Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods (Journal of Computational Mathematics, 2020)
In this paper, we provide a number of new estimates on the stability and convergence ofboth hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By usingthe standard Brezzi theory on mixed methods, we carefully define appropriate norms forthe various discretization variables and then establish that the stability and error estimateshold uniformly with respect to stabilization and discretization parameters. As a result, bytaking appropriate limit of the stabilization parameters, we show that the HDG methodconverges to a primal conforming method and the WG method converges to a mixedconforming method.
Keywords: Uniform Stability; Uniform Error Estimate; Hybrid Discontinuous Galerkin; Weak Galerkin.