A priori and a posteriori error analysis of a staggered discontinuous Galerkin method for convection dominant diffusion equations

An upwind staggered discontinuous Galerkin (upwind-SDG) method for convection dominant diffusion problems is developed. Optimal a priori error estimates can be achieved for both the scalar and vector functions approximated by the method. To efficiently capture the layer problems, we propose a robust a posteriori error estimator for upwind-SDG method measured in a natural norm and a semi-norm associated with the convective derivative. The semi-norm can be neglected when the mesh Péclet number is sufficiently small. The key is to bound the conforming contribution by the dual norm of the residual, and the robustness arises from the incorporation of the semi-norm associated with the convective derivative. Finally, various numerical examples are tested to illustrate the performance of the robust a posteriori error estimator. The results indicate that optimal convergence rates can be achieved and the singularity can be well-captured by the proposed error estimator.