Convergence of renormalized finite element methods for heat flow of harmonic maps

Abstract

A linearly implicit renormalized lumped mass finite element method is considered for solving the equations describing heat flow of harmonic maps, of which the exact solution naturally satisfies the pointwise constraint |ｍ| = 1. At every time level, the method first computes an auxiliary numerical solution by a linearly implicit lumped mass method and then renormalizes it at all finite element nodes before proceeding to the next time level. It is shown that such a renormalized finite element method has an error bound of O(Ｔ+ hr+1) for tensor-product finite elements of degree r ≽ 1. The proof of the error estimates is based on a geometric relation between the auxiliary and renormalized numerical solutions. The extension of the error analysis to triangular mesh is straightforward and discussed in the conclusion section.