A High Order HDG Method for Stokes Flow in Curved Domains

Abstract

We propose and analyze a high order hybridizable discontinuous Galerkin (HDG) method for the Stokes equations in a curved domain. It is based on approximating the domain by a polyhedral computational subdomain where an HDG solution is computed. To obtain a high order approximation of the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the approximation of the gradient. In addition, we first seek for a discrete pressure having zero-mean in the computational domain and then the zero-mean condition in the entire domain is recovered by a post-process that involves an extrapolation of the discrete pressure. We prove that the method provides optimal order of convergence for the approximations of the pressure, the velocity and its gradient, that is, order $$h^{k+1}$$ if the local discrete spaces are constructed using polynomials of degree k and the meshsize is h. We present numerical experiments validating the method.