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Localized pointwise error estimates for mixed finite element methods

Author: Alan Demlow
Journal: Math. Comp. 73 (2004), 1623-1653
MSC (2000): Primary 65N30, 65N15
Published electronically: March 23, 2004
MathSciNet review: 2059729
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Abstract: In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.

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  • [BDM85] Franco Brezzi, Jim Douglas, Jr., and Donatella Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217-235. MR 87g:65133
  • [Dem02] Alan Demlow, Suboptimal and optimal convergence in mixed finite element methods, SIAM J. Numer. Anal. 39 (2002), no. 6, 1938-1953. MR 2003e:65216
  • [DR82] Jim Douglas, Jr. and Jean E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91-103. MR 84b:65111
  • [DR85] -, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. MR 86b:65122
  • [GN88] Lucia Gastaldi and Ricardo H. Nochetto, On $L^\infty$-accuracy of mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 7 (1988), no. 1, 13-39. MR 90d:65192a
  • [GN89] -, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic problems, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103-128. MR 91b:65125
  • [GT98] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1998.
  • [HSWW01] Wolfgang Hoffmann, Alfred H. Schatz, Lars B. Wahlbin, and Gabriel 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 2002a:65178
  • [HTB95] Kenneth H. Huebner, Earl A. Thornton, and Ted G. Byrom, The finite element method for engineers, 3rd ed., John Wiley and Sons, Inc., New York, 1995.
  • [Kra69] Ju. P. Krasovskii, Properties of Green's functions and generalized solutions of elliptic boundary value problems, Soviet Math. Dokl. 10 (1969), 54-58.
  • [Sch83] Reinhard Scholz, Optimal $L^\infty$-estimates for a mixed finite element method for second order elliptic and parabolic problems, Calcolo 20 (1983), 355-377. MR 86j:65164
  • [Sch98] 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 98j:65082
  • [SF73] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. MR 56:1747
  • [SW95] Alfred H. Schatz and Lars B. Wahlbin, Interior maximum-norm estimates for finite element methods, Part II, Math. Comp 64 (1995), no. 211, 907-928. MR 95j:65143
  • [Wah91] Lars B. Wahlbin, Local behavior in finite element methods, Finite Element Methods. Part 1 (P. G. Ciarlet and J.-L. Lions, eds.), Handbook of Numerical Analysis, vol. II, North-Holland, Amsterdam, 1991, pp. 353-522.

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Additional Information

Alan Demlow
Affiliation: Abteilung für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104 Freiburg, Germany

Keywords: Mixed finite element methods, pointwise error estimates
Received by editor(s): May 22, 2002
Published electronically: March 23, 2004
Additional Notes: This material is based on work supported under a National Science Foundation graduate fellowship and under NSF grant DMS-0071412
Article copyright: © Copyright 2004 American Mathematical Society

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