Maximum norm estimates for energy-corrected finite element method

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1 Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de, wohlmuth@ma.tum.de Abstract. Nonsmoothness of the boundary of polygonal domains limits the regularity of the solutions of elliptic problems. This leads to the presence of the so-called pollution effect in the finite element approximation, which results in a reduced convergence order of the scheme measured in the L 2 and L -norms, compared to the best-approximation order. We show that the energy-correction method, which is known to eliminate the pollution effect in the L 2 -norm, yields the same convergence order of the finite element error as the best approximation also in the L -norm. We confirm the theoretical results with numerical experiments. 1 Introduction Let Ω R 2 be an open and simply connected polygonal domain. Consider a model Poisson problem u = f in Ω, and u = 0 on Ω. (P) The regularity of the solution of such a problem, when considered on a smooth domain, depends only on the regularity of the given data f. This, however, is not true anymore, when polygonal domains are considered, since then singular functions of the form s i (r, φ) = η(r)r λi sin(λ i φ), λ i = iπ/θ, (1) enter the solution [7][Section 2.1.1]. Here, Θ is the angle of the considered corner and (r, φ) are polar coordinates defined around it. Also, η(r) is a smooth cut-off function equal to 1 for r < r and equal to 0, when r > r, for some positive constants r, r. Note that r can be chosen so that the cutoff functions around different corners have disjoint supports. For the sake of notational simplicity, we focus our attention on one of the corners only. This is justified since all the results presented here have a local nature and naturally extend to the case of multiple corners. Maximum norm error estimates for the standard finite element approximation of elliptic problems are well-studied, e.g., in [8,10,12] in multiple contexts. It was shown in [11] that the presence of the non-smooth corners in the computational domain reduces the convergence order of the finite element method in the maximum norm. A similar phenomenon, known as a

2 2 Swierczynski, Wohlmuth pollution effect, see, e.g., [2,3], is also known when the error is measured in the L 2 -norm. The standard approach for mitigating the pollution effect in the finite element solution includes mesh grading, for which maximum error estimates exist, see, e.g., [1]. Recently an alternative approach was developed - the energy-correction based on a local modification of the bilinear form governing the problem [4]. This work was inspired by similar considerations in the context of finite difference method [13] and has been later extended to piecewise polynomial approximation spaces in [5]. We build on the analysis presented there and show that the energy-correction method converges optimally in the sense of the best approximation property when measured in the weighted L -norm. The forthcoming analysis of the energy correction scheme is carried out in weighted Lebesgue and weighted Sobolev spaces, which constitute a very convenient framework for describing the regularity of elliptic problems on polygonal domains [7]. We define for R, l Z + {0} and 1 p the weighted Sobolev spaces W l,p (Ω), as spaces of measurable functions v : Ω R, for which the following norms are finite v W l, (Ω) := m l r l+ m D m v L (Ω), and v p := r l+ m D m v p W l,p (Ω) L p (Ω), for 1 p <, m l where m is a multi-index with m = 2 i=1 m i, and D m represents the m- th generalized derivative, see, e.g., [6]. We also use the standard notation W 0,p (Ω) = Lp l,2 (Ω) and W Furthermore, we define N l,σ (Ω) = Hl (Ω). (Ω) spaces of weighted Hölder continuous functions, where σ (0, 1), consisting of all l-times continuously differentiable functions, for which the following norm is finite v N l,σ (Ω) = r σ l+ m D m v C 0 (Ω) m l + sup x m =l 1,x 2 Ω r(x 1 ) D m v(x 1 ) r(x 2 ) D m v(x 2 ) x 1 x 2 σ. Note that for l Z + such an embedding holds N l,σ l, (Ω) W (Ω). The following regularity result is a consequence of [7][Theorem 2.6.5]. Theorem 1. Let us for any ɛ > 0 take max(0, k + 1 λ 1 + ɛ), k λ 1 < α < k and α = α k + 1. Assume that f N k 1,σ (Ω) H k 1 α (Ω) for some σ > 0. Then u W k+1, (Ω) Hα k+1 (Ω) H0 1 (Ω) is continuous. Moreover, in the neighborhood of the corner, the solution admits the following expansion u = U + k i s i, i: λ i<k+1

3 Maximum norm estimates for EC FEM 3 and the regular part of the solution satisfies U W k+1, (Ω) H k+1 α (Ω) H 1 0 (Ω). Furthermore, for some c > 0 u W k+1, (Ω) c f N k 1,σ (Ω), and u Hα k+1 (Ω) c f H k 1 2 Energy corrected finite element α (Ω). In this section, we give a brief overview of the energy-correction techniques used for improving the convergence order in the finite element approximations of elliptic problems on non-convex domains. Let T be a given uniform triangulation of the computational domain Ω. We define Vh k, to be a space of globally continuous, piecewise polynomial functions of order k Z +, which have values 0 on the boundary Ω. In order to remove the pollution effect from the standard finite element approximation, we introduce a modification of the bilinear form a(, ), which mitigates the stiffness of the problem in the vicinity of the singularity. The modified finite element approximation of (P) reads then: find u m h V h k such that a h (u m h, v h ) = f, v h Ω for all v h V k h, (2) where the bilinear form is defined as a h (u, v) := a(u, v) c h (u, v) and a(u, v) = Ω u v. We assume that a h(, ) is bilinear, continuous and elliptic, and that c h (, ) is symmetric. In [4], the modification was defined on a one-element patch around the re-entrant corner. We use an extension of this idea, so that the correction of the bilinear form is supported in K element layers from the corner. Let S 1 h = {T T h : 0 T }, S i h = {T T : T S i 1 h }, i = 1,..., K. Moreover, we assume that the one element patch around the corner consists of identical isosceles triangles. We can now define the modification as c h (u, v) := K γ i i=1 S i h u v dx. (3) Note that with each layer of elements there is an associated correction parameter γ i, which still needs to be determined. Asymptotically a unique optimal sequence of parameters γ = {γ} K i=1 on a given correction patch exist. Moreover, the sequence of correction parameters depends on the number and shape of the elements T of the correction patch and on the angle Θ of the re-entrant corner only. Several effective procedures for finding it, based on nested Newton strategies, were proposed in [9] for piecewise linear finite element. This modification, together with an alternative version in case of higher order spaces, is discussed in [5]. Due to the choice of the c h (, ), we preserve the

4 4 Swierczynski, Wohlmuth sparsity structure of the stiffness matrix, as only a fixed number of its entries needs to be suitably scaled. The following theorem, providing sufficient conditions for the optimal convergence of the energy-corrected method in the weighted L 2 -norms, was proven in [5][Theorem 2]. Theorem 2. Let k denote the polynomial degree of the finite element space, k λ 1 < α < k, α = α k + 1 and f H k 1 α (Ω). Let the modification c h(, ) be defined as above and satisfy a(s i s m i,h, s i s m i,h) c h (s m i,h, s m i,h) = O(h k+1 ), for all i K. Then for the energy-corrected finite element solution, we obtain the following optimal error estimates (u u m h ) L 2 α (Ω) ch k f H k 1 α (Ω), u um h L 2 α (Ω) ch k+1 f H k 1 α (Ω). The following result describing the convergence of the energy-corrected finite element in negatively weighted norms can be found in [5][Lemma 4] Remark 3. Let u m h V h k be the energy-corrected approximation (2). Suppose that the modification c h (, ) is chosen as in Theorem 2 and let u H α k+1 (Ω) H0 1 (Ω) for some 0 < α < 1. Then for some c > 0 u u m h L 2 α (Ω) ch k+1 u H k+1 α (Ω). 3 Maximum norm error estimates In this section, we investigate the maximum norm error estimates of the energy-corrected finite element scheme introduced above. We begin by stating some auxiliary results. Lemma 4 (Inverse inequality). For all v h Vh k and all α 1 the following estimate holds. Moreover, when the triangulation is uniform, the constant is independent of the choice of the element. v h L α (T ) ch 1 v h L 2 α (T ) for all T T. Proof. The proof follows from the standard scalling argument and the equivalence of finite dimensional norms. Lemma 5 (Interpolation error). Let I h : C(Ω) Vh k denote the standard nodal interpolation operator. Then for any function u W k+1, (Ω) C( Ω) and > 0 the following estimate holds for some constant c > 0 independent of h u I h u L (Ω) ch k+1 u W k+1, (Ω).

5 Maximum norm estimates for EC FEM 5 Lemma 6. Let α > 1 and T T be a single element of the triangulation that lies close to the corner located at the origin, namely max x T x < ch for constant c > 0. Then the following estimate holds for all v h V k h h α v h L 2 (T ) c v h L 2 α (T ). Proof. Let first α 0. If the element T is not attached to the corner, then it lies in the distance of at least h from it and hence the estimate is obvious. Suppose then that the triangle T shares a vertex with the corner. Note that since v h (0) = 0, then the norm v h L (T ) is attained at some point P max, which is one of the remaining nodal points of the finite element basis. Notice also that for some constant ρ k depending only on the order of the finite element space, we have ρ k h r max, where r max is the distance of the point P max from the origin. Thus, we can write h α v h L2 (T ) h α T 1/2 v h L (T ) ρ α k T 1/2 r α max v h (P max ) ρ α k T 1/2 v h L α (T ) c v h L 2 α (T ). In the last step we used the inverse inequality from Lemma 4. Suppose now that α ( 1, 0). Since r ch, we have h α 1 c α r α and h α v h L 2 (T ) c v h L 2 α (T ) Finally, we are in a position to state main result. Theorem 7. Let max(0, k + 1 λ 1 + ɛ), with ɛ > 0 and also k λ 1 < α < k, α = α k + 1. Assume that f N k 1,σ (Ω) H k 1 α (Ω) for some σ > 0. Then the energy corrected finite element approximation (2) of (P) admits the following convergence property (Ω) ch k+1 log h s ( f N k 1,σ (Ω) + f H k 1 (Ω)), α where s = 1, when k = 1 and s = 0 otherwise. Proof. Without loss of generality we assume that the corner lies at the origin. We also assume that Θ > π k+1 since otherwise standard methods provide the desired convergence order of the scheme [11]. Following the line of proof presented in [1,10,11] we introduce a diadic decomposition around this corner of the domain Ω. Let R > 0 and for j = 0,..., I we set Ω J = {x Ω : d J+1 < x < d J }, where d J = 2 J R and d I c h, d I+1 = 0. Moreover, d I is chosen so that the correction patch of (3) is contained in Ω I. We also define Ω 1 = Ω \ I J=0 Ω J. Finally, we set Ω J = Ω J+1 Ω J Ω J 1. In the proof we consider cases J = I 1, I and the case J < I 1 separately. For J < I 1 we can rely on results proven in [10], namely ( u u m h L (Ω J ) c log h s inf χ V k h u χ L (Ω J ) + d 1 J u um h L2 (Ω J ) ).

6 6 Swierczynski, Wohlmuth This result was proven under the Galerkin orthogonality assumption a(u u m h, v h) = 0. Although this does not hold globally for the energy-corrected finite element scheme, it is true for functions v h V k h with support in Ω J. This is sufficient for this result to hold and there is no need to use a more general version provided in [12]. As a consequence, we immediately obtain (Ω J ) d J u um h L (Ω J ) ( c log h s inf u χ L χ Vh k (Ω J ) + u u m h L 2 1 (Ω ). J ) (4) Now, we can move our investigations to subregions, which are close to the corners of the domain, namely, we consider J = I 1, I. Let T T denote the element in the domain s triangulation, in which the maximum error of the scheme, when measured on Ω J only, is attained. Note also that T Ω J. Then, for any χ V k h (Ω J ) (T ) u χ L (Ω J ) + χ u m h L (T ) (5) We now focus our attention on the second term in this estimate. Applications of the inverse inequality between L and L 2 norms of discrete functions and Lemma 6 give for some constants c J, c J > 0 χ u m h L (T ) c J h χ u m h L (T ) c J h 1 χ u m h L 2 (T ) c J χ u m h L 2 1 (T ). A simple application of the Hölder inequality for satisfying > > k + 1 λ 1 leads to Hence, for c J > 0 we have u χ L 2 1 (Ω J ) c u χ L (Ω J ). c J χ u m h L 2 1 (T ) c J u u m h L 2 1 (Ω J ) + c J u χ L 2 1 (Ω J ) c J u u m h L 2 1 (Ω J ) + c J u χ L (Ω J ) Therefore, we obtain for all χ Vh k using (5) ( ) (Ω J ) c u χ L (Ω J ) + u u m h L 2 1 (Ω J ). (6) Combining (4) and (6) we see that for some c > 0 and for any > k + 1 λ 1 ( ) (Ω) c log h s inf u χ L χ Vh k (Ω) + u u m h L 2 1 (Ω). Application of the interpolation error estimate from Lemma 5 and existing energy corrected finite element estimates in weighted L 2 norm stated in Theorem 2 completes the proof. Note that the weight α = 1 is exactly the one π required there. However, when the considered angle satisfies k+1 < Θ < π k, then 1 < 1 < 0 and the results from Remark 3 need to be used instead.

7 Maximum norm estimates for EC FEM 7 Remark 8. The energy-correction needs to be applied to all corners, for which the singular functions (1) influence the regularity stated in Theorem 1. This are exactly the corners, for which Θ > π k+1, where k is the order of polynomials used in the finite element discretization. A similar condition appears also in schemes employing mesh grading on domains with corners, see [12]. 4 Numerical results We consider two examples with known analytical solutions. First, we choose the L-shape domain Ω = ( 1, 1) 2 \ ( [0, 1] [ 1, 0] ) with the largest interior angle of size Θ = 3π/2. In the second example, we focus our attention on the domain Ω = ( 1, 1) 2 \ { x y 0}, which has the angle Θ = 7π/4 located at the origin. We choose a known exact solution u = s 1 + s 2 + s 3, where s i are known singular functions (1) corresponding to the considered angles. For the sake of presentation, it is chosen so that the energy-correction needs to be applied only around the origin, see Remark 8 in a more general case Weighted L error of the ECFEM 10 1 Weighted L error of the ECFEM Error Mesh size h P 1 P 2 P 3 P 1 with EC P 2 with EC P 3 with EC Error Mesh size h P 1 P 2 P 3 P 1 with EC P 2 with EC P 3 with EC Fig. 1. Comparison of weighted L errors of piecewise polynomial finite element schemes with and without energy-correction (EC) for Θ = 3π/2 (left) and Θ = 7π/4 (right). Doted lines present the errors obtained with the standard finite element resulting in 2π/Θ orders of convergence. In the energy-corrected scheme, we set for corresponding discretization orders γ 1 = , γ 2 = ( , ), γ 3 = ( , ) for Θ = 3π/2, and γ 1 = , γ 2 = (0.0737, ), γ 3 = ( , , ) for Θ = 7π/4. Energy-correction improves the convergence properties of the scheme and yields optimal convergence orders in the weighted norms, namely 2, 3 and 4 for respective discretization orders. The parameters γ in the modification (3) are computed using a version of the Newton algorithm described in [5] and in the experiments we choose the weight = k + 1 λ 1, where k = 1, 2, 3 is the order of polynomials used in the discretizations and λ 1 = π/θ. This choice induces a slightly stronger norm than assumed in Section 3 but the optimal convergence order of the

8 8 Swierczynski, Wohlmuth energy-corrected scheme can be observed regardless of this. These numerical tests confirm the theoretical results of Theorem 7. Acknowledgments We gratefully acknowledge the support of the German Research Fundation (DFG) through the grant WO 671/11-1 and, together with the Austrian Science Fund, through the IGDK1754 Training Group. We would also like to thank Dr Johannes Pfefferer for many fruitful and helpful discussions. References 1. T. Apel and J. Pfefferer and A. Rösch, Finite element error estimates on the boundary with application to optimal control, Math. Comput. 84:291 (2014), H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing 28:1 (1982), P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis, vol. II. Finite Element Methods (Part 1), North Holland (1991). 4. H. Egger and U. Rüde and B. Wohlmuth, Energy-corrected finite element methods for corner singularities, SIAM J. Numer. Anal. 52:1 (2014), T. Horger and P. Pustejovska and B. Wohlmuth, Higher order energycorrected finite element methods (submitted), (in review) 6. V.A. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967), V. Kozlov and V.G. Maz ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI 85 (2001). 8. J. Nitsche and A.H. Schatz, Inerior estimates for Ritz-Galerkin methods, Math. Comput. 28 (1974), U. Rüde and C. Waluga and B. Wohlmuth, Nested Newton Strategies for Energy-Corrected Finite Element Methods, SIAM J. on Sci. Com. 36:4 (2014), A1359 A A. H. Schatz and L. B. Wahlbin, Interior Maximum Norm Estimates for Finite Element Methods, Math. Comput. 31:138 (1977), A. H. Schatz and L. B. Wahlbin, Maximum Norm Estimates in the Finite Element Method on Plane Polygonal Domains. Part 1, Math. Comput. 32:141 (1978), A. H. Schatz and L. B. Wahlbin, Maximum Norm Estimates in the Finite Element Method on Plane Polygonal Domains. Part 2, Refinements, Math. Comput. 33:146 (1979), C. Zenger and H. Gietl, Improved difference schemes for the Dirichlet problem of Poisson s equation in the neighbourhood of corners, Numer. Math. 30:3 (1978),