In this talk, we consider a diffusion problem, referred to as the Ventcel problem, involving a second order term on the domain boundary (the Laplace-Beltrami operator).
The focus is on obtaining error estimations expressed with respect to the finite element degree k >= 1 and to the mesh order r >= 1. Indeed a crucial point concerns the construction of high order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one. This lift is defined in a way as to satisfy adapted properties on the boundary, relatively to the trace operator. Once the theoretical a priori error estimates depending on the two parameters k and r have been obtained, we perform numerical experiments which validate these results. Lastly, an eigenvalue problem with Ventcel boundary conditions is introduced. A similar procedure is used to estimate the eigenvalues and eigenvectors errors. Numerical experiments in 2d and 3d are presented validating the theoretical estimations.