Hp analysis of a hybrid DG method for Stokes flow

Abstract

This manuscript deals with the hp error analysis of a hybrid discontinuous Galerkin method for incompressible flow. Besides the usual coercivity and boundedness estimates, we establish inf–sup stability for the discrete incompressibility constraint with a constant which is only suboptimal by one-half order of the polynomial degree. This result holds on irregular and hybrid meshes in two and three spatial dimensions, and its proof is based on a new stability estimate for the L2 projection on simplex elements. The sharp estimate for the inf–sup constants in turn allows a priori estimates to be derived that are optimal with respect to the mesh size and only slightly suboptimal with respect to the polynomial degree. In addition to the a priori results, we also present a rigorous hp analysis of a residual-type a posteriori error estimator. Reliability and efficiency are proved and the explicit dependence of the estimates on the polynomial approximation order is elaborated. The validity of the theoretical results is demonstrated in numerical tests.