In this work is considered an elliptic problem, referred to as the Ventcel problem, involvinga second order term on the domain boundary (the Laplace-Beltrami operator). A variationalformulation of the Ventcel problem is studied, leading to a finite element discretization. Thefocus is on the construction of high order curved meshes for the discretization of the physicaldomain and on the definition of the lift operator, which is aimed to transform a functiondefined on the mesh domain into a function defined on the physical one. This lift is definedin a way as to satisfy adapted properties on the boundary, relatively to the trace operator.The Ventcel problem approximation is investigated both in terms of geometrical error and offinite element approximation error. Error estimates are obtained both in terms of the meshorder r $\ge$ 1 and to the finite element degree k $\ge$ 1, whereas such estimates usually have beenconsidered in the isoparametric case so far, involving a single parameter k = r. The numericalexperiments we led, both in dimension 2 and 3, allow us to validate the results obtained andproved on the a priori error estimates depending on the two parameters k and r. A numericalcomparison is made between the errors using the former lift definition and the lift defined inthis work establishing an improvement in the convergence rate of the error in the latter case.

翻译：本文研究了一个涉及边界二阶项（拉普拉斯-贝尔特拉米算子）的椭圆问题，称为Ventcel问题。通过研究Ventcel问题的变分形式，导出了有限元离散方案。研究重点在于构建用于物理域离散化的高阶弯曲网格，以及定义提升算子——该算子旨在将定义在网格域上的函数转换为定义在物理域上的函数。此提升算子的定义方式需满足边界上关于迹算子的适应性条件。本文从几何误差和有限元逼近误差两个角度研究了Ventcel问题的近似。分别获得了关于网格阶数r $\ge$ 1和有限元次数k $\ge$ 1的误差估计，而此类估计以往通常仅在等参情形（单参数k = r）下被考虑。我们在二维和三维情况下进行的数值实验验证了所得结果，并证明了依赖于双参数k和r的先验误差估计。通过对比传统提升算子定义与本工作定义下的误差，证实了后者在误差收敛速率上的改进。