Convergence analysis of the intrinsic surface finite element method

TL;DR

This paper provides a comprehensive convergence analysis and stability estimates for the Intrinsic Surface Finite Element Method (ISFEM), validating its theoretical properties through rigorous proofs and numerical experiments.

Contribution

It offers the first complete convergence theory and stability analysis for ISFEM, enhancing understanding of its mathematical properties.

Findings

Proved convergence and stability of ISFEM.

Derived error estimates involving geometric quantities.

Numerical results confirm theoretical predictions.

Abstract

The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and makes direct use of the resulting covariant basis. Starting from a shape-regular triangulation of the surface, existence of a local parametrization for each triangle is exploited to approximate relevant quantities on the local chart. Standard two-dimensional FEM techniques in combination with surface quadrature rules complete the ISFEM formulation thus achieving a method that is fully intrinsic to the surface and makes limited use of the surface embedding only for the definition of basis functions. However, theoretical properties have not yet been proved. In this work we complement the original derivation of ISFEM with its complete convergence theory and propose the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities. Numerical experiments are included to support the theoretical results.