TY - JOUR
T1 - Combination of Global and Local Approximation Schemes for Harmonic Maps into Spheres
AU - Sören Bartels
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, Iterative methods, Pointwise constraint.
AB - <p style="text-align: justify;">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.</p>