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Weak series BV convergence of a moving finite-element method for singular axisymmetric harmonic maps. (English) Zbl 1109.65103

The aim of the paper is to prove the convergence of an optimal mesh method in the presence of a consistency error. The solution of the continuous problem considered here [cf. F. Alouges and M. Pierre [Numer. Math. 101, No. 3, 391–414 (2005; Zbl 1088.65106)] minimizes a relaxed Dirichlet energy among axisymmetric maps from the disc into the sphere. The convergence of the moving finite element method for conforming elements, under the assumption that the discrete energy is computed exactly is proved. For nonconforming elements, an external approximation (cf. [A. Quarteroni, R. Sacco and F. Saleri [Numerical mathematics, Texts in Applied Mathematics. 37. (New York), NY: Springer. (2000; Zbl 0957.65001)] is introduced in the \(BV\) space, which enables the authors to prove the convergence.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
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