COMBINATION OF GLOBAL AND LOCAL APPROXIMATION SCHEMES FOR HARMONIC MAPS INTO SPHERES  

作　　者：Sren Bartels 

机构地区：[1]Institute for Numerical Simulation,Rheinische Friedrich-Wilhelms-Universitt Bonn,Wegelerstraβe 6,53115 Bonn,Germany

出　　处：《Journal of Computational Mathematics》2009年第2期170-183,共14页计算数学（英文）

基　　金：Supported by Deutsche Forschungsgemeinschaft through the DFG Research Center MATHEON‘Mathematics for key technologies’in Berlin;The authors wish to thank C.Melcher for pointing out the Example 4.1.

摘　　要：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.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.

关 键 词：Harmonic maps Iterative methods Pointwise constraint 

分 类 号：O241.5[理学—计算数学]