The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

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...

The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the ...

Discontinuous Galerkin Methods for Advection-diffusion-reaction Problems

We apply the weighted-residual approach recently introduced in [7] to derive dis-continuous Galerkin formulations for advection-diffusion-reaction problems. We devise the basic ingredients to ensure stability and optimal error estimates in suitable norms, and propose two new methods. 1. Introduction. In recent years Discontinuous Galerkin methods have become increasingly popular, and they have ...

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

Discontinuous Galerkin Methods for Fractional Diffusion Equations

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems, characterized by having fractional derivatives, parameterized by β ∈ [1, 2]. We show through analysis that one can construct a numerical flux which results in a scheme that exhibit optimal order of convergence O(hk+1) in the continuous range between pure advection (β = 1) and pure...