On a-Posteriori Error Estimate of Approximate Solutions to a Mildly Nonlinear Nonpotential Elliptic Boundary Value Problem
✍ Scribed by Juraj Weisz
- Publisher
- John Wiley and Sons
- Year
- 1991
- Tongue
- English
- Weight
- 236 KB
- Volume
- 153
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
The paper deals with the construction of a computable a-posteriori error estimate of the approximate solution to some nonpotential nonlinear elliptic boundary value problems. The convergence of the presented error estimate to the true error is proved. The method is illustrated on some numerical examples.
Introduction. The aim of this paper is to construct an a-posteriori error estimate of approximate solutions to a two-dimensional nonpotential mildly nonlinear elliptic boundary value problem. A-posteriori error estimates for potential mildly nonlinear elliptic boundary value problems -Au + g(u) = f in SZ, ulan = 0, were studied in [4] (the one-dimensional case) and in [5] (the two-dimensional case). In [5] the function g : R 4 R was supposed to be continuous, increasing, surjective with polynomial growth. Two-dimensional potential problems with nonlinearity also in the main part of the equation were considered in [6]. In all mentioned papers is the error estimate based on two-sided estimates of the energy of the exact solution. In this paper we shall extend the method of [5] to the case of nonpotential problems. The function g is supposed to be neither strictly increasing nor surjective. We shall use the theory of conjugate operators and a-posteriori error estimates from [l]. We are interested in convergent a-posteriori error estimates i.e. such a-posteriori error estimates which can be made arbitrary small for sufficiently good approximations of the exact solution. Our method will be illustrated on a numerical example in the last section of the paper.