Mixed discretisation methods for the Discontinuous Galerkin method with analytical test-functions

Abstract

The idea to this paper is a framework for modifying a standard Discontinuous Galerkin method for convection-diffusion-reaction-equations. We develop an abstract theory for the stability and error-estimates in $L^2$-norms for a mixed formulation. For diverse test-functions we apply our abstract theory. We apply the theory for the test-functions coming from analytical solutions of the adjoint problem. With the new test-functions we could improve the approximation. At the end we discuss the application of the new analytical test-functions.