Convergence analysis of symmetric dual-wind discontinuous Galerkin approximation methods for the obstacle problem

Abstract

This paper formulates and analyzes symmetric dual-wind discontinuous Galerkin (DG) methods for second order elliptic obstacle problem. These new methods follow from the DG differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation (PDE). We establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular. These results are also shown to hold for some non-positive penalty parameters, with the emphasis on zero penalization across all interior and boundary edges. Numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods.