Ricardo Oyarzua, Manuel Solano, Paulo Zuñiga:
A high order mixed-FEM for diffusion problems on curved domains
We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain Ω with curved boundary Γ. The method is based on approximating Ω by a polygonal subdomain Dh, with boundary Γh, where a high order conforming Galerkin method is considered to compute the solution. To approximate the Dirichlet data on the computational boundary Γh, we employ a transferring technique based on integrating the extrapolated discrete gradient along segments joining Γh and Γ. Considering general finite dimensional subspaces we prove that the resulting Galerkin scheme, which is H(div ;Dh)-conforming, is wellposed 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 Γh and Γ is at most of the order of the meshsize h. We also approximate the solution in Dch := Ω\Dh and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.