Direct Discontinuous Galerkin Method and Its Variations for Diffusion Problems

Discontinuous Galerkin method is a special class of finite element method that use discontinuous piecewise polynomials as the approximate solution space. The imposed discontinuity across the cell interface gives the method the flexibility to handle h-p adaptivity and the advantage to solve problems with discontinuities, for example, the shocks for hyperbolic problems. Due to the lack of up-winding mechanism (characteristics), discontinuous Galerkin method for diffusion problems are not as well studied as for the hyperbolic problems.

In this talk, we will discuss the recent four discontinuous Galerkin methods for diffusion problems; 
1) the Direct discontinuous Galerkin(DDG) method; 
2) the DDG method with interface corrections; 
3) the DDG method with symmetric structure; and 
4) a new DG method with none symmetric structure. 
Major contribution of the DDG method is the introduction of the jumps of second or higher order solution derivatives in the numerical flux formula. The symmetric version of the DDG method helps us obtain the optimal L2(L2) error analysis for the DG solution. For the non-symmetric version, we show that the scheme performs better than the Baumann-Oden scheme or the NIPG method in the sense that optimal convergence is recovered with even-th order polynomial approximations. A series of numerical examples are presented to show the high order accuracy and the capacity of the methods. At the end, we will discuss the recent studies of the maximum-principle-satisfying or the positivity preserving properties of the DDG related methods.