Manuel Solano, Felipe Vargas:
A high order HDG method for Stokes flow in curved domains
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 this by a polygonal/polyhedral domain where an HDG approximation can be computed. In order to obtain a suitable approximation for the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the extrapolated discrete gradient. We also propose to extrapolate the discrete pressure and impose the value of its integral over the computational domain in such a way that the approximated pressure has zero-mean in the entire domain. We show that, if the computational domain is defined through interpolating the boundary of the original domain by a piece-wise linear function, the method provides optimal order of convergence, i.e., order k+1 for the approximations of the pressure, the velocity and its gradient and order k+2 for the numerical trace of the velocity and for the element-by-element post-processed velocity. We also provide numerical experiments validating the theoretical error estimates.