A priori error estimates of a diusion equation with Ventcel boundary conditions on curved meshes

Abstract

In this work is considered a diusion problem, referred to as the Ventcel problem, involving a second order term on the domain boundary (the Laplace-Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a nite element discretization. The focus is on the construction of high order curved meshes for the discretization of the physical domain and on the denition of the lift operator, which is aimed to transform a function dened on the mesh domain into a function dened on the physical one. This lift is dened in a way as to satisfy adapted properties on the boundary, relatively to the trace operator. Error estimations are computed and expressed both in terms of nite element approximation error and of geometrical error, respectively associated to the nite element degree k $\ge$ 1 and to the mesh order r $\ge$ 1. The numerical experiments we led allow us to validate the results obtained and proved on the a priori error estimates depending on the two parameters k and r