A high order discontinuous Galerkin method with skeletal multipliers for convection-diffusion-reaction problems

A discontinuous Galerkin method with skeletal multipliers (DGSM) is developed for diffusion problem. Skeletal multiplier is introduced on the edge/face of each element through the definition of a weak divergence and a weak derivative in the method. The local weak formulation is derived by weakly imposing the Dirichlet boundary condition and continuity of fluxes and solutions on the edges/faces. The global weak formulation is then obtained by adding all the local problems. Equivalence of the weak formulation and the original problem is proved. Stability of DGSM is shown and an error estimate is derived in a broken norm. A DGSM for linear convection-diffusion-reaction problems is also derived. An explanation on algorithmic aspects is given. Some numerical results are presented. Singularities due to discontinuities in the diffusion coefficients are accurately approximated. Internal/boundary layers are well captured without showing spurious oscillations. Robustness of the method in increasingly small diffusivity is demonstrated on the whole domain.