In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{lift} is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of \textit{finite element approximation error} and of \textit{geometric error}, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.

翻译：本文研究了一个涉及域边界上二阶项（Laplace-Beltrami算子）的谱问题。提出了一个变分公式，并由此进行有限元离散。为使Laplace-Beltrami算子在边界上有意义，域是光滑的：因此计算域（通常为多边形域）与物理域不匹配。为此，使用高阶弯曲网格对物理域进行离散，以减小\textit{几何误差}。回顾了\textit{提升算子}，该算子旨在将定义在网格域上的函数变换为定义在物理域上的函数。此\textit{提升}是估计特征值和特征函数误差的关键工具。采用自举方法证明误差估计，这些估计分别以有限元次数$k\ge 1$和网格阶数$r\ge 1$相关的\textit{有限元逼近误差}和\textit{几何误差}表示。在二维和三维的各种光滑域上进行了数值实验，验证了所提出的理论结果。