hp-DGFEM for Partial Differential Equations with Nonnegative Characteristic Form

Abstract

Presented as Invited Lecture at the International Symposium on Discontinuous Galerkin Methods: Theory, Computation and Applications, in Newport, RI, USA.

We develop the error analysis for the hp-version of a discontinuous finite element approximation to second-order partial differential equations with nonnegative characteristic form. This class of equations includes classical examples of second-order elliptic and parabolic equations, first-order hyperbolic equations, as well as equations of mixed type. We establish an a priori error bound for the method which is of optimal order in the mesh size h and 1 order less than optimal in the polynomial degree p. In the particular case of a first-order hyperbolic equation the error bound is optimal in h and 1/2 an order less than optimal in p.