The Optimal Convergence Order of the Discontinuous Finite Element Methods for First Order Hyperbolic Systems (Journal of Computational Mathematics, 2018)
In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems (steady and nonsteady state) is constructed and analyzed by using linear triangle elements, and the $O(h^2)$-order optimal error estimates are derived under the assumption of strongly regular triangulation and the $H^3$-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.
Keywords: First order hyperbolic systems; Discontinuous finite element method; Convergence order estimate.