Convergence analysis of the Q1‐finite element method for elliptic problems with non‐boundary‐fitted meshes

Abstract

AbstractThe aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian‐grid methods. These methods use regular structured meshes (most often Cartesian) for non‐aligned domains. Although non‐boundary‐fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates.In this paper, the convergence of a Q1‐non‐conforming finite element method is analyzed for second‐order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q1‐rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non‐conforming. A stair‐step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non‐boundary‐fitted stair‐stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary‐fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments.The rate of convergence is proved to be in 𝒪(h1/2) for the H1‐norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L2‐norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non‐boundary‐fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.