3). Let f h be a piecewise polynomial approximation of f, possibly discontinuous across elemental edges. For any element T h and interior edge I (T h ), we introduce the residuals R = ( f h + ν u h (u h )u h p h ), R = [(p h I ν u h ) n], respectively. Here, the matrix I is the identity matrix in R. For each T h, we then introduce the local error indicator η where η = η R + η + η J, (3.1) η R = ν 1 h R L (), η = 1 ν 1 h R L (), \Γ (3.) η J = 1 Finally, we introduce the data oscillation term and define The error estimator η is now given by while the data error ω is defined by κ ν [u h ] L (). O = (f f h ), T h, ω = ν 1 h O L (). (3.3) η = ω = ( T h η ( T h ω ) 1, (3.4) ) 1. (3.5)

10 Reliability We now state that the error indicator η gives rise to a reliable energy norm a-posteriori error estimator, up to the data approximation term ω. We assume that the data is small and satisfies The following result holds. max{1, γ 1 }c o c p ν f L () 1. (3.6) Theorem 3.1. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1) and (u h, p h ) V h Q h the discontinuous Galerkin approximation obtained by (.3). Let η and ω be the error estimator and the data approximation term in (3.4) and (3.5), respectively. Then the following a-posteriori error bound holds: (u u h, p p h ) C ( η + ω ), with a constant C > 0 that is independent of the viscosity and the mesh size. The proof of this theorem will be presented in Section 4.1. Next, we state that the local error indicators can be bounded from above by the local energy error, up to data approximation errors. Hence, the estimator η is also efficient. We first consider the residuals η R and η. To that end, we define δ by δ = { T h : and share an edge or a vertex }. Theorem 3.. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1) and (u h, p h ) V h Q h the discontinuous Galerkin approximation obtained by (.3). Let η R, η and ω be defined in (3.) and (3.3), respectively. For any element T h, there holds ) η R C (ν 1 u uh H1 () + ν 1 p ph L () + ω, η C ) (ν 1 u uh H1 () + ν 1 p ph L () + ω, δ with a constant C > 0 that is independent of, the viscosity and the mesh size. The proof of this theorem will be given in Sections 4. and 4.3. Finally, let us note that we trivially have efficiency for the jump residuals η J. Indeed, since the jumps of the exact solution vanish, there holds ηj = 1 κ ν [u u h ] L () + κ ν [u u h ] L (). \Γ Γ As a consequence of the above results, we have the following lower bound of the energy error. Corollary 3.1. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1) and (u h, p h ) V h Q h the discontinuous Galerkin approximation obtained by (.3). Let η and ω be the error estimator and the data approximation term in (3.4) and (3.5), respectively. Then the following efficiency bound holds: η C ( (u u h, p p h ) + ω), with a constant C > 0 that is independent of the viscosity and the mesh size.

11 11 4. Proofs In this section, we present the proofs of Theorems 3.1 and Proof of Theorem 3.1 We introduce the discontinuous RT space Ṽ h = { v L () : v RT k (), T h }. Then, following [0, Section 4], we define V c h = Ṽ h H 1 0 (). The orthogonal complement of V c h in Ṽ h with respect to the norm 1,h is denoted by V h. Hence, we have Ṽ h = V c h V h. If u h is the discontinuous Galerkin velocity approximation, we can decompose it uniquely into u h = u c h + u r h, (4.1) with u c h V c h and ur h V h. Then, since u h V h and u c h V c h V h, we must also have that u r h = u h u c h V h. By employing arguments analogous to those in [0, Section 4] and [4] for the discontinuous space Ṽ h, the following result is obtained: there is c e > 0, independent of the viscosity and the mesh size, such that ν 1 u r h 1,h c e ( T h η J ) 1. (4.) Let now (u, p) be the solution of the Navier-Stokes equations, and (u h, p h ) the DG solution. We denote the error of the DG approximation by and also set e c u = u u c h. (e u, e p ) = (u u h, p p h ), Lemma 4.1. Assume (3.6). Then there is (v, q) H0 1() L 0 () such that (v, q) 1 and for any v h V h. γ (e u, e p ) f (v v h ) dx A h (u h )(u h, p h ; v v h, q) + (c A + γ)c e ( T h η J ) 1, Here, note that, since u h is exactly divergence-free, an approximation q h of q is not needed (and is set to zero). Proof. From the triangle inequality and (4.), we have γ (e u, e p ) γ (e c u, e p ) + γ (u r h, 0) γ (e c u, e ( p) + γc e η ) 1 J T h.

12 1 Furthermore, from the stability estimate (.10), the fact that (3.6) implies condition (.13), the inf-sup condition in Lemma. is applicable to (e c u, e p ). It follows that there is a test function (v, q) H0 1() L 0 () such that (v, q) 1 and γ (e c u, e p) A h (u h )(e c u, e p; v, q) = A h (u h )(e u, e p ; v, q) + A h (u h )(u r h, 0; v, q) A h (u h )(e u, e p ; v, q) + c A (u r h, 0) ( A h (u h )(e u, e p ; v, q) + c A c e T h η J Here, we have also used the continuity of A h in Lemma.1 and the bound (4.). Since we conclude that Due to the continuity of O h assumption (3.6), we have A(u h )(u, p; v, q) = A h (u)(u, p; v, q) O h (e u ; u, v), γ (e u, e p ) A h (u)(u, p; v, q) O h (e u ; u, v) A h (u h )(u h, p h ; v, q) + (c A c e + γc e ) ( T h η J ) 1 ) 1.. (4.3) in (.6), the stability bound (1.) and the smallness O h (e u ; u, v) c o e u 1,h u 1,h v 1,h c o c p ν f L ()ν 1 eu 1,h ν 1 v 1,h γ ν 1 eu 1,h. Hence, this term can be brought to the left-hand side of (4.3). Then, we have that A h (u)(u, p; v, q) = f v dx. Therefore, we obtain γ (e u, e p ) f v dx A h (u h )(u h, p h ; v, q) + (c A c e + γc e ) ( The assertion now follows from this inequality by noting that A h (u h )(u h, p h ; v h, 0) f v h dx = 0, for all v h V h. T h η J ) 1 Lemma 4.. For (v, q) H0 1() L 0 () there is v h V h such that f (v v h ) dx A h (u h )(u h, p h ; v v h, q) Cν 1 v L ()(η + ω), with a constant C > 0 that is independent of the viscosity and the mesh size..

13 13 Proof. We set ξ v = v v h, with v h V c h to be selected. Then, T = f ξ v dx A h (u h )(u h, p h ; ξ v, q) = f ξ v dx A h (u h, ξ v ) O h (u h ; u h, ξ v ) B h (ξ v, p h ). Here, we have used that B h (u h, q) = 0, thanks to the fact that u h is divergence-free. Integration by parts yields A h (u h, ξ v ) = ν ( u h L(u h )) : ξ v dx ( ) ν u h ξ v dx (ν u h )n ξ v ds \Γ T h +ν 1 L(uh ) L ()ν 1 ξv L (), with n denoting the unit outward normal vector on. There holds ( ) 1 ν 1 L(uh ) L () c l, T h η J for a constant c l > 0 that is independent of the viscosity and the mesh size, see [8, Lemma 7. and Lemma 7.4]. We further have B h (ξ v, p h ) = p h ξ v dx (p h n ) ξ v ds, \Γ and T h O h (u h ; u h, ξ v ) = T h Combining the above relations, we conclude that T (R + O ) ξ v dx + T h + c l ( T h η J (u h )u h ξ v dx. I(T h ) ) 1 ν 1 ξv L (). R ξ v ds We now choose v h V c h as the standard Scott-Zhang interpolant of v, see, e.g., [17]. It satisfies ( h v v h L () + (v v h) L ()) C v L (), (4.5) T h and h 1 v v h L () C v L (). (4.6) I(T h ) Note that on an edge with a hanging node we use the interpolant from the unrefined side in order to maintain conformity. The assertion now follows from (4.4), by using the weighted Cauchy-Schwarz inequality and the approximation properties in (4.5)-(4.6). Theorem 3.1 is an immediate consequence of Lemma 4.1 and Lemma 4.. (4.4)

14 fficiency bound for η R In this section, we prove the efficiency bound for η R stated in Theorem 3.. We do so by employing the bubble function technique introduced in [9, 30]. Let be an element of T h. We denote by b the standard polynomial bubble function on. Let now v be a (vector-valued) polynomial function on ; then there exists a constant C > 0 independent of v and such that b v L () C v L (), (4.7) v L () C b 1 v L (), (4.8) (b v) L () Ch 1 v L (), (4.9) b v L () Ch 1 v L (). (4.10) The proof of (4.7) and (4.8) is given in [9, Lemma 4.1]. The proof of (4.9) and (4.10) can be obtained by similar arguments, see [1, Theorems. and.4]. Fix an element T h and set V b = b R. By (4.8), the definition of the data approximation term O, and the fact that (u, p) solves the Navier-Stokes equations, we have C R L () b 1 R L () = R V b dx = (f + ν u h (u h )u h p h ) V b dx O V b dx. Hence, where C R L () T 1 + T + T 3, (4.11) T 1 = ( ν e u + e p ) V b dx, T = ((u )u (u h )u h ) V b dx, T 3 = O V b dx. To bound T 1, we first integrate by parts over and use the Cauchy-Schwarz inequality: ( ) T 1 Cν 1 V b L () ν 1 eu L () + ν 1 ep L (). Upon application of (4.9) we then obtain T 1 Cν 1 h 1 R L () ( ) ν 1 eu L () + ν 1 ep L (). (4.1) For the term T, we proceed as follows. By the Cauchy-Schwarz inequality and (4.10), we have T = ((e u )u + (u h )e u ) V b dx ( ) C V b L () e u H1 () u L () + u h L () Ch 1 R L () e u H 1 () ( u L () + u h L ()).

15 15 The stability bound (1.) and the smallness assumption (3.6) then yield u L () c p ν 1 f L () = c 1 o c o c p ν 1 f L () Cν. From the discrete Poincaré inequality (.11) and the bound (.10), we conclude similarly that Hence, u h L () u h L () c p u h 1,h c p c p ν 1 f L () Cν. T Cν 1 h 1 R L ()ν 1 eu H1 (). (4.13) Finally, for the term T 3, we use (4.7) and conclude that T 3 C f f h L () V b L () C f f h L () R L (). (4.14) We now combine (4.11) with the bounds in (4.1), (4.13) and (4.14), divide the resulting estimate by R L () and multiply it with ν 1 h. This yields ) = ν 1 h R L () C (ν 1 eu H1 () + ν 1 ep L () + ω. (4.15) η R This is the desired bound for η R fficiency bound for η Next, we show the efficiency of η stated in Theorem 3.. Consider an interior edge that is shared by two elements and. We denote by b the standard polynomial bubble function on. Let δ = {, }. If is a regular edge, we set =. If one vertex of is a hanging node, we may assume without loss of generality that is an entire edge of. We then denote by the largest rectangle contained in so that is also an entire edge of. Let now δ = {, }. If w is a (vector-valued) polynomial function on, then there exists a constant C > 0 independent of w and h such that w L () C b 1 w L (). (4.16) Furthermore, there exists an extension w b H 1 0 ( ) such that w b = b w and w b L () Ch 1 w L () δ, (4.17) w b L () Ch 1 w L () δ, (4.18) w b L () Ch 1 w L () δ. (4.19) with a constant C > 0 independent of w and ; cf. [9, Lemma 4.1] and [1, Theorems. and.4]. We extend w b by zero outside the patch formed by the union of and. Let now R be the edge residual over the edge. We denote by W b the extension of b R constructed above. By (4.16), we obtain C R L () b 1 R L () = R W b ds.

16 16 To estimate the latter integral, we first note that the solution (u, p) of the Navier-Stokes equations satisfies [(pi ν u) n] = 0. Using this and Green s formula in each element of the patch δ, we then conclude that R W b ds = ( ) ( ν e u + e p ) W b + (e p I ν e u ) : W b dx. e δ Consequently, using that (u, p) solves the Navier-Stokes equations, we see that R W b ds = (f + ν u h p h (u h )u h ) W b dx Therefore, e δ + e δ + e δ ((u h )u h (u )u) W b dx (e p I ν e u ) : W b dx. C R L () S 1 + S + S 3, (4.0) where S 1 = e δ S = e δ S 3 = δ e ((f h + ν u h (u h )u h p h ) + (f f h )) W b dx, ((u h )u h (u )u) W b dx, (e p I ν e u ) : W b dx. To bound S 1, we employ the Cauchy-Schwarz inequality, the bound (4.17), and take into account the shape-regularity of the mesh. We obtain S 1 Cν 1 h 1 R L () + Cν 1 h 1 R L () e δ ν 1 e δ ν 1 h f h + ν u h (u h )u h p h L () h f f h L (). Obviously, there holds L ( e ) L ( ). Therefore, we conclude that S 1 Cν 1 h 1 R L () (η R + ω ). (4.1) δ

17 17 By proceeding as for the bound of T in (4.13) and using (4.19), we find that S Cν 1 h 1 R 1 L () eu H 1 () Cν 1 h 1 R L () δ f δ ν δ ν 1 eu H1 (). (4.) Similarly, employing (4.18), the sum S 3 can be readily bounded by S 3 Cν 1 h 1 R ( ) L () ν 1 eu L () + ν 1 ep L (). (4.3) By combining (4.0), (4.1), (4.) and (4.3), by dividing the resulting bound by R L () and by multiplying it by ν 1 h 1, we obtain ν 1 1 h R L () C δ + C ( ) ν 1 eu H1 () + ν 1 ep L () δ (η R + ω ). (4.4) The desired bound for η now follows easily from (4.15) and (4.4). This concludes the proof of Theorem Numerical results In this section, we test our error estimate by reproducing known analytical solutions. All the results were obtained using the deal.ii finite element library (see [4]). Let us consider the standard singular solution of the Stokes problem with ν = 1 on an L-shaped domain (see [0, 30]). The singularities at the reentrant corner are of the form r λ and r λ 1 for the velocity and pressure, respectively, where λ First, we use uniform mesh refinement and do not expect convergence orders better than λ/ in the energy norm in terms of the number of degrees of freedom n dofs. This is confirmed in Table 5.1 for RT 3 /Q 3 and RT /Q elements. The a-posteriori error estimate η is of the same order as the actual energy error, thereby confirming the reliability and efficiency results from Theorem 3.1 and Theorem 3., respectively. The ratio between estimator and error is also shown and is around 8 for k = and close to 14 for k = 3. In order to use the estimate η for adaptive mesh refinement, we use a heuristic criterion based on cellwise error indicators only. We simply refine the third of the overall cells with the highest indicators. Strategies like this have been used successfully in many applications, see e. g. [5] and references cited therein. While for this strategy, no quantitative convergence estimates like in [16, 19] are obtained, its implementation is very simple. From the sets η and η, we obtain cell refinement indicators by adding all or half of the face indicator η to its both neighboring cells for boundary and interior faces, respectively. Then, the fraction Θ of the total number of cells is refined. In Figure 1, we present results on adaptively refined meshes for degrees two to four, again for the Stokes solution. First, this figure shows that the graphs for the estimates are parallel to

18 18 RT /Q RT 3 /Q 3 η L (e u, e p ) η (e u,e p) (e u, e p ) η η (e u,e p) Table 5.1. Convergence history for the Stokes problem on an L-shaped domain with uniform meshes RT /Q errors RT /Q estimate RT 3 /Q 3 errors RT 3 /Q 3 estimate RT 4 /Q 4 errors RT 4 /Q 4 estimate e+06 Degrees of freedom Figure 1. rrors and estimates for adaptive refinement. Stokes problem on an L-shaped domain. those of the error, confirming reliability and efficiency of the estimator. Furthermore, adaptive refinement allows us to recover the optimal convergence rate of n k/ dofs ; here, the fraction Θ was chosen as 0.15, 0.1 and 0.05 for degrees to 4, respectively. As a second example, we reproduce the analytical Navier-Stokes solution by ovasznay [5] for a viscosity of ν = 1/10. The estimates and the errors in the norm (u, p) are reported in Figure. Since the solution is smooth, both error graphs are parallel and exhibit first order convergence. Again, the estimate is by a factor of 5 larger than the error, even if the viscosity is smaller, confirming the robustness of our estimate. On the first mesh, the nonlinearity is unresolved and therefore the estimate cannot capture the error. The two meshes in Figure 3 show why adaptive mesh refinement yields much better results for the L-shaped domain than for ovasznay flow: while refinement is concentrated around the reentrant corner in the former case, it extends to a wide area to the left for the latter.

19 Uniform error Uniform estimate Adaptive error Adaptive estimate e+06 Degrees of freedom Figure. nergy error and estimates for ovasznay flow, ν = 0.1, uniform and adaptive refinement with RT 1/Q 1 elements. Figure 3. Adaptive meshes for the L-shaped domain (RT 4/Q 4) and for ovasznay flow (RT 1/Q 1).

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