A TraceFEM $C^0$ Interior Penalty Method for the Surface Biharmonic Equation

TL;DR

This paper introduces a TraceFEM $C^0$ interior penalty method for solving the surface biharmonic equation, achieving optimal convergence rates on a polyhedral surface approximation.

Contribution

It develops a novel TraceFEM discretization using quadratic Lagrange elements and a symmetric interior penalty formulation for the surface biharmonic problem.

Findings

Proves optimal first-order convergence in a discrete $H^2$ norm.

Establishes quadratic convergence in the $L^2$ norm.

Ensures stability through surface edge and bulk-facet penalization.

Abstract

We construct and analyze a TraceFEM discretization for the surface biharmonic problem. The method utilizes standard quadratic Lagrange finite element spaces defined on a three-dimensional background mesh and a symmetric $C^0$ interior penalty formulation posed on a second-order polyhedral approximation of the surface. Stability is achieved through a combination of surface edge penalties and bulk-facet penalization of gradient and Hessian jumps. We prove optimal first-order convergence in a discrete $H^2$ norm and quadratic convergence in the $L^2$ norm.