Error analysis for a Finite Element Discretization of a corotational harmonic map heat flow problem

TL;DR

This paper analyzes an $H^1$-conforming finite element method combined with semi-implicit Euler time stepping for a corotational harmonic map heat flow, providing optimal error bounds and numerical validation.

Contribution

It offers the first error analysis for this discretization approach applied to the corotational harmonic map heat flow, establishing stability and convergence results.

Findings

Optimal order discretization error bounds are proven.

A discrete energy estimate mimicking continuous energy dissipation is developed.

Numerical results confirm the theoretical error bounds.

Abstract

We consider the harmonic map heat flow problem for a corotational case. For discretization of this problem we apply a $H^1$-conforming finite element method in space combined with a semi-implicit Euler time stepping. The semi-implicit Euler method results in a linear problem in each time step. We restrict to the regime of smooth solutions of the continuous problem and present an error analysis of this discretization method. This results in optimal order discretization error bounds. Key ingredients of the analysis are a discrete energy estimate, that mimics the energy dissipation of the continuous solution, and a convexity property that is essential for discrete stability and for control of the linearization error. We also present numerical results that validate the theoretical ones.