A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type

Source abstract

Abstract In this paper, we present a priori and a posteriori analysis of a staggered discontinuous Galerkin (DG) method for quasi-linear second order elliptic problems of nonmonotone type. First, existence is proved by using Brouwer’s fixed point argument and uniqueness is verified utilizing Lipschitz continuity of the discrete solution map. Next, optimal a priori error estimates for both potential and flux variables are derived. Then the residual based a posteriori error estimates on the potential energy error and the flux L 2 {L^{2}} error, respectively, are proposed. The flux error estimator makes use of a Helmholtz-type decomposition for the nonlinear system, which relies on appropriate choice of an auxiliary problem. While a priori error analysis is based on the observation that the staggered DG method can be viewed as a nonconforming approximation of the primal mixed formulation of the problem, a posteriori error estimation takes advantage of the primal formulation which can be obtained from the primal mixed formulation by eliminating the flux variable in the continuous setting. Finally, the theoretical findings are illustrated by numerical experiments.