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 $k = 1$.

翻译：针对边界上包含二阶微分算子的广义Robin边值问题，本文提出了有限元方法的理论分析。当Ω为具有曲线边界的一般光滑区域时，需引入近似区域Ω_h并处理因域扰动Ω≠Ω_h带来的问题。与现有研究中采用的变换方法不同，本文采用在实际计算中更易处理的扩展方法构造数值格式。在假定近似区域与函数空间由k阶等参有限元给定的条件下，我们证明了H^1范数和L^2范数下的最优收敛速率。最后给出分段线性情形(k=1)下的数值算例。