Continuous Linear Finite Element Method for Biharmonic Problems on Surfaces

TL;DR

This paper introduces a continuous linear finite element method for biharmonic problems on surfaces, utilizing a surface gradient recovery operator and stabilization techniques to achieve optimal error estimates despite geometric errors.

Contribution

The paper proposes a novel finite element approach that effectively handles second derivatives on surfaces using gradient recovery and stabilization, with proven stability and optimal error bounds.

Findings

Method achieves optimal error estimates in energy and L2 norms.

Numerical experiments confirm theoretical stability and accuracy.

Approach effectively manages geometric errors in surface biharmonic problems.

Abstract

This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to compute the second-order surface derivative of a piecewise continuous linear function defined on the approximate surface, as conventional notions of second-order derivatives are not directly applicable in this context. By incorporating appropriate stabilizations, we rigorously establish the stability of the proposed formulation. Despite the presence of geometric error, we provide optimal error estimates in both the energy norm and $L^2$ norm. Theoretical results are supported by numerical experiments.