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Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers


Authors: Sören Bartels, Christian Lubich and Andreas Prohl
Journal: Math. Comp. 78 (2009), 1269-1292
MSC (2000): Primary 65M12, 65M60, 35K55, 35Q35
DOI: https://doi.org/10.1090/S0025-5718-09-02221-2
Published electronically: February 18, 2009
MathSciNet review: 2501050
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Abstract: We propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres. The finite-element based schemes preserve a unit length constraint at the nodes by means of approximate discrete Lagrange multipliers, satisfy a discrete energy law, and iterates are shown to converge to weak solutions of the continuous problem. Comparative computational studies are included to motivate finite-time blow-up behavior in both cases.


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  • 1. F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case, SIAM J. Numer. Anal. 34, pp. 1708-1726 (1997). MR 1472192 (98k:82190)
  • 2. F. Alouges, P. Jaisson Convergence of a finite elements discretization for the Landau Lifshitz equations, Math. Models Methods Appl. Sci. 16, pp. 299-316 (2006). MR 2210092 (2007b:65091)
  • 3. S. Bartels, A. Prohl, Constraint preserving implicit finite element discretization of harmonic map flow into spheres, Math. Comp. 76, pp. 1847-1859 (2007). MR 2336271 (2008j:65156)
  • 4. S. Bartels, A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz equation, SIAM J. Numer. Anal. 44, pp. 1405-1419 (2006). MR 2257110 (2007g:65087)
  • 5. S. Bartels, A. Prohl, Stable discretization of scalar and constrained vectorial Perona-Malik equation, Interfaces and Free Boundaries 9, pp. 431-453 (2007). MR 2358212
  • 6. J.W. Barrett, S. Bartels, X. Feng, A. Prohl, A convergent and constraint-preserving finite element method for the $ p$-harmonic flow into spheres, SIAM J. Numer. Anal. 45, pp. 905-927 (2007). MR 2318794 (2008f:65170)
  • 7. S. Bartels, X. Feng, A. Prohl, Finite element approximations of wave maps into spheres, SIAM J. Numer. Anal. 46, pp. 61-87 (2007). MR 2377255 (2008k:35318)
  • 8. S. Bartels, A. Prohl, Convergence of an implicit, constraint preserving finite element discretization of $ p$-harmonic heat flow into the sphere, Numer. Math. 109 (2008), no. 4, 489-507. MR 2407320
  • 9. P. Bizoń, T. Chmaj, Z. Tabor, Dispersion and collapse of wave maps, Nonlinearity 13, pp. 1411-1423 (2000). MR 1767966 (2001g:58047)
  • 10. P. Bizoń, T. Chmaj, Z. Tabor, Formation of singularities for equivariant $ (2+1)$-dimensional wave maps into the $ 2$-sphere, Nonlinearity 14, p. 1041-1053 (2001). MR 1862811 (2003b:58043)
  • 11. K.C. Chang, W.Y. Ding, R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geom. 36, pp. 507-511 (1992). MR 1180392 (93h:58043)
  • 12. Y.M. Chen, M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201, pp. 83-103 (1989). MR 990191 (90i:58031)
  • 13. J.-M. Coron, J.-M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, CR. Acad. Sci., Paris, Ser. I 308, pp. 339-344 (1989). MR 992088 (90g:58026)
  • 14. J.F. Grotowski, J. Shatah, A note on geometric heat flows in critical dimensions, Preprint (2006), downloadable at: http://math.nyu.edu/faculty/shatah/preprints/gs06.pdf.
  • 15. E. Hairer, C. Lubich, G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd edition, Springer (2006). MR 2221614 (2006m:65006)
  • 16. J. Krieger, W. Schlag, D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543-615. MR 2372807
  • 17. M. Kruzik, A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Review 48, pp. 439-483 (2006). MR 2278438 (2008c:82088)
  • 18. I. Rodnianski, J. Sterbenz, On the formation of singularities in the critical $ O(3)$ $ \sigma$-model, preprint (arXiv-series), (2006).
  • 19. J. Shatah, Weak solutions and development of singularities in the $ SU(2)$ $ \sigma$ model, Comm. Pure Appl. Math. 41, pp. 459-469 (1988). MR 933231 (89f:58044)
  • 20. J. Shatah, M. Struwe, Geometric wave equations, New York University, Courant Institute of Mathematical Sciences, New York (1998). MR 1674843 (2000i:35135)
  • 21. R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, AMS (1997). MR 1422252 (98c:47076)
  • 22. M. Struwe, Geometric evolution problems, IAS/Park City Math. Series, vol. 2, pp. 259-339 (1996). MR 1369591 (97e:58057)
  • 23. M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Math. Helv. 60, pp. 558-581 (1985) MR 826871 (87e:58056)
  • 24. B. Tang, G. Sapiro, V. Caselles, Diffusion of generated data on non-flat manifolds via harmonic maps theory: the direction diffusion case. Int. J. Comput. Vision 36, pp. 149-161 (2000).
  • 25. B. Tang, G. Sapiro, V. Caselles, Color image enhancement via chromaticity diffusion, IEEE Trans. Image Proc. 10, pp. 701-707 (2001).
  • 26. D. Tataru, The wave maps equation, Bull. Amer. Math. Soc. 41, pp. 185-204 (2004). MR 2043751 (2005h:35245)
  • 27. L.A. Vese, S.J. Osher, Numerical methods for $ p$-harmonic flows and applications to image processing, SIAM J. Numer. Anal. 40, pp. 2085-2104 (2002). MR 1974176 (2004k:65102)

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Additional Information

Sören Bartels
Affiliation: Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany
Email: bartels@ins.uni-bonn.de

Christian Lubich
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: lubich@na.uni-tuebingen.de

Andreas Prohl
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: prohl@na.uni-tuebingen.de

DOI: https://doi.org/10.1090/S0025-5718-09-02221-2
Received by editor(s): April 10, 2007
Received by editor(s) in revised form: April 30, 2008
Published electronically: February 18, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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