A High Order Mixed-FEM for Diffusion Problems on Curved Domains

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 $$\mathrm {D}_{h}$$ , 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 $${\mathbf {H}}(\mathrm {div}\,; \mathrm {D}_{h})$$ -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 $$k\ge 0$$ 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 $$\mathrm {D}_{h}^{c}\,{:}{=}\,\Omega \backslash \overline{\mathrm {D}_{h}}$$ and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.