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Based on a quantitative version of the inverse function theorem and anappropriate saddle-point formulation we derive a quasi-optimal error estimatefor the finite element approximation of harmonic maps into spheres with a nodaldiscretization of the unit-length constraint. The estimate holds under naturalregularity requirements and appropriate geometric stability conditions onsolutions. Extensions to other target manifolds including boundaries ofellipsoids are discussed.

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