Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes

Abstract

L2 stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction of a local projection. In this paper, we first complete the one-dimensional analysis by providing optimal error estimates that match all numerical observations in that paper. The key ingredient is to construct an optimal global projection with the characteristic decomposition. We then extend the analysis on optimal error estimate to multidimensions by constructing a global projection on unstructured meshes, which can be considered as a perturbation away from the local projection studied by Cockburn, Gopalakrishnan, and Sayas in [Math. Comp. 79 (2010), pp. 1351–1367] for hybridizable discontinuous Galerkin methods. As a main contribution, we use a novel energy argument to prove the optimal approximation property of the global projection. This technique does not require explicit assembly of the matrix for the perturbed terms and hence can be easily used for unstructured meshes in multidimensions. Finally, numerical tests in two dimensions are provided to validate our analysis is sharp and at least one of the unknowns will degenerate to suboptimal rates if the assumptions are not satisfied.