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Stability and Convergence of Finite-Element Approximation Schemes for Harmonic Maps

Published: 01 January 2005 Publication History

Abstract

This article discusses stability and convergence of approximation schemes for harmonic maps. A finite-element discretization of an iterative algorithm due to F. Alouges is introduced and shown to be stable and convergent in general only on acute-type triangulations. An\break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement. Numerical\break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency.

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  1. Stability and Convergence of Finite-Element Approximation Schemes for Harmonic Maps

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      cover image SIAM Journal on Numerical Analysis
      SIAM Journal on Numerical Analysis  Volume 43, Issue 1
      2005
      454 pages

      Publisher

      Society for Industrial and Applied Mathematics

      United States

      Publication History

      Published: 01 January 2005

      Author Tags

      1. adaptive refinement
      2. finite element method
      3. harmonic maps
      4. iterative algorithm
      5. liquid crystals
      6. weak convergence

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      • (2024)A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry ProcessingACM Transactions on Graphics10.1145/367365243:5(1-26)Online publication date: 9-Aug-2024
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      • (2019)Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamicsAdvances in Computational Mathematics10.1007/s10444-019-09667-z45:3(1329-1368)Online publication date: 1-Jun-2019
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      • (2018)$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifoldsNumerische Mathematik10.1007/s00211-017-0941-3139:2(381-410)Online publication date: 1-Jun-2018
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      • (2015)Optimal A Priori Discretization Error Bounds for Geodesic Finite ElementsFoundations of Computational Mathematics10.1007/s10208-014-9230-z15:6(1357-1411)Online publication date: 1-Dec-2015
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