In this work is provided a numerical study of a diffusion problem involving a second order term on the domain boundary (the Laplace-Beltrami operator) referred to as the \textit{Ventcel problem}.A variational formulation of the Ventcel problem is studied, leading to a finite element discretization.The focus is on the resort to high order curved meshes for the discretization of the physical domain.The computational errors are investigated both in terms of geometrical error and of finite element approximation error, respectively associated to the mesh degree $r\ge 1$ and to the finite element degree $k\ge 1$. The numerical experiments we led allow us to formulate a conjecture on the \textit{a priori} error estimates depending on the two parameters $r$ and $k$. In addition, these error estimates rely on the definition of a functional \textit{lift} with adapted properties on the boundary to move numerical solutions defined on the computational domain to the physical one.

翻译：本文针对域边界上包含二阶项（Laplace-Beltrami算子）的扩散问题（即Ventcel问题）进行了数值研究。首先研究了Ventcel问题的变分形式，进而推导出有限元离散格式。研究重点在于采用高阶曲边网格对物理域进行离散化。计算误差分别从几何误差和有限元逼近误差两个维度进行探讨，前者与网格阶数$r\ge 1$相关，后者与有限元阶数$k\ge 1$相关。通过数值实验，我们提出了一个关于依赖参数$r$和$k$的先验误差估计的猜想。此外，这些误差估计依赖于定义在边界上具有适应属性的函数提升（lift），该提升用于将计算域上的数值解映射至物理域。