Convergence of multistep projection methods for harmonic map heat flows into general surfaces

Abstract

We propose a high-order multistep projection method for the harmonic map heat flow from a bounded domain Ω⊂Rd into a given N-dimensional smooth surface Γ⊂RN+1. At every time level, an auxiliary numerical solution is solved by a multistep backward difference formula with a mass-lumping finite element method in space, and then projected onto the surface Γ. The projected numerical solution is used in the backward difference formula and the extrapolation of nonlinearities in the following time levels. Such projection algorithms are convenient in computation while still preserving the pointwise geometric constraint of the solution to stay on the target surface Γ. The convergence of some low-order single-step projection algorithms based on the backward Euler and Crank–Nicolson schemes have been studied in many articles for harmonic map heat flow and related models into the unit sphere, while the convergence of high-order multistep projection methods still remains open. In this article, we propose a high-order multistep projection method for harmonic map heat flows into a general smooth surface (not necessarily the unit sphere) and prove its optimal-order convergence by combining four techniques, i.e., decomposition of the Nevanlinna–Odeh multiplier technique into approximately normal and tangential components separately, an almost orthogonal relation between the error functions associated to the auxiliary and projected numerical solutions, pointwise L∞ error estimates, the use of orthogonal projection onto the target surface Γ. Numerical results are provided to support the theoretical analysis on the convergence of the high-order multistep projection methods.