Local convergence of the method of pseudolinear equations for quasilinear elliptic boundary-value problems

Abstract

A variant of the method of pseudolinear equations, an iterative method of solving quasilinear partial differential equations, is described for quasilinear elliptic boundary-value problems of the type -[ p 1( u x )] x - [ p 2( u y )] y = f on a bounded simply connected two-dimensional domain D. A theorem on local convergence in C 2, λ( D) of this variant, which has constant coefficients, is proved. Three other method of solving quasilinear elliptic boundary-value problems, namely. Newton's method, the Kačanov method and a variant of the method of successive approximations that has constant coefficients, are briefly discussed. Results of a series of numerical experiments in a finite-difference setting of solving quasilinear Dirichlet problems of the above-mentioned type by the method of pseudolinear equations and these three methods are given. These results show that Newton's method converges for stronger nonlinearities than do the other methods, which, in order thereafter, are the Kačanov method, the method of pseudolinear equations and, last, the method of successive approximations, which converges only for relatively weak nonlinearities. From fastest to slowest, the methods are: the method of successive approximations, the method of pseudolinear equations, Newton's method, the Kačanov method.