On the numerical analysis of a nonlinear elliptic problem via mixed-FEM and Lagrange multipliers

Abstract

We apply a mixed finite element method to solve a nonlinear second order elliptic equation in divergence form with mixed boundary conditions. Our approach introduces the trace of the solution on the Neumann boundary as a further unknown that acts also as a Lagrange multiplier. We show that the resulting variational formulation and an associated discrete scheme defined with Raviart–Thomas spaces are well posed, and derive the usual a priori estimates and the corresponding rate of convergence. In addition, we develop a Bank–Weiser type a posteriori error analysis and provide an implicit reliable and quasi-efficient estimate, and a fully explicit reliable one. Several numerical results illustrate the suitability of the explicit a posteriori estimate for the adaptive computation of the discrete solutions.