# Trace Finite Element Methods for Surface Vector-Laplace Equations

## TL;DR

This paper analyzes trace finite element methods for surface vector-Laplace equations, focusing on different techniques for enforcing the tangential constraint and providing detailed error bounds considering geometric and solution approximations.

## Contribution

It presents a comprehensive analysis of three methods for enforcing the tangent condition in TraceFEM for surface vector-Laplace problems, including error estimates.

## Key findings

- Error bounds depend on polynomial degrees and penalty parameters.
- All three techniques effectively enforce the tangent condition.
- The analysis guides optimal parameter choices for accurate discretization.

## Abstract

In this paper we analyze a class of trace finite element methods (TraceFEM) for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (``tangent condition''). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. A main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of Lagrange multiplier.