A posteriori error estimates for approximate solutions of nonlinear equations with weakly stable operators. (English) Zbl 0903.65051
Whereas a posteriori error estimates for the approximate solution to linear elliptic boundary value problems have been studied quite well, those for nonlinear or nonstationary problems wait for more attention in the literature.
With the paper in hand, a general approach to a posteriori error estimates for the approximation (or perturbation, resp.) of an operator equation with a not necessarily uniformly monotone operator is presented. The technique is a generalization of one of the basic ones by I. Babuška and W. C. Rheinboldt [SIAM J. Numer. Anal. 15, 736-754 (1978; Zbl 0398.65069)].
The author derives upper and lower bounds for energy-type norms of the error. The original operator is assumed to be weakly stable. Furthermore, the author describes how to compute the error indicators in practice. Finally, the extension of the approach presented to time-dependent problems is considered.
With the paper in hand, a general approach to a posteriori error estimates for the approximation (or perturbation, resp.) of an operator equation with a not necessarily uniformly monotone operator is presented. The technique is a generalization of one of the basic ones by I. Babuška and W. C. Rheinboldt [SIAM J. Numer. Anal. 15, 736-754 (1978; Zbl 0398.65069)].
The author derives upper and lower bounds for energy-type norms of the error. The original operator is assumed to be weakly stable. Furthermore, the author describes how to compute the error indicators in practice. Finally, the extension of the approach presented to time-dependent problems is considered.
Reviewer: E.Emmrich (Magdeburg)
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 65Z05 | Applications to the sciences |
| 47J25 | Iterative procedures involving nonlinear operators |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35K55 | Nonlinear parabolic equations |
Citations:
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