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A posteriori error estimates for nonlinear problems. (English) Zbl 0869.65067

Hebeker, Friedrich-Karl (ed.) et al., Numerical methods for the Navier-Stokes equations. Proceedings of the international workshop held, Heidelberg, Germany, October 25-28, 1993. Braunschweig: Vieweg. Notes Numer. Fluid Mech. 47, 288-297 (1994).
Summary: We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from below and from above by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained.
The general results are applied to finite element discretizations of scalar quasilinear elliptic partial differential equations of 2nd order and yield residual a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global lower bounds on the error of the numerical solution.
For the entire collection see [Zbl 0837.00018].

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
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