Quasi-optimal error estimates for the approximation of stable harmonic maps

TL;DR

This paper develops quasi-optimal error estimates for finite element approximations of harmonic maps into spheres, utilizing a saddle-point formulation and inverse function theorem, with potential extensions to other manifolds.

Contribution

It introduces a new error estimation framework for harmonic maps with nodal constraints, improving accuracy analysis under natural regularity and stability conditions.

Findings

Error estimates are quasi-optimal for harmonic maps into spheres.

The approach applies under natural regularity and geometric stability.

Extensions to other target manifolds are feasible.

Abstract

Based on a quantitative version of the inverse function theorem and an appropriate saddle-point formulation we derive a quasi-optimal error estimate for the finite element approximation of harmonic maps into spheres with a nodal discretization of the unit-length constraint. The estimate holds under natural regularity requirements and appropriate geometric stability conditions on solutions. Extensions to other target manifolds including boundaries of ellipsoids are discussed.