Volume 27, Issue 2-3
Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres

Sören Bartels

DOI:

J. Comp. Math., 27 (2009), pp. 170-183.

Published online: 2009-04

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  • Abstract

It is well understood that a good way to discretize a pointwise length constraint in partial differential equations or variational problems is to impose it at the nodes of a triangulation that defines a lowest order finite element space. This article pursues this approach and discusses the iterative solution of the resulting discrete nonlinear system of equations for a simple model problem which defines harmonic maps into spheres. An iterative scheme that is globally convergent and energy decreasing is combined with a locally rapidly convergent approximation scheme. An explicit example proves that the local approach alone may lead to ill-posed problems; numerical experiments show that it may diverge or lead to highly irregular solutions with large energy if the starting value is not chosen carefully. The combination of the global and local method defines a reliable algorithm that performs very efficiently in practice and provides numerical approximations with low energy.

  • Keywords

Harmonic maps Iterative methods Pointwise constraint

  • AMS Subject Headings

65N12 65N30 35J60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-170, author = {}, title = {Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {170--183}, abstract = {

It is well understood that a good way to discretize a pointwise length constraint in partial differential equations or variational problems is to impose it at the nodes of a triangulation that defines a lowest order finite element space. This article pursues this approach and discusses the iterative solution of the resulting discrete nonlinear system of equations for a simple model problem which defines harmonic maps into spheres. An iterative scheme that is globally convergent and energy decreasing is combined with a locally rapidly convergent approximation scheme. An explicit example proves that the local approach alone may lead to ill-posed problems; numerical experiments show that it may diverge or lead to highly irregular solutions with large energy if the starting value is not chosen carefully. The combination of the global and local method defines a reliable algorithm that performs very efficiently in practice and provides numerical approximations with low energy.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8566.html} }
TY - JOUR T1 - Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres JO - Journal of Computational Mathematics VL - 2-3 SP - 170 EP - 183 PY - 2009 DA - 2009/04 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8566.html KW - Harmonic maps KW - Iterative methods KW - Pointwise constraint AB -

It is well understood that a good way to discretize a pointwise length constraint in partial differential equations or variational problems is to impose it at the nodes of a triangulation that defines a lowest order finite element space. This article pursues this approach and discusses the iterative solution of the resulting discrete nonlinear system of equations for a simple model problem which defines harmonic maps into spheres. An iterative scheme that is globally convergent and energy decreasing is combined with a locally rapidly convergent approximation scheme. An explicit example proves that the local approach alone may lead to ill-posed problems; numerical experiments show that it may diverge or lead to highly irregular solutions with large energy if the starting value is not chosen carefully. The combination of the global and local method defines a reliable algorithm that performs very efficiently in practice and provides numerical approximations with low energy.

Sören Bartels. (2019). Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres. Journal of Computational Mathematics. 27 (2-3). 170-183. doi:
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