A priori and a posteriori error analyses of a high order unfitted mixed-FEM for Stokes flow

Abstract

We propose and analyze a high order unfitted mixed finite element method for the pseudostress-velocity formulation of the Stokes problem with Dirichlet boundary condition on a fluid domain Ω with curved boundary Γ. The method consists of approximating Ω by a polyhedral subdomain Dh, with boundary Γh, where a Galerkin method is applied to approximate the solution, and on a transferring technique, based on integrating the extrapolated discrete gradient of the velocity, to approximate the Dirichlet boundary data on the computational boundary Γh. The associated Galerkin scheme is defined by Raviart–Thomas elements of order k≥0 for the pseudostress and discontinuous polynomials of degree k for the velocity. Provided suitable hypotheses on the mesh near the boundary Γ, we prove well-posedness of the Galerkin scheme by means of the Babuška–Brezzi theory and establish the corresponding optimal convergence of O(hk+1). Moreover, for the case when Γh is constructed through a piecewise linear interpolation of Γ, we propose a reliable and quasi-efficient residual-based a posteriori error estimator. Its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough. Numerical experiments illustrate the performance of the scheme, show the behavior of the associated adaptive algorithm and validate the theory.