Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

Abstract

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is O(h^p+2) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.