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

TL;DR

This paper develops a priori error estimates for a diffusion equation with Ventcel boundary conditions on curved meshes, emphasizing high-order mesh construction, lift operator definition, and validation through numerical experiments in 2D and 3D.

Contribution

It introduces a novel lift operator and derives error estimates considering both mesh order and finite element degree, extending beyond the isoparametric case.

Findings

Error estimates depend on mesh order r and finite element degree k.

Numerical experiments validate theoretical error bounds in 2D and 3D.

Improved convergence rates with the new lift operator compared to previous methods.

Abstract

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.