A new approach to Richardson extrapolation in the finite element method for second order elliptic problems

Authors:
M. Asadzadeh, A. H. Schatz and W. Wendland

Journal:
Math. Comp. **78** (2009), 1951-1973

MSC (2000):
Primary 65N15, 65N30, 35J25

DOI:
https://doi.org/10.1090/S0025-5718-09-02241-8

Published electronically:
February 11, 2009

MathSciNet review:
2521274

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in , . The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given.

**1.**Ivo Babuška and Michael B. Rosenzweig,*A finite element scheme for domains with corners*, Numer. Math.**20**(1972/73), 1–21. MR**323129**, https://doi.org/10.1007/BF01436639**2.**H. Blum,*Numerical treatment of corner and crack singularities*, Finite element and boundary element techniques from mathematical and engineering point of view, CISM Courses and Lect., vol. 301, Springer, Vienna, 1988, pp. 171–212. MR**1002579**, https://doi.org/10.1007/978-3-7091-2826-8_4**3.**H. Blum,*On Richardson extrapolation for linear finite elements on domains with reentrant corners*, Z. Angew. Math. Mech.**67**(1987), no. 5, T351–T353. MR**907630**, https://doi.org/10.1002/zamm.19870670503**4.**H. Blum, Q. Lin, and R. Rannacher,*Asymptotic error expansion and Richardson extrapolation for linear finite elements*, Numer. Math.**49**(1986), no. 1, 11–37. MR**847015**, https://doi.org/10.1007/BF01389427**5.**Klaus Böhmer,*Asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions*, Math. Z.**177**(1981), no. 2, 235–255. MR**612877**, https://doi.org/10.1007/BF01214203**6.**Susanne C. Brenner and L. Ridgway Scott,*The mathematical theory of finite element methods*, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR**1894376****7.**A. Cameron,*Weighted -based negative norm estimates for the finite methods for second order elliptic problems*. In preparation.**8.**Chuan Miao Chen and Qun Lin,*Extrapolation of finite element approximation in a rectangular domain*, J. Comput. Math.**7**(1989), no. 3, 227–233. MR**1017183****9.**H. Chen and R. Rannacher,*Local error expansions and Richardson extrapolation for the streamline diffusion finite element method*, East-West J. Numer. Math.**1**(1993), no. 4, 253–265. MR**1318805****10.**Yan Heng Ding and Qun Lin,*Finite element expansion for variable coefficient elliptic problems*, Systems Sci. Math. Sci.**2**(1989), no. 1, 54–69. MR**1110121****11.**W. Hoffmann, A. H. Schatz, L. B. Wahlbin, and G. Wittum,*Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. I. A smooth problem and globally quasi-uniform meshes*, Math. Comp.**70**(2001), no. 235, 897–909. MR**1826572**, https://doi.org/10.1090/S0025-5718-01-01286-8**12.**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**13.**Lin Qun,*Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners*, Numer. Math.**58**(1991), no. 6, 631–640. MR**1083525**, https://doi.org/10.1007/BF01385645**14.**Lin Qun and Tao Lü,*Asymptotic expansions for finite element approximation of elliptic problem on polygonal domains*, Computing methods in applied sciences and engineering, VI (Versailles, 1983) North-Holland, Amsterdam, 1984, pp. 317–321. MR**806787****15.**Qun Lin and Jun Ping Wang,*Some expansions of the finite element approximation*, Shuli Kexue [Mathematical Sciences. Research Reports IMS], vol. 15, Academia Sinica, Institute of Mathematical Sciences, Chengdu, 1984. MR**777686****16.**Jun Ping Wang and Qun Lin,*Asymptotic expansions and extrapolation for the finite element method*, J. Systems Sci. Math. Sci.**5**(1985), no. 2, 114–120 (Chinese, with English summary). MR**841406****17.**Qun Lin and Rui Feng Xie,*Error expansion for FEM and superconvergence under natural assumption*, J. Comput. Math.**7**(1989), no. 4, 402–411. MR**1149709****18.**Qun Lin and Qi Ding Zhu,*Asymptotic expansion for the derivative of finite elements*, J. Comput. Math.**2**(1984), no. 4, 361–363. MR**869509****19.**R. Rannacher,*Richardson extrapolation for a mixed finite element approximation of a plate bending problem*, Z. Angew. Math. Mech.**67**(1987), no. 5, T381–T383. MR**907636****20.**R. Rannacher,*Extrapolation techniques in the finite element method (survey),*of Mathematics, Report C7, Feb. 1988, pp. 80-113.**21.**Rolf Rannacher and Ridgway Scott,*Some optimal error estimates for piecewise linear finite element approximations*, Math. Comp.**38**(1982), no. 158, 437–445. MR**645661**, https://doi.org/10.1090/S0025-5718-1982-0645661-4**22.**U. Rüde,*The hierarchical basis extrapolation method*, SIAM J. Sci. Statist. Comput.**13**(1992), no. 1, 307–318. MR**1145188**, https://doi.org/10.1137/0913016**23.**U. Rüde,*Extrapolation and related techniques for solving elliptic equation,*Bericht I-9135, Institut fur Informatik der TU Munchen, (1991).**24.**Alfred H. Schatz,*Perturbations of forms and error estimates for the finite element method at a point, with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point*, SIAM J. Numer. Anal.**42**(2005), no. 6, 2342–2365. MR**2139396**, https://doi.org/10.1137/S0036142902408131**25.**Alfred H. Schatz,*Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates*, Math. Comp.**67**(1998), no. 223, 877–899. MR**1464148**, https://doi.org/10.1090/S0025-5718-98-00959-4**26.**Alfred H. Schatz,*Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. II. Interior estimates*, SIAM J. Numer. Anal.**38**(2000), no. 4, 1269–1293. MR**1786140**, https://doi.org/10.1137/S0036142997324800**27.**A. H. Schatz, I. H. Sloan, and L. B. Wahlbin,*Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point*, SIAM J. Numer. Anal.**33**(1996), no. 2, 505–521. MR**1388486**, https://doi.org/10.1137/0733027**28.**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**502065**, https://doi.org/10.1090/S0025-5718-1978-0502065-1**29.**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**502065**, https://doi.org/10.1090/S0025-5718-1978-0502065-1**30.**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**436617**, https://doi.org/10.1090/S0025-5718-1976-0436617-2**31.**Jun Ping Wang,*Asymptotic expansions and 𝐿^{∞}-error estimates for mixed finite element methods for second order elliptic problems*, Numer. Math.**55**(1989), no. 4, 401–430. MR**997230**, https://doi.org/10.1007/BF01396046**32.**Wolfgang Wasow,*Discrete approximations to elliptic differential equations*, Z. Angew. Math. Phys.**6**(1955), 81–97. MR**80369**, https://doi.org/10.1007/BF01607295**33.**Rui Feng Xie,*Pointwise estimates for finite-element approximations to Green functions on a concave polygonal domain, and finite-element extrapolation*, Math. Numer. Sinica**10**(1988), no. 3, 232–241 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl.**10**(1988), no. 4, 19–29. MR**985474**

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Additional Information

**M. Asadzadeh**

Affiliation:
Department of Mathematics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden

Address at time of publication:
Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853

Email:
mohammad@chalmers.se, asadzadeh@math.cornell.edu

**A. H. Schatz**

Affiliation:
Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853

Email:
schatz@math.cornell.edu

**W. Wendland**

Affiliation:
Institute for Applied Analysis and Numerical Simulations, University of Stuttgart, Pfaffenwaldring 57, D-750550, Germany

Email:
wendland@mathematik.uni-stuttgart.de

DOI:
https://doi.org/10.1090/S0025-5718-09-02241-8

Keywords:
Richardson extrapolation,
local estimates,
asymptotic error expansion inequalities,
similarity of subspaces,
scalings,
finite element method,
elliptic equations

Received by editor(s):
November 21, 2007

Received by editor(s) in revised form:
October 11, 2008

Published electronically:
February 11, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.