# A comparative study of finite element methods for a class of harmonic map heat flow problems

## TL;DR

This paper systematically compares three finite element methods for solving harmonic map heat flow problems from a disk to a sphere, focusing on convergence, efficiency, and stability within a unified framework.

## Contribution

It provides a comprehensive comparison and analysis of three finite element discretization methods, including a new stability result for one method.

## Key findings

- All methods show expected convergence rates for smooth solutions
- Computational efficiency varies among the methods
- A discrete inf-sup stability result is established for one method

## Abstract

In this paper, we review and systematically compare three finite element discretization methods for a harmonic map heat flow problem from the unit disk in $\mathbb{R}^2$ to the unit sphere in $\mathbb{R}^3$ in an unified framework. Numerical tests validate the convergence rates in a regime of smooth solutions and are used to compare the methods in terms of computational efficiency. For one of the methods a discrete inf-sup stability result is derived.