Mathematics of Computation

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

Author(s): Sören Bartels.
Journal: Math. Comp.
MSC (2000): Primary 65N12, 65N22, 58E20
Posted: September 16, 2009
Retrieve article in: PDF

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.

1.: ALOUGES, F.
A new algorithm for computing liquid crystal stable configurations: The harmonic mapping case.
SIAM J. Numer. Anal. 34, 5 (1997), 1708-1726. MR 1472192 (98k:82190)
2.: ALOUGES, F.
A new finite element scheme for Landau-Lifschitz equations.
Discrete Contin. Dyn. Syst. Ser. S 1, 2 (2008), 187-196. MR 2379897 (2009a:65244)
3.: 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).
4.: BARRETT, J. W., BARTELS, S., FENG, X., AND PROHL, A.
A convergent and constraint-preserving finite element method for the -harmonic flow into spheres.
SIAM J. Numer. Anal. 45, 3 (2007), 905-927. MR 2318794 (2008f:65170)
5.: BARRETT, J. W. AND NüRNBERG, R.
Finite element approximation of a Stefan problem with degenerate Joule heating.
M2AN Math. Model. Numer. Anal. 38, 4 (2004), 633-652. MR 2087727 (2005g:80014)
6.: BARTELS, S.
Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices.
M2AN Math. Model. Numer. Anal. 39, 5 (2005), 863-882. MR 2178565 (2006g:35114)
7.: BARTELS, S.
Stability and convergence of finite-element approximation schemes for harmonic maps.
SIAM J. Numer. Anal. 43, 1 (2005), 220-238 (electronic). MR 2177142 (2006j:65336)
8.: BARTELS, S.
Finite element approximation of harmonic maps between surfaces.
Habilitation thesis, Humboldt-Universität zu Berlin, Germany, 2008.
9.: BARTELS, S. AND PROHL, A.
Constraint preserving implicit finite element discretization of harmonic map flow into spheres,
Math. Comp. 76, 260 (2007), 1847-1859. MR 2336271 (2008j:65156)
10.: BETHUEL, F.
On the singular set of stationary harmonic maps.
Manuscripta Math. 78, 4 (1993), 417-443. MR 1208652 (94a:58047)
11.: BREZIS, H., CORON, J.-M., AND LIEB, E. H.
Harmonic maps with defects.
Comm. Math. Phys. 107, 4 (1986), 649-705. MR 868739 (88e:58023)
12.: CHRISTODOULOU, D. AND TAHVILDAR-ZADEH, A. S.
On the regularity of spherically symmetric wave maps.
Comm. Pure Appl. Math. 46, 7 (1993), 1041-1091. MR 1223662 (94e:58030)
13.: CIARLET, P. G.
The finite element method for elliptic problems, vol. 40 of Classics in Applied Mathematics.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1930132
14.: 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).
15.: CLéMENT, P.
Approximation by finite element functions using local regularization.
RAIRO Analyse Numérique 9, R-2 (1975), 77-84. MR 0400739 (53:4569)
16.: DE GENNES, P.-G., AND PROST, J.
The Physics of Liquid Crystals.
Oxford Science Publications, Oxford, 1993.
second ed.
17.: DEMLOW, A. AND DZIUK, G.
An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces.
SIAM J. Numer. Anal. 14 (2007), 421-442. MR 2285862 (2008c:65320)
18.: 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).
19.: DZIUK, G.
Finite elements for the Beltrami operator on arbitrary surfaces.
In Partial differential equations and calculus of variations, vol. 1357 of Lecture Notes in Math. Springer, Berlin, 1988, pp. 142-155. MR 976234 (90i:65194)
20.: EELLS, J. AND LEMAIRE, L.
On the construction of harmonic and holomorphic maps between surfaces.
Math. Ann. 252, 1 (1980), 27-52. MR 590546 (81k:58030)
21.: EVANS, L. C.
Partial regularity for stationary harmonic maps into spheres.
Arch. Rational Mech. Anal. 116, 2 (1991), 101-113. MR 1143435 (93m:58026)
22.: FRANK, F. C.
On the theory of liquid crystals.
Discuss. Faraday Soc. 25 (1958), 19-28.
23.: FREIRE, A., MüLLER, S., AND STRUWE, M.
Weak compactness of wave maps and harmonic maps.
Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 6 (1998), 725-754. MR 1650966 (2000a:58045)
24.: HARDT, R. M.
Singularities of harmonic maps.
Bull. Amer. Math. Soc. (N.S.) 34, 1 (1997), 15-34. MR 1397098 (98b:58046)
25.: HéLEIN, F.
Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne.
C. R. Acad. Sci. Paris Sér. I Math. 312, 8 (1991), 591-596. MR 1101039 (92e:58055)
26.: HéLEIN, F.
Harmonic maps, conservation laws and moving frames, second ed., vol. 150 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 2002. MR 1913803 (2003g:58024)
27.: JäGER, W. AND KAUL, H.
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 (85f:58031)
28.: JOST, J.
On the existence of harmonic maps from a surface into the real projective plane.
Compositio Math. 59, 1 (1986), 15-19. MR 850117 (87k:58069)
29.: 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.
30.: LIN, F.-H.
A remark on the map $x/\vert x\vert$ .
C. R. Acad. Sci. Paris Sér. I Math. 305, 12 (1987), 529-531. MR 916327 (89b:58056)
31.: LIN, S. Y. AND LUSKIN, M.
Relaxation methods for liquid crystal problems.
SIAM J. Numer. Anal. 26, 6 (1989), 1310-1324. MR 1025090 (90m:65106)
32.: LIONS, P.-L.
The concentration-compactness principle in the calculus of variations. The limit case. I.
Rev. Mat. Iberoamericana 1, 1 (1985), 145-201. MR 834360 (87c:49007)
33.: MORREY, JR., C. B.
Multiple integrals in the calculus of variations.
Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York, 1966. MR 0202511 (34:2380)
34.: MüLLER, S., STRUWE, M., AND ŠVERáK, V.
Harmonic maps on planar lattices.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, 3-4 (1997), 713-730 (1998). MR 1655538 (2000c:58030)
35.: OSEEN, C. W.
The theory of liquid crystals.
Trans. Faraday Soc. 29 (1933), 883-899.
36.: RIVIèRE, T.
Everywhere discontinuous harmonic maps into spheres.
Acta Math. 175, 2 (1995), 197-226. MR 1368247 (96k:58059)
37.: RIVIèRE, T.
Conservation laws for conformally invariant variational problems.
Invent. Math. 168, 1 (2007), 1-22. MR 2285745 (2008d:58010)
38.: RUDIN, W.
Real and complex analysis, third ed.,
McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002)
39.: SCHOEN, R., AND UHLENBECK, K.
A regularity theory for harmonic maps.
J. Differential Geom. 17, 2 (1982), 307-335. MR 664498 (84b:58037a)
40.: STRUWE, M.
Variational methods, third ed., vol. 34 of Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics.
Springer-Verlag, Berlin, 2000. MR 1736116 (2000i:49001)
41.: VESE, L. A. AND OSHER, S. J.
Numerical methods for -harmonic flows and applications to image processing.
SIAM J. Numer. Anal. 40, 6 (2002), 2085-2104 (electronic) (2003). MR 1974176 (2004k:65102)
42.: VIRGA, E. G.
Variational theories for liquid crystals, vol. 8 of Applied Mathematics and Mathematical Computation.
Chapman & Hall, London, 1994. MR 1369095 (97m:73001)

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N22, 58E20

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS