The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional LDG method, the scheme, called the direct discontinuous Galerkin (DDG) method, is based on the direct weak formulation for solutions of parabolic equations in each computational cell, and let cells communicate via the numerical flux ûx ONLY. We propose a general numerical flux formula for the solution derivative, which is consistent, and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one dimensional and multi-dimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the one dimensional linear case, kth order accuracy in an energy norm is proven when using k−th degree polynomials. The DDG method has the advantage of easier formulation and implementation, and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes.