Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension method

TL;DR

This paper develops a finite element method using the extension approach for a generalized Robin boundary problem on curved domains, proving optimal convergence rates and demonstrating with a numerical example.

Contribution

It introduces an extension-based finite element scheme for curved domains with a second-order boundary operator, showing optimal convergence rates.

Findings

Optimal convergence in H^1 and L^2 norms.

Extension approach simplifies practical computation.

Numerical validation for linear elements.

Abstract

A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\Omega$ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $\Omega_h$ and to address issues owing to the domain perturbation $\Omega \neq \Omega_h$. In contrast to the transformation approach used in existing studies, we employ the extension approach, which is easier to handle in practical computation, in order to construct a numerical scheme. Assuming that approximate domains and function spaces are given by isoparametric finite elements of order $k$, we prove the optimal rate of convergence in the $H^1$- and $L^2$-norms. A numerical example is given for the piecewise linear case $k = 1$.