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Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers. (English) Zbl 1198.65178
Summary: 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.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35Q35 PDEs in connection with fluid mechanics
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[1] François Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case, SIAM J. Numer. Anal. 34 (1997), no. 5, 1708 – 1726. · Zbl 0886.35010
[2] François Alouges and Pascal Jaisson, Convergence of a finite element discretization for the Landau-Lifshitz equations in micromagnetism, Math. Models Methods Appl. Sci. 16 (2006), no. 2, 299 – 316. · Zbl 1102.35333
[3] Sören Bartels and Andreas Prohl, Constraint preserving implicit finite element discretization of harmonic map flow into spheres, Math. Comp. 76 (2007), no. 260, 1847 – 1859. · Zbl 1124.65089
[4] Sören Bartels and Andreas Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405 – 1419. · Zbl 1124.65088
[5] Sören Bartels and Andreas Prohl, Stable discretization of scalar and constrained vectorial Perona-Malik equation, Interfaces Free Bound. 9 (2007), no. 4, 431 – 453. · Zbl 1147.35011
[6] John W. Barrett, Sören Bartels, Xiaobing Feng, and Andreas Prohl, A convergent and constraint-preserving finite element method for the \?-harmonic flow into spheres, SIAM J. Numer. Anal. 45 (2007), no. 3, 905 – 927. · Zbl 1155.35055
[7] Sören Bartels, Xiaobing Feng, and Andreas Prohl, Finite element approximations of wave maps into spheres, SIAM J. Numer. Anal. 46 (2007/08), no. 1, 61 – 87. · Zbl 1160.65050
[8] Sören Bartels and Andreas Prohl, Convergence of an implicit, constraint preserving finite element discretization of \?-harmonic heat flow into spheres, Numer. Math. 109 (2008), no. 4, 489 – 507. · Zbl 1155.65068
[9] Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, Dispersion and collapse of wave maps, Nonlinearity 13 (2000), no. 4, 1411 – 1423. · Zbl 0963.35121
[10] Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere, Nonlinearity 14 (2001), no. 5, 1041 – 1053. · Zbl 0988.35010
[11] Kung-Ching Chang, Wei Yue Ding, and Rugang Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), no. 2, 507 – 515. · Zbl 0765.53026
[12] Yun Mei Chen and Michael Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), no. 1, 83 – 103. · Zbl 0652.58024
[13] Jean-Michel Coron and Jean-Michel Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 12, 339 – 344 (French, with English summary). · Zbl 0679.58017
[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] Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. · Zbl 1094.65125
[16] J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543 – 615. · Zbl 1139.35021
[17] Martin Kružík and Andreas Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006), no. 3, 439 – 483. · Zbl 1126.49040
[18] I. Rodnianski, J. Sterbenz, On the formation of singularities in the critical \( O(3)\) \( \sigma\)-model, preprint (arXiv-series), (2006). · Zbl 1213.35392
[19] Jalal Shatah, Weak solutions and development of singularities of the \?\?(2) \?-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459 – 469. · Zbl 0686.35081
[20] Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. · Zbl 0993.35001
[21] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. · Zbl 0870.35004
[22] Michael Struwe, Geometric evolution problems, Nonlinear partial differential equations in differential geometry (Park City, UT, 1992) IAS/Park City Math. Ser., vol. 2, Amer. Math. Soc., Providence, RI, 1996, pp. 257 – 339. · Zbl 0847.58012
[23] Michael Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), no. 4, 558 – 581. · Zbl 0595.58013
[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). · Zbl 1037.68792
[26] Daniel Tataru, The wave maps equation, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185 – 204. · Zbl 1065.35199
[27] Luminita A. Vese and Stanley J. Osher, Numerical methods for \?-harmonic flows and applications to image processing, SIAM J. Numer. Anal. 40 (2002), no. 6, 2085 – 2104 (2003). · Zbl 1035.65065
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