Error analysis for a Finite Element Discretization of a corotational harmonic map heat flow problem
This work offers rigorous error bounds for a numerical method applied to a specific nonlinear PDE, which is incremental for the numerical analysis community.
The authors provide an error analysis for a finite element discretization of a corotational harmonic map heat flow problem, proving optimal order error bounds and validating them with numerical experiments.
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.