Stability and Convergence of Finite-Element Approximation Schemes for Harmonic Maps

This article discusses stability and convergence of approximation schemes for harmonic maps. A finite-element discretization of an iterative algorithm due to F. Alouges is introduced and shown to be stable and convergent in general only on acute-type triangulations. An a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement. Numerical experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency.