An unfitted HDG method for Oseen equations

Abstract

We propose and analyze a high order unfitted hybridizable discontinuous Galerkin method to numerically solve Oseen equations in a domain Ω having a curved boundary. The domain is approximated by a polyhedral computational domain not necessarily fitting Ω. The boundary condition is transferred to the computational domain through line integrals over the approximation of the gradient of the velocity and a suitable decomposition of the pressure in the computational domain is employed to obtain an approximation of the pressure having zero-mean in the domain Ω. Under assumptions related to the distance between the computational boundary and Ω, we provide stability estimates of the solution that will lead us to the well-posedness of the scheme and also to the error estimates. In particular, we prove that the approximations of the pressure, velocity and its gradient are of order hk+1, where h is the meshsize and k the polynomial degree of the local discrete spaces. We provide numerical experiments validating the theory and also showing the performance of the method when applied to the steady-state incompressible Navier–Stokes equations.