×

A posteriori error estimates for nonlinear problems. \(L^r\)-estimates for finite element discretizations of elliptic equations. (English) Zbl 0920.65064

The author presents an extension of his earlier paper [Math. Comput. 62, No. 206, 445-475 (1994; Zbl 0799.65112)] for the derivation of a posteriori error estimates for approximate solutions of nonlinear elliptic problems. This extension yields \(L^r\)-error estimates. A mathematical analysis of the error estimates with upper and lower bounds is presented without numerical experiments.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35Q30 Navier-Stokes equations

Citations:

Zbl 0799.65112
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] R. A. ADAMS, Sobolev Spaces. Academic Press, New York, 1975. Zbl0314.46030 MR450957 · Zbl 0314.46030
[2] I. BABUŠKA and W. C. RHEINBOLDT, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978). Zbl0398.65069 MR483395 · Zbl 0398.65069
[3] I. BABUŠKA and W. C. RHEINBOLDT, A posteriori error estimates for the finite element method. Int. J. Numer. Methods in Engrg. 12, 1597-1615 (1978). Zbl0396.65068 · Zbl 0396.65068
[4] E. BÄNSCH and K. G. SIEBERT, A posteriori error estimation for nonlinear problems by dual techniques. Preprint, Universität Freiburg, 1995.
[5] C. BERNARDI, B. MÉTIVET and R. VERFÜRTH, Analyse numérique d’indicateurs d’erreur. Preprint R 93025, Université Paris VI, 1993.
[6] P. G. CIARLET, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[7] [7] P. CLÉMENT, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77-84 (1975). Zbl0368.65008 MR400739 · Zbl 0368.65008
[8] M. DAUGE, Elliptic Boundary Value Problems on Corner Domains. Springer, Lecture Notes in Mathematics 1341, Berlin, 1988. Zbl0668.35001 MR961439 · Zbl 0668.35001
[9] K. ERIKSSON, An adaptive finite element method with efficient maximum norm error control for elliptic problems. Math. Models and Math. in Appl. Sci. 4, 313-329 (1994). Zbl0806.65106 MR1282238 · Zbl 0806.65106
[10] K. ERIKSSON and C. JOHNSON, An adaptive finite element method for linear elliptic problems. Math. Comput. 50, 361-383 (1988). Zbl0644.65080 MR929542 · Zbl 0644.65080
[11] K. ERIKSSON and C. JOHNSON, Adaptive finite element methods for parabolic problems I. A linear model problem. SIAM J. Numer. Anal. 28, 43-77 (1991). Zbl0732.65093 MR1083324 · Zbl 0732.65093
[12] K. ERIKSSON and C. JOHNSON, Adaptive finite element methods for parabolic problems IV. Nonlinear problems. Chalmers University of Göteborg, Preprint 1992, 44 (1992). Zbl0835.65116 MR1360457 · Zbl 0835.65116
[13] V. GIRAULT and P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations. Computational Methods in Physics, Springer, Berlin, 2nd édition, 1986. Zbl0413.65081 MR548867 · Zbl 0413.65081
[14] P. GRISVARD, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. Zbl0695.35060 MR775683 · Zbl 0695.35060
[15] C. JOHNSON and P. HANSBO, Adaptive finite element methods in computational mechanics. Comp. Math. Appl. Mech. Engrg. 101, 143-181 (1992). Zbl0778.73071 MR1195583 · Zbl 0778.73071
[16] R. H. NOCHETTO, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes. Math. Comput. 64, 1-22 (1995). Zbl0920.65063 MR1270622 · Zbl 0920.65063
[17] J. POUSIN and J. RAPPAZ, Consistency, stability, a priori, and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69, 213-231 (1994). Zbl0822.65034 MR1310318 · Zbl 0822.65034
[18] R. VERFÜRTH, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. (206), 445-475 (1994). Zbl0799.65112 MR1213837 · Zbl 0799.65112
[19] R. VERFÜRTH, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic problems. Bericht Nr. 180, Ruhr-Universität Bochum, 1995. Zbl0869.65067 · Zbl 0869.65067
[20] R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner Series in advances in numerical mathematics, Stuttgart, 1996. Zbl0853.65108 · Zbl 0853.65108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.