Convergence and Error Estimates for the Method of Lines for Certain Nonlinear Elliptic and Elliptic-Parabolic Equations

Abstract

Differential inequality theory for systems of second order ordinary differential equations is applied to obtain convergence and error estimates for the method of lines (MOL). Comparison theorems for boundary value problems for systems of ordinary differential equations are developed which are then used to compare solutions to elliptic and elliptic-parabolic boundary value problems with the solutions to their MOL analogs. Included are some results on equations defined on unbounded domains. These theorems extend some results of W. Walter on parabolic equations to the elliptic and elliptic-parabolic cases.