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 ω with curved boundary. The domain is embedded in a rectangular domain R on which uniform mesh and linear continuous elements are employed. The numerical scheme is based on an extension of the differential equation from ω to R by regularization with a small parameter (for Neumann and Robin problems), or penalty with a large parameter −1 (for Dirichlet problem), or a mixture of these (for mixed boundary value problem). For Neumann and Robin problems, we prove that when ≤ h (that is the mesh size), the error in the H(ω) norm is of the optimal order O(h). For Dirichlet problem, when ≤ h, the error is O(h1/2) that is not optimal. If the mesh is adjusted around ∂ω by moving the near-by nodes onto it and reconnecting some nodes such that a polygonal interpolation of ∂ω is formed in the mesh, then the optimal convergence rate O(h) holds for Dirichlet problem if ω is convex and ≤ h. If ω is not convex, then the convergence rate can only be improved to O(h2/3) by such mesh adjustment with the parameter being = h. In this latter case, a parameter smaller than h thwarts the convergence rate.