A Saddle Point Approach to the Computation of Harmonic Maps

In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed.

[1]  Xue-Cheng Tai,et al.  Noise removal using smoothed normals and surface fitting , 2004, IEEE Transactions on Image Processing.

[2]  S. Yau,et al.  Lectures on Harmonic Maps , 1997 .

[3]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[4]  Sören Bartels,et al.  Stability and Convergence of Finite-Element Approximation Schemes for Harmonic Maps , 2005, SIAM J. Numer. Anal..

[5]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[6]  Jürgen Jost,et al.  Riemannian Geometry and Geometric Analysis, 5th Edition , 2008 .

[7]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[8]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[9]  F. Hélein,et al.  REGULARITE DES APPLICATIONS FAIBLEMENT HARMONIQUES ENTRE UNE SURFACE ET UNE VARIETE RIEMANNIENNE , 1991 .

[10]  F. Alouges A New Algorithm For Computing Liquid Crystal Stable Configurations: The Harmonic Mapping Case , 1997 .

[11]  F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[12]  Douglas N. Arnold,et al.  Preconditioning in Hdiv and applications , 1997, Math. Comput..

[13]  John W. Barrett,et al.  A Convergent and Constraint-Preserving Finite Element Method for the p-Harmonic Flow into Spheres , 2007, SIAM J. Numer. Anal..

[14]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[15]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[16]  Willi Jäger,et al.  Uniqueness and stability of harmonic maps and their Jacobi fields , 1979 .

[17]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[18]  Peter Li,et al.  LECTURES ON HARMONIC MAPS , 2011 .

[19]  Haim Brezis,et al.  The interplay between analysis and topology in some nonlinear PDE problems , 2003 .

[20]  Jinchao Xu,et al.  Global and uniform convergence of subspace correction methods for some convex optimization problems , 2002, Math. Comput..

[21]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[22]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[23]  San-Yih Lin,et al.  Relaxation methods for liquid crystal problems , 1989 .

[24]  D. Arnold,et al.  Preconditioning discrete approximations of the Reissner-Mindlin plate model , 1997 .

[25]  Jie Shen,et al.  Corrigendum: Fourier Spectral Approximation to a Dissipative System Modeling the Flow of Liquid Crystals , 2001, SIAM J. Numer. Anal..

[26]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[27]  Fuzhen Zhang Matrix Theory: Basic Results and Techniques , 1999 .

[28]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[29]  Xiaojun Chen Global and Superlinear Convergence of Inexact Uzawa Methods for Saddle Point Problems with Nondiffer , 1998 .

[30]  高等学校計算数学学報編輯委員会編 高等学校計算数学学報 = Numerical mathematics , 1979 .

[31]  Roland Glowinski,et al.  An operator-splitting method for a liquid crystal model , 2003 .

[32]  Mitchell Luskin,et al.  Minimum Energy Configurations for Liquid Crystals: Computational Results , 1987 .

[33]  Jie Shen,et al.  Corrigendum: Fourier Spectral Approximation to a Dissipative System Modeling the Flow of Liquid Crystals , 2003, SIAM J. Numer. Anal..

[34]  Mitchell Luskin,et al.  Remarks about the mathematical theory of liquid crystals , 1988 .

[35]  E Weinan,et al.  Numerical Methods for the Landau-Lifshitz Equation , 2000, SIAM J. Numer. Anal..

[36]  Stanley Osher,et al.  Numerical Methods for p-Harmonic Flows and Applications to Image Processing , 2002, SIAM J. Numer. Anal..

[37]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[38]  Jun Zou,et al.  Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems , 2006, SIAM J. Optim..

[39]  J. Jost Riemannian geometry and geometric analysis , 1995 .