Numerical study of a diffusion equation with Ventcel boundary condition using curved meshes

TL;DR

This paper presents a numerical analysis of a diffusion equation with Ventcel boundary conditions, emphasizing the use of high-order curved meshes and investigating errors related to geometry and finite element approximation.

Contribution

It introduces a finite element discretization for the Ventcel problem on curved meshes and formulates a conjecture on error estimates based on mesh and element degrees.

Findings

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

Numerical experiments support the conjectured error behavior

A functional lift is used to relate computational and physical solutions

Abstract

In this work is provided a numerical study of a diffusion problem involving a second order term on the domain boundary (the Laplace-Beltrami operator) referred to as the \textit{Ventcel problem}.A variational formulation of the Ventcel problem is studied, leading to a finite element discretization.The focus is on the resort to high order curved meshes for the discretization of the physical domain.The computational errors are investigated both in terms of geometrical error and of finite element approximation error, respectively associated to the mesh degree $r\ge 1$ and to the finite element degree $k\ge 1$. The numerical experiments we led allow us to formulate a conjecture on the \textit{a priori} error estimates depending on the two parameters $r$ and $k$. In addition, these error estimates rely on the definition of a functional \textit{lift} with adapted properties on the boundary to move numerical solutions defined on the computational domain to the physical one.