Discontinuous Galerkin Method for Monotone Nonlinear Elliptic Problems

Abstract

In this paper, we consider the incomplete interior penalty method for a class of second order monotone nonlinear elliptic problems. Using the theory of monotone operators, we show that the corresponding discrete method has a unique solution. The a priori error estimate in an energy norm is developed under the minimal regularity assumption on the exact solution, i.e., $u \in H^1(\Omega)$. Moreover, we propose a residual-based a posteriori error estimator and derive the computable upper and lower bounds on the error in an energy norm.