Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids

We analyze the error of a finite element domain embedding method for elliptic equations on a domain $\omega$ with curved boundary. The domain is embedded in a rectangle $R$ on which uniform mesh and linear continuous elements are employed. The numerical scheme is based on an extension of the differential equation from $\omega$ to $R$ by regularization with a small parameter $\epsilon$ (for Neumann and Robin problems), or penalty with a large parameter $\epsilon^{-1}$ (for the Dirichlet problem), or a mixture of these (for a mixed boundary value problem). For Neumann and Robin problems, we prove that when $\epsilon\le h$ (the mesh size), the error in the $H^1(\omega)$ norm is of the optimal order $\mathcal{O}(h)$. For the Dirichlet problem, when $\epsilon\le h^{1/2}$, the error is $\mathcal{O}(h^{1/2})$ that is not optimal. If the mesh is adjusted around $\partial\omega$ to fit it, then the optimal convergence rate $\mathcal{O}(h)$ holds for the Dirichlet problem if $\omega$ is convex and $\epsilon\le h$. If $\omega$ is not convex, then the convergence rate can only be improved to $\mathcal{O}(h^{2/3})$ by such mesh adjustment, with the parameter being $\epsilon=h^{2/3}$. In this latter case, a parameter smaller than $h^{2/3}$ thwarts the convergence rate, which is verified by a numerical result.