Abstract

We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain Omega with curved boundary Gamma. The method is based on approximating Omega by a polygonal subdomain D-h, with boundary Gamma(h), where a high order conforming Galerkin method is considered to compute the solution. To approximate the Dirichlet data on the computational boundary Gamma(h), we employ a transferring technique based on integrating the extrapolated discrete gradient along segments joining Gamma(h) and Gamma. Considering general finite dimensional subspaces we prove that the resulting Galerkin scheme, which is H(div; D-h)-conforming, is well-posed provided suitable hypotheses on the aforementioned subspaces and integration segments. A feasible choice of discrete spaces is given by Raviart-Thomas elements of order k >= 0 for the vectorial variable and discontinuous polynomials of degree k for the scalar variable, yielding optimal convergence if the distance between Gamma(h) and Gamma is at most of the order of the meshsize h. We also approximate the solution in D-h(c) := Omega(D-h) over bar and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.