Localized pointwise error estimates for mixed finite element methods

Demlow, Alan

doi:10.1090/S0025-5718-04-01650-3

Localized pointwise error estimates for mixed finite element methods
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by Alan Demlow;

Math. Comp. 73 (2004), 1623-1653

DOI: https://doi.org/10.1090/S0025-5718-04-01650-3

Published electronically: March 23, 2004

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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.

References

Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Alan Demlow, Suboptimal and optimal convergence in mixed finite element methods, SIAM J. Numer. Anal. 39 (2002), no. 6, 1938–1953. MR 1897944, DOI 10.1137/S0036142900376900
J. Douglas Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103 (English, with Portuguese summary). MR 667620
Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39–52. MR 771029, DOI 10.1090/S0025-5718-1985-0771029-9
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 (English, with Portuguese summary). MR 965675
Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, DOI 10.1051/m2an/1989230101031
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1998.
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, DOI 10.1090/S0025-5718-01-01286-8
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.
Ju. P. Krasovskii, Properties of Green’s functions and generalized solutions of elliptic boundary value problems, Soviet Math. Dokl. 10 (1969), 54–58.
R. Scholz, Optimal $L_{\infty }$-estimates for a mixed finite element method for second order elliptic and parabolic problems, Calcolo 20 (1983), no. 3, 355–377 (1984). MR 761790, DOI 10.1007/BF02576470
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, DOI 10.1090/S0025-5718-98-00959-4
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. MR 443377
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI 10.1090/S0025-5718-1995-1297478-7
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.