Discontinuous Galerkin methods for solving a hyperbolic inequality

Abstract

In this paper, we study spatially semi‐discrete and fully discrete schemes to numerically solve a hyperbolic variational inequality, with discontinuous Galerkin (DG) discretization in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for four DG schemes, which reaches the optimal convergence order for linear elements. A numerical example is presented, and the numerical results confirm the theoretical error estimates.