A Priori Error Estimates of a Poisson Equation with Ventcel Boundary Conditions on Curved Meshes

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1929-1955, August 2024.
Abstract. In this work is considered an elliptic 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 finite element discretization. The focus is on 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 at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order [math] and to the finite element degree [math], whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter [math]. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters [math] and [math]. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case.