Numerical solutions for nonlinear elliptic problems based on first-order system

Abstract

We present a theoretical analysis of numerical solutions for nonlinear elliptic problems. Our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by Brezzi, Rappaz, and Raviart (BRR). In most cases of finite element analysis, the same spaces are used for the domain and range of the nonlinear operator concerning the application of BRR theory. This results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates. The main contribution of this paper is a theoretical analysis for the optimal convergence rates. This is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of BRR theory. Previously, this idea was used for an analysis of Petrov–Galerkin formulation for the BRR theory. Our analysis is developed for the first-order least-squares finite element methods and the mixed Galerkin methods based on first-order systems of equations.