Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions

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1 Noname manuscript No. (will be inserted by the editor) Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions Dmitriy Leykekhman Buyang Li Received: date / Accepted: date Abstract In this paper we prove that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon-Miranda) maximum principle on convex polygonal domains. Using this result, we establish the stability of the Ritz projection with mixed boundary conditions in L norm up to a logarithmic term. 1 Introduction We consider a finite element approximation of the mixed boundary value problem Δu = f, u = u Γ, n u = 0, in, on Γ, on Γ, (1) where is a convex polygon in R 2, Γ is a union of some edges of (Γ ) and u Γ is a given function defined on Γ. Let S h denote a space of continuous piecewise polynomials of degree r 1 on a quasi-uniform triangulation of of size h and let Sh Γ = S h HΓ 1 (), with H1 Γ () := {u H1 () : u = 0 on Γ }. We define a finite element approximation u h S h of (1) as ( u h, v h ) = (f, v h ), v h S Γ h, (2) with u h = I h u Γ on Γ, where I h denotes the Lagrange interpolation operator. It is well known (cf. [30, Sec. 4.4]) that the finite element solution u h defined this way is stable in H 1 () norm and as a result satisfies the following approximation property u u h H 1 () C u I h u H 1 (). (3) The main goal of this paper is to establish the analogous approximation (stability) result in L () norm, i.e. D. Leykekhman Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA leykekhman@math.uconn.edu u u h L () Cl h u I h u L (), (4) B. Li Mathematisches Institut, Universität Tübingen, Tübingen 72076, Germany. The work of the author was supported in part by Alexander von Humboldt Foundation. li@na.uni-tuebingen.de

2 2 Dmitriy Leykekhman, Buyang Li with a constant C independent of the solution u and the mesh size h, and l h = { 1 + ln h if r = 1, 1 if r 2. (5) The logarithmic factor in the piecewise linear case cannot be removed in general [17] (see also [10]). The estimate (4) by the triangle inequality and the stability of the interpolant implies that the Ritz projection R h : H 1 () C() S h defined by ( R h u, v h ) = ( u, v h ), v h S Γ h, (6) with R h u = I h u on Γ, extends to a bounded linear operator R h : C() S h, and satisfies R h u L () Cl h u L (). (7) Such result has important applications in maximum-norm error estimates of finite element methods for elliptic and parabolic equations [20],[18],[21], [31], discrete resolvent estimates of elliptic operators [3], [6], and maximal L p regularity of finite element solutions of parabolic equations [11], [12],[22]. However, all papers that establish (7) only deal with Dirichlet or Neumann boundary conditions. Thus, (7) has been established for the Dirichlet problem on polygonal domains [27], convex polyhedral domains [19], and smooth domains in R N [29]. The mixed boundary conditions are not covered in the literature so far, however such problems have a number of applications [9]. A similar result in W 1, norm, namely u u h W 1, () C u I h u W 1, () (8) has been established for Dirichlet problems on convex polygonal domains [23], [26] and convex polyhedral domains with quasi-uniform [16] and mildly graded meshes [8] without any logarithmic factor even for the piecewise linear case. The corresponding W 1, estimates were also established for the Stokes problem in convex polyhedra [15] with Dirichlet boundary conditions. Our work was motivated by [27], where the error estimate (4) and as a result (7) were established for Dirichlet problems on polygonal domains. The proof is based on a weak discrete maximum principle (Agnom-Miranda maximum principle), namely for any discrete harmonic function u h satisfying u h L () C u h L (Γ ), (9) ( u h, v h ) = 0, v h S Γ h, (10) with Γ =. Using (9) and the knowledge of the finite element solution on the boundary, the Dirichlet problem in a polygon can be converted to a problem on a larger domain by extending the exact solution out of the polygon. Then (4) can be proved by applying an interior maximum-norm estimates of [28]. Unfortunately, this argument cannot be extended to the Neumann problem or a problem with mixed boundary conditions since the values of the finite element solutions on the boundary in these cases are unknown. In this paper, we prove the weak discrete maximum principle for mixed boundary value problems on convex polygons, where Γ is only a proper subset of the boundary on which the Dirichlet bounder condition is specified. Then we apply this weak discrete maximum principle to prove the maximum-norm error estimate (4) by converting the mixed boundary conditions to a pure Neumann problem. Our main results are presented in the following two theorems. The first theorem establishes a weak discrete maximum principle for the problem with mixed boundary conditions. Theorem 1 (Weak discrete maximum principle) Let be a convex polygon, Γ be the union of some edges of, and u h S h be a discrete harmonic function satisfying (10). Then there exists a constant C, independent of h and u h, such that u h L () C u h L (Γ ).

3 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 3 The second result establishes the best approximation property of the error in L norm. Theorem 2 (Best approximation property) Let be a convex polygon, Γ be the union of some edges of, and u h S h be the finite element solution of (1) defined by (2) with u h = I h u Γ on Γ. Then there exists a constant C independent of u, u h and h such that where l h is given in (5). u u h L () Cl h u I h u L (), Corollary 1 (Maximum-norm stability) By the triangle inequality one can immediately deduce that Theorem 2 is equivalent to the stability of the Ritz projection in L norm, i.e. u h L () := R h u L () Cl h u L (). Remark 1 Since the regularity of the solution of a mixed boundary value problem on convex polygonal domains is similar to the regularity of a Dirichlet problem on nonconvex polygonal domains, our results for mixed boundary value problems on convex polygonal domains are as sharp as the results for Dirichlet problems on nonconvex polygonal domains. The rest of the paper is organized as follows. In the next section we provide some important local and global regularity results for the continuous problem (1) to be used later in the proofs. In Section 3 we provide proofs of Theorems 1 and 2. Finally, Section 4 is devoted to a proof of the stability of the Ritz projection for a pure inhomogeneous Neumann problem which plays a central role in the proofs of the main results Theorem 1 and Theorem 2. 2 Preliminaries Throughout the paper we use the usual notation for Lebesgue and Sobolev spaces. We denote by (, ) the L 2 () inner-product and we will specify a subdomain in the case it is not the whole. Throughout this paper we assume that is a convex polygonal domain. The boundary segments of are denoted by Γ j, j = 1,..., M with the corresponding angles α j, j = 1,..., M. Without loss of generality we assume α M < π to be the largest interior angle of. 2.1 Global Regularity Results In this section we state the regularity results for the Poisson equation with the Neumann boundary conditions { Δu = f, in, with the compatibility condition n u = g, The first result is H 2+s () estimate of the solution to (1). on, f + g = 0. (1) Lemma 1 Let s [0, 1 α M π where ) and u H 1 () be a solution of (1) with f H s () and g H 1/2+s piecewise ( ), H 1/2+s piecewise ( ) = {q L2 ( ) : q H 1/2+s (Γ i ), i = 1,..., M}, with = M i=1 Γ i. Then there exists a constant C such that u H 2+s () C( f H s () + g H 1/2+s piecewise ( )). The case s = 0 can be found in [2]. For s (0, 1 α M π two lemmas. ), the H 2+s regularity is a consequence of the following

4 4 Dmitriy Leykekhman, Buyang Li Lemma 2 For s (0, 1 α M π we have ), suppose that u H 1 () is a solution of (1) with g = 0 and f H s (). Then u H 2+s () C f Hs (). (2) Lemma 3 Let s (0, 1 α M π H 2+s () such that ). Then there exists a bounded linear lifting operator A : H 1/2+s piecewise ( ) n Ag = g. Lemma 2 is a special case of (23.3) in [7] and Lemma 3 is a consequence of Theorem 5.2 and Corollary 5.3 in Chapter 1 of [4]. 2.2 Local Regularity Results For the Neumann problem (1) with g = 0, by using Lemma 1 we can show the following local regularity result. Lemma 4 (Local H 2+s regularity in balls) Let u be a solution to (1) with g = 0 and s [0, 1 α M π ). Let B d (z) B 2d (z) be balls of radius d and radius 2d centered at the point z, respectively, and set d = B d (z) and d = B 2d(z). Then the following estimate holds: ( ) u H 2+s ( d ) Cd s f L 2 ( d ) + d f L 2 ( d ) + u L ( d ). Proof Let z be any point in d and set u = (u u(z))ω. We consider a smooth cut-off function ω with the following properties There holds: ω(x) 1, x B d (z) (3a) ω(x) 0, x B 2d (z) (3b) k ω C k d k, k = 1, 2,... (3c) { Δ u = ((u u(z)) ω) u ω + fω, in, n u = (u u(z)) ω n on, and therefore u satisfies the following equation { Δ u = f, in, n u = g n on, (4) (5) where f = fω 2 u ω (u u(z))δω and g = (u u(z)) ω, and the compatibility condition is satisfied, i.e. f + g n = 0. Note that fω L 2 () C f L 2 ( d ) and fω H 1 () Cd 1 f L 2 ( d ) + C f L 2 ( d ), which imply by the interpolation and Young s inequalities, fω Hs () fω 1 s L 2 () fω s H 1 () Cd s f L2 ( d ) + C f 1 s L 2 ( d ) f s L 2 ( d ) Cd s f L 2 ( d ) + Cd 1 s f L 2 ( d ). (6)

5 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 5 Since the domain d is convex, any two points z 1, z 2 d can be connected by a line of length at most 4d in d, by the properties of ω and the Hölder inequality, we have and (u u(z))δω L 2 () Cd 2 u u(z) L 2 ( d ) Cd 1 u u(z) L ( d ) C u L ( d ) (u u(z))δω H 1 () (u u(z))δω L 2 () + (u u(z))δω L 2 () + (u u(z)) Δω L 2 () which imply Cd 2 u u(z) L 2 ( d ) + Cd 2 (u u(z)) L 2 ( d ) + Cd 3 u u(z) L 2 ( d ) Cd 1 u L ( d ), (u u(z))δω Hs () (u u(z))δω 1 s L 2 () (u u(z))δω s H 1 () Cd s u L ( d ). (7) Similarly, we have u ω L 2 () Cd 1 u L 2 ( d ) and u ω H1 () Cd 2 u L2 ( d ) + Cd 1 u H2 ( d ), which imply by the interpolation and Young s inequalities, u ω Hs () u ω 1 s L 2 () u ω s H 1 () Cd (1 s) u 1 s L 2 ( d )(Cd 2 u L 2 ( d ) + Cd 1 u H 2 ( d ) ) s Cd 1 s u L 2 ( d ) + Cd 1 u 1 s L 2 ( d ) u s H 2 ( d ). (8) By using Lemma 1, (6), (7) and (8), we obtain and for s (0, 1 α M π ) u H 2 ( d ) = u H 2 ( d ) u H 2 () C f L2 () + C g n H 1/2 piecewise ( ) C f L2 () + C g H1 () C f L 2 ( d ) + Cd 1 u L 2 ( d ) + C u L ( d ) C f L2 ( d ) + C u L ( d ), u H 2+s ( d ) = u H 2+s ( d ) u H 2+s () C f H s () + C g n H 1/2+s piecewise ( ) C f H s () + C g H 1+s () Cd s f L2 ( d ) + C f 1 s L 2 ( d ) f s L 2 ( d ) + Cd 1 s u L 2 ( d ) + Cd 1 u 1 s L 2 ( d ) u s H 2 ( d ) + Cd s u L ( d ) Cd s f L 2 ( d ) + Cd 1 s f L 2 ( d ) + Cd 1 s u L2 ( d ) + Cd s u H2 ( d ) + Cd s u L ( d ) Cd s f L 2 ( d ) + Cd 1 s f L 2 ( d ) + Cd s u H 2 ( d ) + Cd s u L ( d ). By substituting the H 2 estimate into the H 2+s estimate, we complete the proof.

6 6 Dmitriy Leykekhman, Buyang Li Since any dyadic annulus R d (x 0 ) = {x R 2 : d x x 0 < 2d} can be covered by a finite number of balls B 2/d (z j ), j = 1,..., M, where z j R d and M is bounded by constant. Lemma 4 immediately implies the following result. Lemma 5 (Local H 2+s regularity in annulus) Let u be a solution to (1) with g = 0 and s [0, 1 α M π ). Let d = {x : d x x 0 < 2d} and d = {x : d/2 x x 0 < 4d}. Then the following estimate holds: ( ) u H 2+s ( d ) Cd s f L 2 ( d ) + d f L 2 ( d ) + u L ( d ). 2.3 Discretization and local energy estimate For h (0, h 0 ], h 0 > 0, let T denote a quasi-uniform triangulation of with a mesh size h, i.e., T = {τ} is a partition of into cells (triangles) τ of diameter h τ such that for h = max τ h τ, diam(τ) h C τ 1 2, τ T, hold. Then the usual nodewise interpolant I h : C() S h has the following approximation properties (cf., e. g., [5, Theorem 3.1.5]) u I h u L q () Ch 2+2( 1 q 1 p ) u W 2,p (), for q p > 1, (9a) u I h u L () Ch 1+α u C 1+α (), for 0 α 1, (9b) where C 1+α () is the space of Hölder continuous functions. We will also require the following superapproximation result [24]. Lemma 6 Assume 0 1 with {x : dist(x, 0 ) < d} 1 and let ω be a smooth cut-off function satisfying ω(x) 1, x 0 (10a) ω(x) 0, x 1 (10b) k ω C k d k, k = 1, 2,... (10c) Then there exists a constant C independent of h and d such that for any χ S h (ω 2 χ I h (ω 2 χ)) L 2 () Ch ( d 1 χ L 2 ( 1) + d 2 χ L 2 ( 1)). We will also require a local energy estimate for a problem with mixed boundary conditions that is valid up to the boundary. Lemma 7 Suppose that u H 1 Γ () and u h S Γ h satisfy ( (u u h ), v h ) = 0 v h S Γ h, and 0 1 with {x : dist(x, 0 ) < d} 1, ω is a smooth cut-off function satisfying ω(x) 1, x 0 (11a) ω(x) 0, x 1 (11b) k ω C k d k, k = 1, 2,... (11c) Then u u h H 1 ( 0) C ( u I h u H 1 ( 1) + d 1 u I h u L 2 ( 1) + d 1 u u h L 2 ( 1)).

7 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 7 Proof Let ω be the cut-off function satisfying (11). Then inserting I h u and using that I h (ω 2 (I h u u h )) S Γ h and the superapproximation Lemma 6, we have (u u h ) 2 L 2 ( 0) ω (u u h) 2 L 2 ( 0) = (ω2 (u u h ), (u u h )) = (ω 2 (u u h ), (u I h u)) + (ω 2 (u u h ), (I h u u h )) ω (u u h ) L2 () ω (u I h u) L2 () + ( (u u h ), [ω 2 (I h u u h ) I h (ω 2 (I h u u h ))]) (2ω (u u h ), (I h u u h ) ω]) 1 2 ω (u u h) 2 L 2 () ω (u I hu) 2 L 2 () + Chd 1 (u u h ) L2 ( 1)( (I h u u h ) L2 ( 1) + d 1 I h u u h L2 ( 1)) + Cd 1 ω (u u h ) L 2 () I h u u h L 2 ( 1) 1 2 ω (u u h) 2 L 2 () ω (u I hu) 2 L 2 () + Chd 1 (u I h u) 2 L 2 ( 1) + Cd 2 u I h u 2 L 2 ( 1) + Chd 1 (u u h ) 2 L 2 ( 1) + Cd 2 u u h 2 L 2 ( 1). Canceling both sides by 1 2 ω (u u h) 2 L 2 () and taking the square root, the above estimate implies u u h H1 ( 0) C u I h u H1 ( 1) + Cd 1 u I h u L2 ( 1) Iterating the argument and using the inverse estimate, we obtain + Ch 1 2 d 1 2 u uh H 1 ( 1) + Cd 1 u u h L 2 ( 1). u u h H1 ( 0) C u I h u H 1 ( 1) + Cd 1 u I h u L 2 ( 1) + Chd 1 u u h H1 ( 1) + Cd 1 u u h L2 ( 1) C u I h u H 1 ( 1) + Cd 1 u I h u L 2 ( 1) + Chd 1 I h u u h H1 ( 1) + Cd 1 u u h L2 ( 1) C u I h u H 1 ( 1) + Cd 1 u I h u L 2 ( 1) + Cd 1 I h u u h L 2 ( 1) + Cd 1 u u h L 2 ( 1) C u I h u H1 ( 1) + Cd 1 u I h u L2 ( 1) + Cd 1 u u h L2 ( 1). The proof of Lemma 7 is complete. 3 Proof of Theorems Proof of Theorem 1 To prove Theorem 1, we choose an arbitrary fixed x 0 and prove that u h (x 0 ) C u h L (Γ ), where the constant C is independent of x 0. Let B d (x 0 ) denote the ball of radius d centered at x 0 and let S d (x 0 ) := B d (x 0 ). Then we have the following lemma, which is a consequence of the maximum norm estimates proved in [28] (also see [27]). Lemma 8 Let ρ = max(dist(x 0, ), 2h). Then we have u h (x 0 ) Cρ 1 u h L2 (S ρ(x 0)).

8 8 Dmitriy Leykekhman, Buyang Li In view of Lemma 8, it suffices to estimate u h L 2 (S ρ(x 0)) in order to prove Theorem 1. By a duality argument, it suffices to estimate (u h, φ) for any given function φ L 2 (S ρ (x 0 )) which is supported in S ρ (x 0 ) and φ L2 (S ρ(x 0)) = 1. For this purpose, we define v HΓ 1 () as the solution of Δv = φ in, v = 0 on Γ, (1) n v = 0 on Γ, and v h S Γ h as the solution of ( v h, ψ h ) = (φ, ψ h ), ψ h S Γ h. (2) Then v W 2,p () H 1 Γ () for any p ( 1, 2 2 π/(2α M )), where αm is the maximal interior angle of the polygon (Theorem of [14]), and v h S Γ h is the finite element approximation of v. Since 2 2 π/(2α M ) > 4 3, there exists some p 0 > 4 3 such that v W 2,p0 () C φ L p0 (). (3) Let w be the solution of Δw = 0 in, w = u h on Γ, n w = 0 on Γ, (4) which satisfies the maximum principle (cf. [25]) w L () w L (Γ ) = u h L (Γ ). (5) Then we have (u h, φ) (u h w, φ) + (w, φ) = ( (u h w), v) + (w, φ) = ( u h, v) + (w, φ) = ( u h, (v v h )) + (w, φ) = ( (u h u h ), (v v h )) + (w, φ) ( (u h u h ), (v v h )) + w L () φ L 1 (S ρ(x 0)) ( (u h u h ), (v v h )) + ρ u h L (Γ ) φ L2 (S ρ(x 0)), (6) where u h can be any function in S Γ h. We simply choose u h to be equal u h at the nodes of Γ and equal zero on Γ. If we define D h = {x : dist(x, Γ ) h}, then we have (u h, φ) (u h u h ) L (D h ) (v v h ) L1 (D h ) + ρ u h L (Γ ) (Ch 1 (v v h ) L 1 (D h ) + ρ) u h L (Γ ). (7) Thus, in order to prove the theorem we need to show (v v h ) L1 (D h ) Cρh. (8) To establish (8), we divide the domain D h into S 8ρ (x 0 ) D h and O j D h, j = 0, 1, 2,, J = log 2 (R 0 /8ρ), with O j = {x : d j+1 x x 0 < d j } and d j = 2 j diam(). Then we have O j D h Chd j and (v v h ) L1 (D h ) C (v v h ) L1 (O j D h ) + (v v h ) L1 (S 8ρ(x 0) D h ) h 1 2 d 1 2 j (v v h) L2 (O j D h ) + Ch 1 2 ρ 1 2 (v vh ) L2 (S 8ρ(x 0) D h ). (9)

9 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 9 The second term on the right-hand side of (9) can be estimated as As a result (v v h ) L2 (S 8ρ(x 0) D h ) (v v h ) L2 () (v I h v) L2 () Ch 2 2 p 0 v H 3 2/p 0 () Ch 2 2 p 0 v W 2,p 0 () Ch 2 2 p 0 φ L p 0 (Sρ(x 0)) Ch 2 2 p 0 ρ 2 p 0 1 φ L 2 (S ρ(x 0)). ( ) 3 h ρ 2 (v vh ) L 2 (S 8ρ(x 0) D h ) Ch p 0 ρ 2 p h 2 2 p 0 = Chρ Chρ, (10) ρ where in last step we used that h ρ and 3 2 > 2 p 0. To estimate the first term on the right-hand side of (9), it suffices to consider the sets O j such that O j D h. For such a set O j we have O j {x : dist(x, Γ ) < 8d j }, and by Lemma 7 we have ( ) (v v h ) L 2 (O j D h ) C (v I h v) L 2 (O j ) + d 1 j v I h v L 2 (O j ) + d 1 j v v h L 2 (O j ) Ch 2 2 p 0 v W 2,p 0 (O j ) + Cd 1 j v v h L 2 (O j ). Since v is harmonic in O j, a classical interior estimate for elliptic equations gives and a standard energy estimate shows Moreover, since v = 0 on Γ and O j The last four inequalities imply 2 p p 3 0 v W 2,p 0 (O j ) Cdj v L2 (O j ) + Cdj v L2 (O j ), (11) v L 2 () Cρ φ L 2 (S ρ(x 0)). (12) {x : dist(x, Γ ) < 8d j }, it follows from Lemma 1.1 of [27] that v L 2 (O j ) Cd j v L 2 (). (13) h 1 2 d 1 2 j (v v h) L 2 (O j D h ) C ( h Cρh + C d j ) p 0 ρh + h 1 2 d 1 2 j v v h L 2 (O j ) h 1 2 d 1 2 j v v h L2 (O j ), (14) where in the last step we used h d j and 3 2 > 2 p 0. The second term on the right-hand side above can be estimated by a duality argument. We consider (v v h, φ) for φ L 2 (O j ). For this purpose, we define w H1 Γ () and w h S Γ h by and Δw = φ in, w = 0 on Γ, n w = 0 on Γ, (15) ( w h, ψ h ) = (φ, ψ h ), ψ h S Γ h, (16)

10 10 Dmitriy Leykekhman, Buyang Li respectively. Then we have (v v h, φ) = ( (v v h ), w) = ( (v v h ), (w w h )) (v v h ) L2 () (w w h ) L2 () (v I h v) L2 () (w I h w) L2 () Ch 4 4 p 0 v 3 2 w 3 H p 2 Ch 4 4 p 0 v 0 () H p W 2,p 0 () w W 2,p 0 () 0 () Ch 4 4 p 0 φ L p 0 (Sρ(x 0)) φ L p 0 (O j ) Ch 4 4 p 0 ρ 2 p 1 2 p 1 0 d 0 j φ L2 (S ρ(x 0)) φ L2 (O j ), (17) that implies Since h d j, h ρ and 3 2 > 2 p 0, we have ( ) 2 2 ( ) 3 h 1 2 d 1 2 h p 0 h 2 2 p 0 j v v h L2 (O j ) Cρh. (18) ρ d j h 1 2 d 1 2 j v v h L 2 (O j ) Cρh, which together with (10) establishes (8). The inequalities (7) and (8) show which together with Lemma 8 implies u h L2 (S ρ(x 0)) Cρ u h L (Γ ), (19) u h (x 0 ) C u h L (Γ ). (20) This completes the proof of Theorem Proof of Theorem 2 For u, the solution of (1), we define u h as the finite element solution of ( (u u h ), ψ h ) = 0, ψ h S h. (21) with the condition (u u h) dx = 0 for the uniqueness of u h. In other words, u h is the finite element approximation of u with certain inhomogeneous Neumann boundary condition. We shall use the following lemma, whose proof is deferred to the next section. Lemma 9 In a convex polygon, we have Now we have u h L () Cl h u L (). (22) u h L () u h u h L () + u h L () u h u h L () + Cl h u L (). (23) From (1), (2) and (21) we see that which together with Theorem 1 implies ( (u h u h ), ψ h ) = 0, ψ h S Γ h, u h u h L () C u h u h L (Γ ) = C I h u u h L (Γ ) Cl h u L (). (24) Overall, (23) and (24) imply Theorem 2.

11 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 11 4 Proof of Lemma 9 To establish the result we require pointwise estimates of the Green s function for the pure Neumann problem. Such estimates on convex polygonal domains are establish for example, in [1]. Lemma 10 Let R 2 be a convex polygonal domain and Γ (x, ξ) be the Green s function defined by { ΔΓ (, ξ) = δ( ξ) 1 in, n Γ (, ξ) = 0 on, (1) where δ(x ξ) is the delta function satisfying δ(x ξ)φ(x) dx = φ(ξ) for any φ C(). We also impose the normalization condition Γ (x, ξ) dx = 0 for the uniqueness of the solution. Then there exists a constant C such that x Γ (x, ξ) C x ξ 1. (2) The following local energy estimate can be proved in the same way as Lemma 7. Lemma 11 Suppose that u H 1 () and u h S h satisfy ( (u u h ), v h ) = 0 v h S h, with (u u h) dx = 0, and 0 1 with {x : dist(x, 0 ) < d} 1, then u u h H1 ( 0) C ( u I h u H1 ( 1) + d 1 u I h u L2 ( 1) + d 1 u u h L2 ( 1)), where I h is the Lagrange interpolant. For a fixed x 0, we shall prove that u h (x 0 ) Cl h u L (), where the constant C does not depend on x 0. Without loss of generality, we suppose that x 0 is contained in a triangle τ 0. We introduce a smooth Delta function [32, Lemma 2.2], which we will denote by δ h x 0 := δ C 2 0(τ 0 ) such that ( δ, χ) τ0 = χ(x 0 ), χ S h, (3a) δ(x) dx = 1, (3b) δ W l,p () Ch l 2(1 1 p ) for 1 p, l = 0, 1, 2. (3c) Thus in particular δ L1 () C, δ L 2 () Ch 1, and δ L () Ch 2. Let G(x, x 0 ) be the regularized Green s function defined by { ΔG(, x0 ) = δ x0 ( ) 1 in, n G(, x 0 ) = 0 on. Since ( δ(x) 1 ) dx = 0, the equation above admits a unique solution up to a constant. Let G h(, x 0 ) S h be the discrete Green s function, i.e. finite element solution of ( G h (, x 0 ), v h ) = v h (x 0 ) 1 v h (x) dx, v h S h. The function G h (, x 0 ) S h is also well defined up to a constant. For the uniqueness, we further impose the condition G(x, x 0 ) dx = G h (x, x 0 ) dx = 0.

12 12 Dmitriy Leykekhman, Buyang Li Then we have u h(x 0 ) (I h u)(x 0 ) 1 = ( G h, ( u h I h u)) = ( G h, (u I h u)) ( u h I h u) dx = ( (G h G), (u I h u)) + ( G, (u I h u)) = ( (G h G), (u I h u)) + u(x 0 ) I h u(x 0 ) 1 (u I h u) dx = ( Δ(G h G), u I h u) τ + ([ n (G h G)], u I h u) e τ T h e E h + u(x 0 ) I h u(x 0 ) 1 (u I h u) dx ( ) Δ(G h G) L 1 (τ) + [ n (G h G)] L 1 (e) + C u I h u L () τ T h e E h C ( G h G W 2,1 h () + h 1 (G h G) L1 () + 1 ) u I h u L (), where we have used the notation and the trace inequality v W 2,1 () := h τ T h v W 2,1 (τ) v L1 (e) C ( h 1 v W 1,1 (τ) + v W 2,1 (τ)). We will estimate G h G W 2,1 h () + h 1 (G h G) L 1 () in the next two subsections for the two cases r = 1 and r 2, respectively, via the decomposition where G h G W 2,1 h () + h 1 (G h G) L1 () G h G W 2,1 h (O*) + h 1 (G h G) L1 (O *) ( + Gh G W 2,1 h (Oj) + ) h 1 (G h G) L 1 (O j) O * = {x : x x 0 < d J+1 }, O j = {x : d j+1 x x 0 < d j }, j = 0, 1,, J, (5b) ( ) with d j = 2 j diam() and J = log diam() 2 κh. Here κ is a positive parameter to be determined later, and in the rest part of this paper we shall keep the generic constant C to be independent of κ until the parameter κ is determined. The following lemma concerns some local estimates of the regularized Green s function, which will be used in the next two subsections. Lemma 12 Let s (0, 1 π α M ), x 0 τ 0 and dist(τ 0, O j ) d j /C. Then the regularized Green s function G satisfies the following estimates G H1 (O j) C, (4) (5a) (6a) G L (O j) + G H 2 (O j) Cd 1 j, (6b) G H 1+s (O j) Cd s j, (6c) G H 2+s (O j) Cd 1 s j. (6d)

13 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 13 Proof We use the Green s function representation ( G(x, x 0 ) = Γ (x, y) δ(y x 0 ) 1 ) dy = Γ (x, y) δ(y) dy, where Γ (x, y) is the Green s function of the Neumann problem, which also satisfies the normalization condition Γ (x, y) dy = 0. For x O j, we have x G(x, x 0 ) x Γ (x, y) δ(y) dy Cd 1 j δ L 1 (τ 0) Cd 1 j, τ 0 where we used Lemma 10 and the fact that dist(x 0, O j ) d j /C. Hence and the Hölder inequality also implies G L (O j) Cd 1 j, (7) G L 2 (O j) C. (8) Clearly, if we define O j := O j 1 O j O j+1, then we have the similar estimates G L (O j ) Cd 1 j and G L2 (O j ) C. (9) To obtain (6d), we apply Lemma 5 to G H 2+s (O j) and using that δ is supported outside of O j, we obtain G H 2+s (O j) Cd s 1 j + L 2 (O j ) Cd s j G L (O j ) Cd 1 s j. (10) 4.1 Linear elements For piecewise linear elements (the case r = 1), we have G h G W 2,1 h () = G W 2,1 (). To estimate G W 2,1 () we use W 2,p regularity for convex domains. Thus for any p > 1, ( ) 1 G W 2,1 () G W 2,p () C p δ L p () ) C p ( δ Lp () + 1 C p h 2(1 1/p), Lp () where the precise behavior of the constant C p 1 p 1 as p 1 can be traced for example from Theorem 9.9 in [13]. Choosing p = 1 + 1/l h we obtain It remains to prove that G W 2,1 () Cl h. (11) (G h G) L 1 () Cl h h. (12) Since O * C(κh) 2 using the approximation theory and H 2 regularity, we have (G G h ) L 1 (O *) Cκh (G G h ) L 2 (O *) C κ h 2 G H 2 () C κ h 2 δ L 2 () C κ h, (13) where we have used the standard energy error estimate and (3c) in the last inequality. By applying the Hölder inequality and Lemma 11, we also obtain (G G h ) L1 (O j) Cd j (G G h ) L2 (O j) Cd j G I h G H1 (O j ) + C G I h G L2 (O j ) + C G G h L2 (O j ) Chd j G H2 (O j ) + C G G h L2 (O j ) Ch + C G G h L 2 (O j ),

14 14 Dmitriy Leykekhman, Buyang Li where we have used Lemma 12 in the last step of the inequality above. We see that (G G h ) L1 () (G G h ) L1 (O *) + Cκh + Ch + Cκh + Cl h h + (G G h ) L1 (O j) C G G h L 2 (O j ) C G G h L 2 (O j ) Cκh + Cl h h + 1 1/2 G G h 2 L 2 (O j ) Cκh + Cl h h + Cl 1/2 h G G h L 2 () Cκh + Cl h h + Cl 1/2 h h2 δ L 2 C(κ + l h )h. This proves (12) and completes the proof of Lemma 9 in the case r = 1. 1/2 (14) 4.2 Higher order elements To estimate G h G W 2,1 () by the triangle inequality and the inverse inequality, we have G h G W 2,1 () G h I h G W 2,1 () + G I h G W 2,1 () Ch 1 G h I h G W 1,1 () + G I h G W 2,1 () Ch 1 ( G h G W 1,1 () + G I h G W 1,1 ()) + G Ih G W 2,1 (). The last two terms in the inequality above are estimated in the following lemma. Lemma 13 For r 2, there exists a constant C independent of h such that h 1 G I h G W 1,1 () + G I h G W 2,1 () C. Proof Using the dyadic decomposition we have h 1 G I h G W 1,1 () + G I h G W 2,1 () h 1 G I h G W 1,1 (O *) + G I h G W 2,1 (O *) + h 1 G I h G W 1,1 (O j) + G I h G W 2,1 (O j). j=1 (15) Using the Cauchy-Schwarz inequality and H 2 regularity, and the properties of δ, we obtain h 1 G I h G W 1,1 (O *) + G I h G W 2,1 (O *) C G I h G H1 (O *) + h G I h G H2 (O *) Ch G H 2 () Ch δ L 2 () C. (16) To estimate the terms in the sum we use the Cauchy-Schwarz inequality together with the approximation theory and (6d) to obtain h 1 G I h G W 1,1 (O j) + G I h G W 2,1 (O j) ( Cd j h 1 ) G I h G H 1 (O j) + G I h G H 2 (O j) = Cd j h s G H 2+s (O j) Cd s j h s, (17)

15 Maximum-norm stability of the finite element Ritz projection with mixed boundary conditions 15 for some 0 < s < 1 π α M. Combing (15) (17), we obtain h 1 G I h G W 1,1 () + G I h G W 2,1 () C + To finish the proof of of Lemma 9, it remains to prove that j=1 h s d s j C. (G h G) L 1 () Ch. (18) Using the dyadic decomposition and the Cauchy-Schwarz inequality again, we obtain (G G h ) L1 () (G G h ) L1 ( *) + (G G h ) L1 (O j) j=1 Ch (G G h ) L2 ( *) + d j (G G h ) L2 (O j). Using the global best approximation property of the Galerkin solution in H 1 norm we obtain Thus we have where (G G h ) L2 ( *) (G G h ) L2 () Ch G H2 () Ch δ L2 () C. (G G h ) L1 () Ch + j=1 M j, (19) j=1 M j := d j (G G h ) L 2 (O j). (20) By the local energy estimates, Lemma 11, and using Lemma 5 and that h d j, we obtain ) d j (G G h ) L 2 (O j) C ( d j (G I h G) L 2 (O j ) + G I h G L 2 (O j ) + G G h L 2 (O j ( ) ) C d j h 1+s G H 2+s (O j ) + G G h L 2 (O j ) Cd s j h 1+s + C G G h L 2 (O j ). Since j d s j h s C, we only need to estimate G G h L 2 (O j ). We will accomplish that by a duality argument. By duality G G h L 2 (O j ) = sup ψ C 0 (O j ) ψ L 2 () 1 (G G h, ψ). For each such ψ, we define w as the solution of { Δw = ψ ψ in, (22) n w = 0 on, where ψ = 1 ψ(x)dx. Such a solution w exists and is unique if we impose the condition w(x) dx = 0. Thus, (G G h, ψ) = (G G h, ψ ψ) = (G G h, Δw) = ( (G G h ), w) = ( (G G h ), (w I h w)) = ( (G G h ), (w I h w)) O j + ( (G G h), (w I h w)) O j := I 1 + I 2. Using the Cauchy-Schwarz inequality and H 2 regularity I 1 (G G h ) L2 (O j ) (w I h w) L2 () Ch (G G h ) L2 (O j ). (21)

16 16 Dmitriy Leykekhman, Buyang Li To estimate the second term we the Hölder inequality, the approximation theory and embedding H 2+s C 1+s, to obtain I 2 = ( (G G h ), (w I h w)) O j (G G h) L1 () (w I h w) L ( O j ) for some s (0, 1 π α M ). Using the representation w(x) = Γ (x, y) ( ψ(y) ψ(y) ) dy = similarly to the Lemma 12 for x O j That implies that Therefore, (G G h ) L 1 ()Ch s w C 1+s ( O j ) (G G h ) L 1 ()Ch s w H 2+s ( O j ), Γ (x, y)ψ(y) dy, and since O j is separated from O j by at least d j, we obtain w H 2+s ( O j ) Cd s j. ( (G G h ), (w I h w)) O j Chs d s j (G G h ) L1 (). G G h L 2 (O j ) Ch s d s j (G G h ) L 1 () + Ch (G G h ) L 2 (O j ). To summarize, for some s (0, 1 π α M ), we have Summing over j we obtain M j Ch(h/d j ) s + C(h/d j ) s (G G h ) L 1 () + Ch (G G h ) L 2 (O j ). M j Ch κ s where we have used that (h/d j ) s h s Clearly, + C κ s (G G h) L 1 () + C κ d j (G G h ) L2 (O j ) C 2 js Ch s 2 sj Cκ s and d 1 j C d j (G G h ) L 2 (O j ), d 1 J. M j + Cκh (G G h ) L2 (O *) M j + Cκh. Thus, using that h/d J κ 1, and taking κ large enough we have M j Ch + C κ s (G G h) L1 (). Therefore, if we plug this result into (19) we get Again by choosing κ large enough we can conclude (G G h ) L1 () Ch + C κ s (G G h) L1 (). (G G h ) L1 () Ch. This proves (18) and completes the proof of Lemma 9 in the case r 2.

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