Constraint preserving implicit finite element discretization of harmonic map flow into spheres

Mathematics of Computation

Volumn 76, Issue 260, 2007, Pages 1847-1859

  Bartels, Sören ^a   Prohl, Andreas ^b  

^a HUMBOLDT UNIVERSITY   (Germany)

^b UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Convergence; Finite element method; Fully discrete scheme; Harmonic map flow

Indexed keywords

EID: 38349060008     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-07-02026-1     Document Type: Article

References (18)

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). MR1472192 (98k:82190)
2. F. Alouges, P. Jaisson, Convergence of a finite element discretization for the Landau Lifshitz equations, Math. Models Methods Appl. Sci. 16, pp. 299-316 (2006). MR2210092 (2007b:65091)
3. S. Bartels, Stability and convergence of finite element approximation schemes for harmonic maps, SIAM J. Numer. Anal. 43, pp. 220-238 (2005). MR2177142 (2006j:65336)
4. S. Bartels, A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Num. Anal. 44, pp. 1405-1419 (2006). MR2257110
5. 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. Num. Anal. (accepted).
6. 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. 507-515 (1992). MR1180392 (93h:58043)
7. Y. Chen, The weak solutions to the evolution problems of harmonic maps, Math. Z. 201, pp. 69-74 (1989). MR990189 (90i:58030)
8. Y. Chen, M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201, pp. 83-103 (1999). MR990191 (90i:58031)
9. I. Cimrak, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA J. Num. Anal. 25, pp. 611-634 (2005). MR2153750 (2006h:82100)
10. L.C. Evans, Weak convergence methods for nonlinear partial differential equations, C.B.M.S. Regional Conf. Series in Mathematics 74, Providence R. I. (1990). MR1034481 (91a:35009)
11. A. Freire, Uniqueness for the harmonic map flow in two dimensions, Cale. Var. PDE 3, pp. 95-105 (1995). MR1384838 (97d:58058)
12. V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: theory and algorithms, Springer-Verlag, Berlin, Heidelberg, New York (1981). MR851383 (88b:65129)
13. B. Guo, M.-C. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. 1, pp. 311-334 (1993). MR1261548 (94m:58059)
14. M. Kružík, A. Prohl, Recent Developments in Modeling, Analysis and Numerics of Ferromagnetism, SIAM Review 48, pp. 439-483 (2006).
15. S. Y. Lin, M. Luskin, Relaxation methods for liquid crystal problems, SIAM J. Numer. Anal. 26, pp. 1310-1324 (1989). MR1025090 (90m:65106)
16. A. Prohl, Computational micromagnetism, Teubner (2001). MR1885923 (2004e:82067)
17. M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Math. Helv. 60, pp. 558-581 (1985). MR826871 (87e:58056)
18. M. Struwe, Geometric evolution problems, IAS/Park City Math. Series, vol. 2, pp. 259-339 (1996). MR1369591 (97e:58057)