Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces

Bartels, Sören

doi:10.1090/S0025-5718-09-02300-X

Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces
HTML articles powered by AMS MathViewer

by Sören Bartels;

Math. Comp. 79 (2010), 1263-1301

DOI: https://doi.org/10.1090/S0025-5718-09-02300-X

Published electronically: September 16, 2009

PDF | Request permission

Abstract:

This article studies the numerical approximation of harmonic maps into surfaces, i.e., critical points for the Dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold. An iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised, and its global convergence to a discrete harmonic map is proved under general conditions. Weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations. Numerical simulations illustrate the performance of the algorithm for curved domains.

References

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. MR 1472192, DOI 10.1137/S0036142994264249
François Alouges, A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187–196. MR 2379897, DOI 10.3934/dcdss.2008.1.187
Ball, J. M. Orientable and non-orientable director fields for liquid crystals. In ICIAM 07 – 6th International Congress on Industrial and Applied Mathematics (Zurich, Switzerland, July 2007).
John W. Barrett, Sören Bartels, Xiaobing Feng, and Andreas Prohl, A convergent and constraint-preserving finite element method for the $p$-harmonic flow into spheres, SIAM J. Numer. Anal. 45 (2007), no. 3, 905–927. MR 2318794, DOI 10.1137/050639429
John W. Barrett and Robert Nürnberg, Finite element approximation of a Stefan problem with degenerate Joule heating, M2AN Math. Model. Numer. Anal. 38 (2004), no. 4, 633–652. MR 2087727, DOI 10.1051/m2an:2004030
Sören Bartels, Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices, M2AN Math. Model. Numer. Anal. 39 (2005), no. 5, 863–882. MR 2178565, DOI 10.1051/m2an:2005038
Sören Bartels, Stability and convergence of finite-element approximation schemes for harmonic maps, SIAM J. Numer. Anal. 43 (2005), no. 1, 220–238. MR 2177142, DOI 10.1137/040606594
Bartels, S. Finite element approximation of harmonic maps between surfaces. Habilitation thesis, Humboldt-Universität zu Berlin, Germany, 2008.
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. MR 2336271, DOI 10.1090/S0025-5718-07-02026-1
Fabrice Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), no. 4, 417–443. MR 1208652, DOI 10.1007/BF02599324
Haïm Brezis, Jean-Michel Coron, and Elliott H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), no. 4, 649–705. MR 868739
Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46 (1993), no. 7, 1041–1091. MR 1223662, DOI 10.1002/cpa.3160460705
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Clarenz, U., and Dziuk, G. Numerical methods for conformally parametrized surfaces. In CPDw04 - Interphase 2003: Numerical Methods for Free Boundary Problems (The Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, April 2003).
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. no. , no. R-2, 77–84 (English, with French summary). MR 400739
de Gennes, P.-G., and Prost, J. The Physics of Liquid Crystals. Oxford Science Publications, Oxford, 1993. second ed.
Alan Demlow and Gerhard Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45 (2007), no. 1, 421–442. MR 2285862, DOI 10.1137/050642873
DeSimone, A., Kohn, R. V., Müller, S., and Otto, F. Recent analytical developments in micromagnetics. in: Science of Hysteresis, Elsevier, G. Bertotti and I Magyergyoz, Eds. (2005).
Gerhard Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial differential equations and calculus of variations, Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988, pp. 142–155. MR 976234, DOI 10.1007/BFb0082865
J. Eells and L. Lemaire, On the construction of harmonic and holomorphic maps between surfaces, Math. Ann. 252 (1980), no. 1, 27–52. MR 590546, DOI 10.1007/BF01420211
Lawrence C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113. MR 1143435, DOI 10.1007/BF00375587
Frank, F. C. On the theory of liquid crystals. Discuss. Faraday Soc. 25 (1958), 19–28.
Alexandre Freire, Stefan Müller, and Michael Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 6, 725–754 (English, with English and French summaries). MR 1650966, DOI 10.1016/S0294-1449(99)80003-1
Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 1, 15–34. MR 1397098, DOI 10.1090/S0273-0979-97-00692-7
Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591–596 (French, with English summary). MR 1101039
Frédéric Hélein, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR 1913803, DOI 10.1017/CBO9780511543036
Willi Jäger and Helmut Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math. 343 (1983), 146–161. MR 705882
Jürgen Jost, On the existence of harmonic maps from a surface into the real projective plane, Compositio Math. 59 (1986), no. 1, 15–19. MR 850117
Landau, L. D. and Lifschitz, E. M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sovietunion 8 (1935), 153–169.
Fang-Hua Lin, A remark on the map $x/|x|$, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 12, 529–531 (English, with French summary). MR 916327
San Yih Lin and Mitchell Luskin, Relaxation methods for liquid crystal problems, SIAM J. Numer. Anal. 26 (1989), no. 6, 1310–1324. MR 1025090, DOI 10.1137/0726076
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 202511
Stefan Müller, Michael Struwe, and Vladimir Šverák, Harmonic maps on planar lattices, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 713–730 (1998). Dedicated to Ennio De Giorgi. MR 1655538
Oseen, C. W. The theory of liquid crystals. Trans. Faraday Soc. 29 (1933), 883–899.
Tristan Rivière, Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), no. 2, 197–226. MR 1368247, DOI 10.1007/BF02393305
Tristan Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), no. 1, 1–22. MR 2285745, DOI 10.1007/s00222-006-0023-0
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geometry 17 (1982), no. 2, 307–335. MR 664498
Michael Struwe, Variational methods, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34, Springer-Verlag, Berlin, 2000. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1736116, DOI 10.1007/978-3-662-04194-9
Luminita A. Vese and Stanley J. Osher, Numerical methods for $p$-harmonic flows and applications to image processing, SIAM J. Numer. Anal. 40 (2002), no. 6, 2085–2104 (2003). MR 1974176, DOI 10.1137/S0036142901396715
Epifanio G. Virga, Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation, vol. 8, Chapman & Hall, London, 1994. MR 1369095, DOI 10.1007/978-1-4899-2867-2