Formula omitted]2 convergence for positive finite-difference solutions of the diffusive logistic equation in two-dimensional bounded domains

Abstract

This article considers finite-difference approximations of the nonlinear homogeneous Dirichlet boundary-value problem for the equation Δw + w[ p( x, y) − qw] = 0 in a two-dimensional bounded domain. The equation is of fundamental importance for more elaborate study of ecological prey-predator diffusive models. This problem has a positive solution as well as the trivial one. By means of the Rayleigh quotient characterization of the first eigenvalue, we give a simple proof of the L 2 convergence of the positive solution of the discrete problem to the positive solution of the continuous problem. Simple criteria for this to happen in a large family of domains are given. Some computational results of interest are also presented.