Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres (Journal of Computational Mathematics, 2018)
It is well understood that a good way to discretize a pointwise length constraint inpartial differential equations or variational problems is to impose it at the nodes of atriangulation that defines a lowest order finite element space. This article pursues thisapproach and discusses the iterative solution of the resulting discrete nonlinear systemof equations for a simple model problem which defines harmonic maps into spheres. Aniterative scheme that is globally convergent and energy decreasing is combined with alocally rapidly convergent approximation scheme. An explicit example proves that thelocal approach alone may lead to ill-posed problems; numerical experiments show that itmay diverge or lead to highly irregular solutions with large energy if the starting value isnot chosen carefully. The combination of the global and local method defines a reliablealgorithm that performs very efficiently in practice and provides numerical approximationswith low energy.