Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions

TL;DR

This paper develops a finite element method for a spectral problem involving boundary diffusion with high order curved meshes, providing error estimates and validating results through numerical experiments in 2D and 3D.

Contribution

It introduces a finite element discretization for a spectral problem with boundary Laplace-Beltrami operator on curved meshes, including error analysis and validation.

Findings

Error estimates depend on finite element degree and mesh order

Curved meshes significantly reduce geometric errors

Numerical results confirm theoretical error bounds

Abstract

In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{lift} is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of \textit{finite element approximation error} and of \textit{geometric error}, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.