Numerical methods for quasi-linear elliptic equations with nonlinear boundary conditions

The purpose of this paper is to give a numerical treatment for a class of quasi-linear elliptic equations under nonlinear boundary conditions, including the three basic types of linear boundary conditions. The quasi-linear equation is discretized by the finite difference method, and the method of upper-lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system. This method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes. It also leads to the existence of a maximal and a minimal finite difference solution, including the uniqueness of the solution, and the convergence of the finite difference solution to the corresponding continuous solution. Applications are given to two physical problems in heat conduction and combustion theory, and numerical results for the heat-conduction problem are given, and are compared with the known true continuous solution.