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

Abstract

Abstract. 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 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 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 (the mesh size), the error in the H 1 (ω) norm is of the optimal order O(h). For Dirichlet problem, when ≤ h 1/2 , the error is O(h 1/2 ) that is not optimal. If the mesh is adjusted around ∂ω to fit it, 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(h 2/3 ) by such mesh adjustment with the parameter being = 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.

Analysis of the fictitious domain method with penalty for elliptic problems

“…Here, n is the unit outward normal to viewed as a boundary of , and n − is opposite to n. We found the analysis in [12] to be complicated and not directly. Our method for estimating of u − u ,h is simpler, which is to find some interpolation of u , denoted as v h , and then estimate u − v h by using a regularity theorem of (Q ).…”

“…Although there exist some ways to derive the sharp error estimates for elliptic problems (cf. [8,11,12]), it seems none of them has been applied to parabolic problem such that the sharpness of the error estimates are maintained. Our motivation lies in the study of the penalty fictitious domain method which can be applied to these time-dependent moving-boundary problems maintaining the sharpness of the error boundary.…”

“…In [7], it is bounded by C √ (C is some constant, so as in the following), and in [8,11,12] the sharp estimate C is achieved. In this paper, we give a new way to derive the sharp estimate.…”

“…In the literature, there are several works devoted to the study of the H 1 error between u ,h and u in . For example, in [12], it is proved that the H 1 error is bounded by…”

“…The analysis of [12] is to consider the estimate of u ,h − u 0 − u 1 to derive the final estimate of u − u ,h , where u 0 is the zero extension of u, and u 1 is the solution of problem:…”