Analysis of a continuous finite element method for $$H(\mathrm{curl},\mathrm{div})$$ -elliptic interface problem

Abstract

In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local $$L^2$$ projector to the curl operator and a pseudo-local $$L^2$$ projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate $$\mathcal{O }(h^r)$$ in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in $$\prod _{l=1}^L (H^r(\Omega _l))^3$$ for some $$r\in (1/2,1]$$ due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in $$\Omega =\bigcup _{l=1}^L \Omega _l$$ .