Convergence of renormalized finite element methods for heat flow of harmonic maps
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We study the finite element approximation for the heat flow of harmonic maps, a parabolic partial differential equation with a strongly nonlinear term. The exact solution naturally satisfies a pointwise unit-length constraint. To address this, we consider a linearly implicit renormalized lumped mass finite element method. At each time step, an auxiliary solution is first computed using a linearly implicit lumped mass scheme and then renormalized at all finite element nodes. This approach preserves the structure of the numerical solution. Based on a newly established geometric relationship between the auxiliary and renormalized solutions, we prove the optimal-order error bounds of this method for tensor-product finite elements of any order. Extensions to triangular meshes in space and high-order multistep methods in time are also discussed.