DGFEMs for Second-Order PDEs of Mixed-Type

Abstract

This chapter is devoted to studying the stability and convergence properties of the discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of a general class of second-order partial differential equations (PDEs) with nonnegative characteristic form. While this class of second-order equations naturally includes parabolic equations, we also pursue the analysis of parabolic time-dependent PDEs in a separate manner in order to admit the use of local time-stepping algorithms. The general analysis of DGFEMs for PDEs with nonnegative characteristic form is pursued under the assumption that the number of faces each element possesses remains bounded under mesh refinement. For the special subclass of parabolic problems we adopt the assumption which permits an arbitrary number of faces per element, thereby highlighting that both assumptions lead to rigorous a priori error estimates for DGFEMs applied to parabolic problems.