Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems

Transcription

1 Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems Sebastian Franz MATH-M July 9, 2012

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3 Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems Sebastian Franz July 9, 2012 Abstract Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß-Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems. AMS subject classification (2000): 6512, 6530, Key words: singular perturbation, layer-adapted meshes, superconvergence, postprocessing 1 Introduction Consider the convection-diffusion problem given by u bu x + cu = f, in Ω = (0,1) 2 (1.1a) u = 0, on Ω (1.1b) where b β > 0, c b x γ > 0 and 0 < 1. ote that condition on c can always be fulfilled by a transformation v = exp(κx)u for a suitably chosen κ. In [5] it was shown that for bilinear elements and a standard Galerkin method its solution u fulfils on a piecewise uniform Shishkin mesh with mesh cells in each coordinate direction the estimates u u C 1 ln, and u I u C( 1 ln) 2, where u I is the standard piecewise bilinear interpolant of u and the energy-norm is defined as u = ( u γ u 2 0) 1/2. Institut für umerische Mathematik, Technische Universität Dresden, Dresden, Germany. sebastian.franz@tu-dresden.de 3

4 Here and throughout the paper we denote by 0 the standard L 2 -norm on Ω and by C a generic constant independent of and. A property like the one above is called supercloseness and by a simple postprocessing routine P [5] it can be exploited as interpolantwise superconvergence (for the naming convention see [19]) in u Pu C( 1 ln) 2. In [6] a similar result was obtained for a streamline diffusion method [11] under some restrictions on the stabilisation parameters. For higher order methods using Q p -elements with p > 1 so far only for the streamline diffusion method supercloseness results are known. In [3] it was proven that π C( 1 ln) p+1/2 ln holds for the streamline diffusion solution u in the case of Q p -elements on a suitable piecewise uniform Shishkin mesh and conditions on the stabilisation parameters. The interpolant π p is a so called vertex-edge-cell interpolant [9, 15]. In [4] the higher order case was investigated numerically and three unproven phenomena were shown. First, there seems to be a supercloseness result for pointwise interpolation w.r.t. Gauß- Lobatto points. Second, the supercloseness order is actually p + 1 and not only p + 1/2. And finally, these results do also hold for standard, unstabilised Galerkin FEM. In the present paper we prove the first of these numerical results. The theoretical results presented in this paper require regularity of the exact solution u and use solution decompositions given e.g. in [12, 13, 16]. We assume the necessary compatibility conditions and smoothness of the data to be fulfilled. The paper is organised as follows. In Section 2 we introduce a class of layer-adapted meshes used in discretising the differential equation. Moreover, a solution decomposition exploited later in the analysis is presented. Section 3 contains the definition and analysis of some properties of two different interpolation operators. In Section 4 we prove supercloseness properties of our two interpolation operators and in the following Section 5 superconvergent numerical solutions are generated by postprocessing. Finally, Section 6 contains a numerical example verifying the theoretical results. 2 The Mesh and a Solution Decomposition We define the underlying mesh as a member of the general class of S-type meshes [18]. Let the number 4 of mesh cells in each direction be divisible by 4 and a user-chosen positive parameter σ > 0 be given. Assume 1 (4σ ln) 2. (2.1) In practice, this assumption is no restriction as otherwise would be exponentially large compared with. In the latter case the analysis could be done in a standard way. We now define mesh transition parameters by λ x := σ β ln 1 and λ y := σ ln (2.2)

5 1 1 λ y λ y Ω 11 := [λ x,1] [λ y,1 λ y ], Ω 12 := [0,λ x ] [λ y,1 λ y ], Ω 21 := [λ x,1] ( [0,λ y ] [1 λ y,1] ), Ω 22 := [0,λ x ] ( [0,λ y ] [1 λ y,1] ) 0λ x 1 Figure 1: Shishkin mesh T 8 of Ω, the bold lines indicate the boundaries of the subdomains. The domain Ω is dissected by a tensor product mesh according to { σ x i := β φ ( ) i 1, i = 0,...,/2, 1 2(1 λ x )(1 i ), i = /2,...,, σ ( ) φ 2 j, j = 0,...,/4, y j := (1 2λ y )( 2 j 1) + 1 2, j = /4,...,3/4, 1 σ ( ) φ 2 2 j, j = 3/4,...,, and the final mesh T is constructed by drawing lines parallel to the coordinate axes through these mesh points. The function φ is a monotonically increasing, mesh-generating function satisfying φ(0)=0 and φ(1/2)=ln. Given an arbitrary function φ fulfilling these conditions, an S-type mesh is defined and the domain Ω is divided into the subdomains Ω 11, Ω 12, Ω 21, and Ω 22 as shown in Figure 1. Related to the mesh-generating function φ, we define the meshcharacterising function ψ = e φ whose derivative yields information on the approximation quality of the mesh, usually expressed in terms of max ψ where the maximum is taken over t [0,1/2]. Several examples can be found in [18]. In this paper we refer to two of them repeatedly. Those are the piecewise uniform Shishkin-mesh with φ(t) = 2t ln and ψ(t) = 2t, and the Bakhvalov Shishkin mesh with φ(t) = ln(1 2t(1 1 )) and ψ(t) = 1 2t(1 1 ). Besides above properties, we assume the function φ also to fulfil max φ (t) C and min φ t [0,1/2] i=1,...,/2 ( i ) φ ( ) i 1 C 1. We denote by τ i j = [x i 1,x i ] [y j 1,y j ] a specific element and by τ a generic mesh rectangle. ote that the mesh cells are assumed to be closed. Let h i := x i x i 1, k j := y j y j 1 be the dimensions of τ i j and h := ote that it holds, [18, eq. (3.5)] max h i, k := max k j and h min := min h i. i=1,...,/2 j=1,...,/4 i=1,...,/2 h i C 1 max ψ e βx/(σ), i = 1,...,/2, x [x i 1,x i ], (2.3)

6 and similarly for k j, where here and further-on max ψ := max t [0,1/2] ψ (t). Let for a fixed polynomial degree p 2 the finite element space be given by V = { v H 1 0 (Ω) : v τ Q p (τ) τ T }. Assumption 2.1. The solution u of (1.1) can be decomposed as u = v + w 1 + w 2 + w 12, where for all x,y [0,1] and 0 i + j p + 1 the pointwise estimates i+ j v x i y j (x,y) C, i+ j w 1 x i y j (x,y) C i e βx/, i+ j w 2 x i y j (x,y) ( ) C j/2 e y/ + e (1 y)/, i+ j w 12 x i y j (x,y) ( ) C (i+ j/2) e βx/ e y/ + e (1 y)/ and for i + j = p + 2 the L 2 -norm bounds i x j y v 0,Ω C, i x j y w 2 0,Ω C j/2+1/4, i x j y w 1 0,Ω C i+1/2, i x j y w 12 0,Ω C (i+ j/2)+3/4 (2.4) (2.5) hold. Here w 1 covers the exponential boundary layer, w 2 the characteristic boundary layers, w 12 the corner layers, and v is the regular part. Remark 2.2. In [12,13] Kellogg and Stynes proved the validity of Assumption 2.1 for constant functions b,c under certain compatibility and smoothness conditions on f. 3 Interpolation We define two different interpolation operators. The first one [9, 15] is the vertex-edge-cell interpolation operator ˆπ p : C( ˆτ) Q p ( ˆτ), defined locally on the reference element ˆτ := [ 1,1] 2 by ( ˆπ p ˆv ˆv)(â i ) = 0, i = 1,...,4, (3.1a) ( ˆπ p ˆv ˆv) ˆq = 0, i = 1,...,4, ˆq P p 2 (ê i ), (3.1b) ê i ( ˆπ p ˆv ˆv) ˆq = 0, ˆq Q p 2 ( ˆτ), (3.1c) ˆτ where â i are the vertices and ê i the edges of ˆτ. Using the bijective reference mapping F τ : ˆτ τ, this operator can be extended to the global interpolation operator π p : C(Ω) V by (π p v) τ := ( ˆπ p (v F τ )) F 1 τ, τ T, v C(Ω). (3.1d)

7 The second interpolation operator is of Lagrange-type. Let 1 = t 0 < t 1 < < t p = 1, be the zeros of (1 t 2 )L p(t) = 0, t [ 1,1], (3.2a) where L p is the Legendre polynomial of degree p. These points are also used in the Gauß- Lobatto quadrature rule of approximation order 2p 1. Therefore, we refer to them as Gauß- Lobatto points. In literature they are also named Jacobi points [14] as they are also the zeros of the orthogonal Jacobi-polynomials P p (1,1) of order p. The operator Î p : C( ˆτ) Q p ( ˆτ) is then defined on the reference element ˆτ by point evaluations (Î p ˆv)(t i,t j ) = ˆv(t i,t j ), i, j = 0,..., p. (3.2b) With an extension like above we obtain the global interpolation operator I p : C(Ω) V. The interpolation error for both operators can be bounded according to [7,8] using Assumption 2.1. Theorem 3.1. Let σ p + 1. Then it holds for the solution u of (1.1) u I p u C( h + k + 1 max ψ ) p and u π p u C( h + k + 1 max ψ ) p. Remark 3.2. ote that max ψ { 2 ln, Shishkin mesh, 2, Bakhvalov Shishkin mesh, which shows the improvement of the bounds using graded meshes near the boundaries. Lemma 3.3. Let ˆπ p and Î p be the local interpolation operators into Q p ( ˆτ) on the reference element ˆτ defined in (3.1) and (3.2), respectively. Furthermore, let ˆπ p+1 be defined similarly to (3.1) interpolating into the local space Q p+1 ( ˆτ). Then it holds ˆπ p = Î p ˆπ p+1. (3.3) Moreover, if Îp+1 is a Lagrange-interpolant into Q p+1( ˆτ) using the interpolation nodes (ti,t j ) where {ti }, i = 0,..., p + 1 consists of the p + 1 Gauß-Lobatto nodes {t i}, i = 0,..., p from (3.2a) plus one arbitrary node tp+1 ( 1,1), then it follows Î p = ˆπ p Î p+1. (3.4) Proof. We extend an idea given in [10]. For any function v C( ˆτ) holds ˆπ p v Q p ( ˆτ) and Î p v Q p ( ˆτ). To prove the equivalence (3.3) we only have to show that Î p ˆπ p+1 shares the same degrees of freedom as ˆπ p. The definitions (3.1) and (3.2) imply ( ˆπ p v)(â i ) = (Î p ˆπ p+1 v)(â i ) = v(â i ), i = 1,...,4, (3.5a)

8 where â i are the vertices of ˆτ. Furthermore, it holds for ˆq P p 2 (ê i ) (v Î p ˆπ p+1 v) ˆq = (v Î p ˆπ p+1 v (v ˆπ p+1 v)) ˆq = ( ˆπ p+1 v Î p ( ˆπ p+1 v)) ˆq =: V, ê i ê i ê i ê i where V P 2p 1 (ê i ) and ê i is any of the four edges of ˆτ. ow, the integral can be rewritten using a quadrature rule that is exact for polynomials of order 2p 1. We use the Gauß-Lobatto rule and obtain ê i (v Î p ˆπ p+1 v) ˆq = p w j (( ˆπ p+1 v Î p ( ˆπ p+1 v)) ˆq) êi (t j ) = 0, i = 1,...,4, (3.5b) j=0 where {w j } are the weights of the quadrature rule. The last equality comes from (3.2b). In a similar fashion it follows for ˆq Q p 2 ( ˆτ) p (v Î p ˆπ p+1 v) ˆq = w i, j (( ˆπ p+1 v Î p ( ˆπ p+1 v)) ˆq)(t i,t j ) = 0, ˆτ i, j=0 (3.5c) due to the integrand being a polynomial in Q 2p 1 ( ˆτ), the Gauß-Lobatto rule on the rectangle ˆτ and again (3.2b). Comparing (3.5) to (3.1) one concludes (3.3). The second equivalence (3.4) can be concluded easily by the first one: ˆπ p Î p+1 = Î p ˆπ p+1 Î p+1 = Î p Î p+1 = Î p, where we use the property of Îp+1 and ˆπ p+1 being projections into Q p+1 ( ˆτ) in the second step. The last equality holds because Î p uses a subset of interpolation nodes of Îp+1 in its definition. 4 Supercloseness Analysis Let us now come to the numerical method. We define the Galerkin bilinear form by a Gal (v,w) := ( v, w) + (cv bv x,w) and a stabilisation bilinear-form of the streamline-diffusion method [11] by a stab (v,w) := τ T δ τ ( v + bv x cv,bw x ) τ, where the parameters δ τ 0 are user chosen and influence both stability and convergence. They are taken constant in each sub-domain of Ω, i.e. δ τ = δ i j for any τ Ω i j. The streamline-diffusion bilinear-form is then defined as a SD (v,w) := a Gal (v,w) + a stab (v,w) and the streamline-diffusion formulation of (1.1) is given by

9 Find u V such that where a SD (u,v ) = f SD (v ), v V (4.1) f SD (v) := ( f,v) τ T δ τ ( f,bv x ) τ. Theorem 4.1. For Q p -elements, σ p + 1 and under the restrictions on the stabilisation parameters δ 11 = C 1, δ 21 C max{1, 1/2 ( 1 max ψ )}( 1 max ψ ) 2, δ 12 = δ 22 = 0 holds the estimate π C( h + k + 1 max ψ ) p+1/2 ln. Proof. In [3] this result is given for the standard Shishkin-mesh. Together with techniques for S-type meshes, see e.g. [7, 18], the desired bound follows. ote that the additional logarithmic factor is caused by the estimation of the convective term inside the characteristic layers. Remark 4.2. By a close inspection of the proof of Theorem 4.1 in [3, Theorem 13] the sharper result π C( h + k + 1 max ψ ) p+1/2 can be given under the modified restriction δ 21 C( 1 max ψ ) 2. Corollary 4.3. Combining Theorems 3.1 and 4.1 yields the convergence result u u C( h + k + 1 max ψ ) p. To analyse the supercloseness behaviour of the Gauß-Lobatto interpolation operator, consider I π + I p (π p+1 u u) (π p+1 u u) + π p+1 u u. (4.2) Its first term can be estimated by the supercloseness result of Theorem 4.1 and its last term by the interpolation error result of Theorem 3.1 adapted to the case of elements of order p + 1. Thus, we only have to estimate the energy norm of We start with some basic estimates for R u. R u := I p (π p+1 u u) (π p+1 u u).

10 Lemma 4.4. For any w C(τ i j ) holds the stability estimate R w L (τ i j ) C w L (τ i j ). (4.3a) For any w H p+2 (Ω), 1 t p and τ i j T we have the anisotropic error estimates R t+2 w 0,τi j C h t+2 r i k r j x t+2 r y r w 0,τi j, (4.3b) (R w) x 0,τi j C and analogously for (R w) y 0,τi j. t+1 h t+1 r i k r j x t+2 r y r w 0,τi j, (4.3c) Proof. The stability estimate (4.3a) is a direct consequence of the stability of the interpolation operators Ip and πp+1 in L. Their stability holds because all degrees of freedom are pointevaluations or integrals. For (4.3b) we use anisotropic error estimates [1, 7, 17] to obtain ] R w 0,τi j C [ h i (π p+1 w w) x 0,τi j + k j (π p+1 w w) y 0,τi j [ t+1 C h t+2 r i k r j t+2 r x y r t+1 w 0,τi j + h t+1 r i k 1+r j t+1 r x r+1 y w 0,τi j ], which gives (4.3b). For (4.3c) we need additionally anisotropic estimates for the second order derivatives, see again [1] ] (R w) x 0,τi j C [ h i (π p+1 w w) xx 0,τi j + k j (π p+1 w w) xy 0,τi j which gives (4.3c). t C[ h 1+t r i k r j t+2 r x r y w 0,τi j + t h t r i k 1+r j t+1 r x r+1 y w 0,τi j ] Theorem 4.5. Let us assume k C 1/4, σ p + 1 and 1 (max ψ ) 2 C. For R defined above holds R u C( (σ 1/2) (1 + h mesh ) + ( h + k + 1 max ψ ) p+1 ) (4.4) where h mesh := 1 (ln) 1/2 /h min. Remark 4.6. For the mesh specific value h mesh holds on a Shishkin mesh on a Bakhvalov Shishkin mesh and on a general S-type mesh C 1 (ln) 1/2 h mesh C 2 (ln) 1/2, C 1 (ln) 1/2 h mesh C 2 (ln) 1/2, C 1 (ln) 1/2 h mesh C 2 (ln) 1/2.

11 Proof of Theorem 4.5. Let us start with the L 2 -norm estimate. We use the solution decomposition of Assumption 2.1 and start with the regular part v. By (4.3b) with t = p 1 1 we obtain R v 0,Ω C( h + k + 1 ) p+1. (4.5a) Estimate (4.3b) can also be used to bound w 1 in Ω 12 Ω 22, where h i can be estimated by (2.3). R w 1 2 0,Ω 12 Ω 22 C τ i j Ω 12 Ω 22 p+1 h p+1 r i k r j x p+1 r y r w 1 2 0,τ i j p+1 C ( 1 max ψ e βx/(σ) ) p+1 r ( k + 1 ) r (p+1 r) e βx/ 2 0,Ω 12 Ω 22 C( k + 1 max ψ ) p+1 2(p+1) C( k + 1 max ψ ) 2(p+1), e βx ( p+1 r σ 1) 2 0,Ω 12 Ω 22 where we used (p + 1)/σ 1 0. In Ω 11 Ω 21 we use the stability (4.3a) to obtain (4.5b) R w 1 0,Ω11 Ω 21 C w 1 L (Ω 11 Ω 21 ) C σ. (4.5c) The other two layer terms can be estimated similarly and combining these results proves the L 2 -estimates. For the H 1 -component we use (4.3c) with t = p and its counterpart for the y- derivative for the regular solution component v to obtain R v 0,Ω C( h + k + 1 ) p+1. (4.6a) Similarly, using the type of analysis as above, we show R w 1 0,Ω12 Ω 22 C 1/2 ( k + 1 max ψ ) p+1, R w 2 0,Ω21 Ω 22 C 1/4 ( h + 1 max ψ ) p+1, R w 12 0,Ω22 C 1/4 ( 1 max ψ ) p+1. (4.6b) (4.6c) (4.6d) On the other domains we use the decay of the layer terms. We show the analysis exemplary for the three terms yielding the largest bounds and invoking the most assumptions. Let us start with (R w 12 ) x in Ω 12. By (4.3c) with t = 1 we obtain (R w 12 ) x 0,Ω12 C 2 ( 1 max ψ ) 2 r r (3 r) r/2 e βx ( 2 r σ 1) (e y 1/2 + e 1 y 1/2 ) 0,Ω12 C 1/2 σ [( 1 max ψ ) 2 3/4 ]. On the other hand, a triangle and an inverse inequality give (R w 12 ) x 0,Ω12 C [ h 1 min ( I p (π p+1 w 12 w 12 ) 0,Ω12 + π p+1 w 12 0,Ω12 ) + (w 12 ) x 0,Ω12 ].

12 While the last term of the right-hand side can be estimated directly, we use an idea from [20] incorporating the stability of Ip and πp+1 for the other two terms. Ip (πp+1 w 12 w 12 ) 2 0,Ω 12 + πp+1 w ,Ω 12 C (Ip πp+1 w 12) 2 + (Ip w 12 ) 2 + (πp+1 w 12) 2 Ω 12 C C xi i=/2+1 x i 1 [ xi e i=/2+1 x i 1 λx [ x1 C e 2βx 0 dx + 0 C [ h + ] 3/4 y j j=/4+1 2βx i 1 dx 0 y j 1 (w 2 12(x i 1,y j 1 ) + w 2 12(x i 1,y j ))dydx e 2βx ][ 3/4 y j j=/4+1 y j 1 dx ] [ y/4+1 y /4 ( e 2y j 1 1/2 + e 2(1 y j ) 1/2 )dy e 2λy 1/2 dy + [ 1 + 1/2] 2σ C( 1/2 + 1 ) 2σ. ] 1/2 e 2y 1/2 dy λ y Here we have used the symmetry of the pointwise bound of w 12 w.r.t. y. Thus, a second bound for (R w 12 ) x in Ω 12 holds: (R w 12 ) x 0,Ω12 C [h mesh 1 (ln) 1/2 σ ( 1/4 + 1/2 ) 1/2 + 1/4 σ] Combining these two estimates we obtain C 1/2 σ [ 1/4 + h mesh ( 1/4 + 1/2 )]. (R w 12 ) x 0,Ω12 C 1/2 σ min{( 1 max ψ ) 2 3/4,h mesh ( 1/4 + 1/2 )} C 1/2 (σ 1/2) (1 + h mesh ), ] (4.6e) where we used 1 (max ψ ) 2 C in estimating the minimum. The second term we want to look at is (R w 2 ) x in Ω 12. This one highlights in cancelling the logarithmic term, why h mesh is defined as it is. We obtain the two estimates (R w 2 ) x 0,Ω12 C 1/2 σ [( h + 1 ) 2 1/4 ], (R w 2 ) x 0,Ω12 C[h mesh 1 (ln) 1/2 σ ( 1/4 + 1/2 ) + 1/4 σ ] 1/2 (ln) 1/2 and therefore C 1/2 σ h mesh [ 1/4 + 1/2 ] (R w 2 ) x 0,Ω12 C 1/2 σ min{( h + 1 ) 2 1/4,h mesh [ 1/4 + 1/2 ]} C 1/2 (σ 1/2) (1 + h mesh ). As a third term we look at (R w 1 ) x in Ω 21. A similar analysis as above gives for k C 1/4 (4.6f) (R w 1 ) x 0,Ω21 C 1/2 σ min{ 2 k 2,1 + () 1/2 + } C 1/2 σ (1 + ( k) 2/3 ) C 1/2 (σ 1/2). (4.6g)

13 All other terms remaining can be estimated by similar steps. Combining the results gives finally the statement of the theorem. Remark 4.7. Following the proof of Theorem 4.5 and neglecting the restriction on k, one can still show with (4.6g) for σ p + 1 R u C( (σ 2/3) + ( h + k + 1 max ψ ) p+1 ). Theorem 4.8. Let σ p + 2. Then it holds for the streamline-diffusion solution u under the restrictions on the stabilisation parameters given in Theorem 4.1 I C( h + k + 1 max ψ ) p+1/2 ln. Proof. Consider again (4.2) I π + I p (π p+1 u u) (π p+1 u u) + π p+1 u u. Theorem 4.1 gives under conditions on the stabilisation parameters and σ p + 1 π C( h + k + 1 max ψ ) p+1/2 ln, Remark 4.7 gives for σ p + 5/3 I p (π p+1 u u) (π p+1 u u) C( h + k + 1 max ψ ) p+1, and Theorem 3.1 yields for σ p + 2 π p+1 u u C( h + k + 1 max ψ ) p+1. Combining the three estimates completes the proof. 5 Superconvergence by Postprocessing By utilising the supercloseness results of Theorems 4.1 and 4.8 we can construct postprocessing operators. They improve our numerical solution with little additional computational effort to higher convergence order. Suppose is divisible by 8. We construct a coarser macro mesh T /2 composed of macro rectangles M, each consisting of four rectangles of T. The construction of these macro elements M is done such that the union on them covers Ω and none of them crosses the transition lines at x = λ x and at y = λ y or y = 1 λ y, see Figure 2. Remark that in general T /2 T /2 due to different transition points λ x and λ y and the mesh generating function φ. We now define postprocessing operators locally for one macro element M T /2. The first one was presented in 1d in [21] and is a modification of an operator given in [15]. Let ˆv be the linearly mapped function v from any interval [x i 1,x i+1 ] onto the reference interval [ 1,1]. ote that x i is not necessarily mapped onto 0, but to a value a ( 1,1). In [2] the following condition on the underlying mesh is given, that guarantees the non-degenerate

14 1 1 λ y λ y 0 λ x 1 Figure 2: Macroelements M of T /2 constructed from T behaviour of the macro elements and of the operators defined on it: There exists a constant q 1 independent of and such that max{h i,h i+1 } q, min{h i,h i+1 } for all i = 1,...,/2 1, max{k j,k j+1 } q, min{k j,k j+1 } for all j = 1,...,/4 1 and j = 3/4 + 1,..., 1. A Shishkin mesh has q = 1 while a Bakhvalov Shishkin mesh has q = ln(3)/ln(5/3). For many more S-type meshes this condition holds and we assume it further-on. Define the reference operator P vec : C[ 1,1] P p+1 [ 1,1] by for p = 2: while for p 3: 1 1 a P vec ˆv( 1) = v(x i 1 ), P vec ˆv(a) = v(x i ), P vec ˆv(1) = v(x i+1 ), ( P vec ˆv ˆv) = 0, ( P vec ˆv ˆv) = 0, 1 a ( P vec ˆv ˆv) = 0, ( P vec ˆv ˆv)p = 0, p P p 2 [ 1,1] \ R. By using the reference mapping and the tensor product structure we obtain the full postprocessing operator P vec,m : C(M) Q p+1 (M) on each macro element. Then, this piecewise projection is extended to a global, continuous function by setting ( P p+1 vec v ) (x,y) := ( P vec,m v ) (x,y) for (x,y) M. The second postprocessing operator is defined by using only point evaluations. Let {( x i,ỹ j )}, i, j = 0,...,2p denote the ordered sample of Gauß-Lobatto points of the four rectangles that M consists of. Let P GL,M : C(M) Q p+1 (M) denote the projection/interpolation operator fulfilling P GL,M v( x i,ỹ j ) = v( x i,ỹ j ), i, j = 0, 1, 3, 5,...,2p 3,2p 1, 2p.

15 Then, this piecewise projection is extended to a global, continuous function by setting ( P p+1 GL v) (x,y) := ( P GL,M v ) (x,y) for (x,y) M. Lemma 5.1. For the postprocessing operators P p+1 GL and Pvec p+1 defined above holds P p+1 GL I p v = P p+1 GL P p+1 GL v C v, v, Pp+1 vec πp v = Pvec p+1 v, for all v C(Ω), (5.1a) P p+1 vec v C v, for all v V. (5.1b) Let u be the solution of (1.1), Assumption 2.1 be true and σ p + 2. Then it holds P p+1 GL u u + P vec p+1 u u C( h + k + 1 max ψ ) p+1. (5.1c) Proof. The consistency (5.1a) is a direct consequence of the definitions of P p+1 GL and Pvec p+1. The stability (5.1b) can be shown for both operators similarly. Therefore, let P M be any of the local operators P GL,M and P vec,m. The stability (5.1b) then follows by showing P M v 0,M C v 0,M and P M v 1,M C v 1,M for any M T /2 and any v V (M). The operator P M is a linear operator from the finite dimensional space V (M) into the finite dimensional space Q p+1 (M). Thus it is continuous and with v v 0,M being a norm in both spaces we obtain P M v 0,M C v 0,M, for all v V (M). Similarly v v 1,M is a norm on the quotient space V (M)\R and v v 1,M is a norm on Q p+1 (M). Therefore, P M v 1,M P M v 1,M C v 1,M for all v V (M) \ R. Finally, the interpolation error (5.1c) follows by Assumption 2.1 and standard anisotropic estimates for interpolation [1, 8]. Theorem 5.2. Let σ p + 2. Then it holds for the streamline-diffusion solution u under the restrictions on the stabilisation parameters given in Theorem 4.1 u P p+1 GL u + u Pvec p+1 u C( h + k + 1 max ψ ) p+1/2 ln. Proof. Using the consistency and stability of P p+1 GL we obtain u P p+1 GL u u P p+1 GL u + P p+1 GL I p u P p+1 GL u u P p+1 GL u +C Ip u u. Similarly one can show u P p+1 vec u u P p+1 vec u +C π. ow the statement follows by (5.1c) and the supercloseness results of Theorems 4.1 and 4.8.

16 Table 1: Convergence and closeness errors for Q 3 -elements on a Shishkin mesh for = 10 6 with corresponding rates r S u u π 3 u u I 3 u u J 3 u u e e e e e e e e e e e e e e e e e e e e e e e e-06 Table 2: Convergence and closeness errors for Q 3 -elements on a Bakhvalov Shishkin mesh for = 10 6 with corresponding rates r B u u π 3 u u I 3 u u J 3 u u e e e e e e e e e e e e e e e e e e e e e e e e-08 6 umerical Example Let us consider the singularly perturbed problem given by u (2 x)u x + 3/2u = f, in Ω = (0,1) 2 u = 0, on Ω, with a constructed right-hand side, such that ( ) u(x,y) = cos(xπ/2) e x/ e 1/ (1 e y/1/2 )(1 e (1 y)/1/2 ) 1 e 1/ 1 e 1/1/2 is the exact solution. The following calculations were done in Matlab and the linear systems solved by its direct \ - solver. We fix the polynomial degree to p = 3 and the parameter for the Shishkin mesh to σ = p + 2 = 5. Moreover, we set = 10 6, sufficiently small to generate the sharp boundary layers we are interested in. ote that additional computations were done with different polynomial degrees p and varied perturbation parameters supporting the same conclusions. In the following tables, the experimental rates of of convergence for given measured errors e are calculated by p S ln(e /e 2 ) = ln(2ln()/ln(2)), pb = ln(e /e 2 ), ln(2)

17 Table 3: Superconvergence errors for Q 3 -elements on a Shishkin mesh for = 10 6 with corresponding rates r S u P vec u u P GL u e e e e e e e e e e e e-06 Table 4: Superconvergence errors for Q 3 -elements on a Bakhvalov Shishkin mesh for = 10 6 with corresponding rates r B u P vec u u P GL u e e e e e e e e e e e e-08 assuming e = C( 1 ln) ps on a Shishkin mesh and e = C pb on a Bakhvalov Shishkin mesh. The numerical method is the SDFEM given in (4.1) with stabilisation parameters according to the upper bounds in Theorem 4.1, where C is set to 1. Tables 1 and 2 present the convergence and closeness results. We observe third order convergence of the numerical method as predicted by Corollary 4.3. Moreover, the results on a graded mesh like the Bakhvalov Shishkin mesh are much better compared with the piecewise equidistant Shishkin mesh. There are two orders of magnitude difference in the final line for the energy error. The closeness results for the vertex-edge-cell interpolant π3 and the Gauß-Lobatto interpolant I3 as well as a pointwise interpolation operator J 3 using equidistantly spaced interpolation points are also given. Supercloseness of a better order than p + 1/2 = 3.5 can clearly be seen for π3 and I 3, whereas for J 3 the rate is not as high. On the Bakhvalov Shishkin mesh, we observe a clear order 4 for the first two operators and an order 3 for the equidistant interpolant. Tables 3 and 4 now show the results of the postprocessed numerical solutions. We observe for both postprocessing operators on both meshes convergence rates of order p + 1 = 4, which compared with the convergence results presented in Tables 1 and 2 is an increase of a full order. Theorem 5.2 only predicted half an order increase. Thus, it seems that the supercloseness result of Theorem 4.1 is not sharp. By improving this estimate, the improvement of Theorem 5.2

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