Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods
Provides theoretical foundations for understanding the relationship between HDG and WG methods and their conforming limits, benefiting numerical analysts working on finite element methods.
The paper establishes uniform stability and error estimates for HDG and WG methods, showing that HDG converges to a primal conforming method and WG to a mixed conforming method as stabilization parameters are varied.
In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converge to a mixed conforming method.