Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii

We prove the existence of weak solutions to the harmonic map heat flow, and wave maps into spheres of nonconstant radii. Weak solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite elements, where discrete Lagrange multipliers are employed to exactly meet the sphere constraint at mesh-points. Computational studies are included to motivate interesting dynamics in two and three spatial dimensions.

[1]  Andreas Prohl,et al.  Constraint preserving implicit finite element discretization of harmonic map flow into spheres , 2007, Math. Comput..

[2]  Rugang Ye,et al.  Finite-time blow-up of the heat flow of harmonic maps from surfaces , 1992 .

[3]  Michael Struwe,et al.  On the evolution of harmonic mappings of Riemannian surfaces , 1985 .

[4]  S. Bartels,et al.  Stable discretization of scalar and constrained vectorial Perona–Malik equation , 2007 .

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

[6]  Guillermo Sapiro,et al.  Color image enhancement via chromaticity diffusion , 2001, IEEE Trans. Image Process..

[7]  P. Bizo'n,et al.  Dispersion and collapse of wave maps , 2000 .

[8]  M. Struwe,et al.  ON GEOMETRIC HEAT FLOWS IN CRITICAL DIMENSIONS , 2006 .

[9]  J. Coron,et al.  Explosion en temps fini pour le flot des applications harmoniques , 1989 .

[10]  Jalal Shatah,et al.  Weak solutions and development of singularities of the SU(2) σ‐model , 1988 .

[11]  Andreas Prohl,et al.  A Convergent Implicit Finite Element Discretization of the Maxwell-Landau-Lifshitz-Gilbert Equation , 2008, SIAM J. Numer. Anal..

[12]  Andreas Prohl,et al.  Modeling of Thermally Assisted Magnetodynamics , 2008, SIAM J. Numer. Anal..

[13]  Yunmei Chen,et al.  The weak solutions to the evolution problems of harmonic maps , 1989 .

[14]  Piotr Bizo'n,et al.  Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere , 2001 .

[15]  Andreas Prohl,et al.  Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers , 2009, Math. Comput..

[16]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[17]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

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

[19]  J. Shatah,et al.  Geometric wave equations , 1998 .

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

[21]  Daniel Tataru,et al.  The wave maps equation , 2004 .

[22]  Andreas Prohl,et al.  Convergence of an Implicit Finite Element Method for the Landau-Lifshitz-Gilbert Equation , 2006, SIAM J. Numer. Anal..

[23]  Andreas Prohl,et al.  Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism , 2006, SIAM Rev..

[24]  Guillermo Sapiro,et al.  Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case , 2000, International Journal of Computer Vision.

[25]  Kunibert G. Siebert,et al.  ALBERT---Software for scientific computations and applications. , 2001 .

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

[27]  Andreas Prohl,et al.  Finite Element Approximations of Wave Maps into Spheres , 2007, SIAM J. Numer. Anal..

[28]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[29]  I. Rodnianski,et al.  On the formation of singularities in the critical $O(3)$ $\sigma$-model , 2006, math/0605023.

[30]  Andreas Prohl,et al.  Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow , 2008, SIAM J. Numer. Anal..

[31]  Michael Struwe,et al.  Geometric evolution problems , 1995 .

[32]  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..

[33]  Wilhelm Schlag,et al.  Renormalization and blow up for charge one equivariant critical wave maps , 2006, math/0610248.

[34]  François Alouges,et al.  CONVERGENCE OF A FINITE ELEMENT DISCRETIZATION FOR THE LANDAU¿LIFSHITZ EQUATIONS IN MICROMAGNETISM , 2006 .