Finite element approximation of a nonlinear heat conduction problem in anisotropic media

Abstract

This paper is a survey of results which we have obtained in solving a stationary nonlinear heat conduction problem by the finite element method. In particular, we present uniqueness theorems for the classical and weak solutions, a comparison theorem, existence theorems for the weak and finite element solutions, approximation of a curved boundary and numerical integration, a discrete maximum principle, convergence without any regularity assumptions, a priori error estimates, nonlinear boundary conditions and numerical algorithms.