Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T22:23:10.830Z Has data issue: false hasContentIssue false
  • English
  • Français

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

Published online by Cambridge University Press:  28 May 2015

Ronald D. Haynes ,
Weizhang Huang  and
Paul A. Zegeling
Ronald D. Haynes*
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada
Weizhang Huang*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
Paul A. Zegeling*
Affiliation:
Department of Mathematics, Utrecht University, Utrecht, The Netherlands
*
Corresponding author.Email address:rhaynes@mun.ca
Corresponding author.Email address:huang@math.ku.edu
Corresponding author.Email address:P.A.Zegeling@uu.nl
Get access

Abstract

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.

Keywords

35K9165M5065M06Heat flowharmonic mapblowupmoving mesh methodfinite difference
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 2 , May 2013 , pp. 364 - 383
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baňas, L., Bartels, S., and Prohl, A.. A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation. SIAM J. Numer. Anal, 46:1399–1422, 2008.CrossRefGoogle Scholar
[2]Bandle, C. and Brunner, H.. Blowup in diffusion equations: a survey. J. Comput. Appl. Math., 97:3–22, 1998.CrossRefGoogle Scholar
[3]Barrett, J. W., Bartels, S., Feng, X., and Prohl, A.. A convergent and constraint-preserving finite element method for the p-harmonic flow into spheres. SIAM J. Numer. Anal, 45:905–927, 2007.CrossRefGoogle Scholar
[4]Bartels, S.. Combination of global and local approximation schemes for harmonic maps into spheres. J. Comput. Math., 27:170–183, 2009.Google Scholar
[5]Bartels, S.. Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces. Math. Comp., 79:1263–1301, 2010.CrossRefGoogle Scholar
[6]Bartels, S., Feng, X., and Prohl, A.. Finite element approximations of wave maps into spheres. SIAM J. Numer. Anal, 46:61–87, 2007/08.CrossRefGoogle Scholar
[7]Bartels, S., Lubich, C., and Prohl, A.. Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers. Math. Comp., 78:1269–1292, 2009.CrossRefGoogle Scholar
[8]Bartels, S. and Prohl, A.. Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation. SIAM J. Numer. Anal, 44:1405–1419 (electronic), 2006.CrossRefGoogle Scholar
[9]Bartels, S. and Prohl, A.. Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres. Numer. Math., 109:489–507, 2008.CrossRefGoogle Scholar
[10]Baňas, L., Prohl, A., and Schätzle, R.. Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii. Numer. Math., 115:395–432, 2010.CrossRefGoogle Scholar
[11]Berger, M. and Kohn., R. VA rescaling algorithm for the numerical calculation of blowing-up solutions. Comm. Pure Appl. Math, 41:841–863, 1988.CrossRefGoogle Scholar
[12]Bertsch, M., Dal Passo, R., and Hout, R. van der. Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Ration. Mech. Anal, 161(2):93–112, 2002.Google Scholar
[13]Bertsch, M., Podio-Guidugli, P., and Valente, V. On the dynamics of deformable ferromagnetic solids: I. Global weak solutions for soft ferromagnets at rest. Ann. Mat. Pura Appl., 179:331–360, 2001.CrossRefGoogle Scholar
[14]Budd, C. J., Galaktionov, V A., and Williams, J. F.. Self-similar blow-up in higher-order semi-linear parabolic equations. SIAM J. Appl. Math., 64:1775–1809 (electronic), 2004.CrossRefGoogle Scholar
[15]Budd, C. J., Huang, W., and Russell, R. D.. Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput, 17:305–327, 1996.CrossRefGoogle Scholar
[16]Chang, K.-C.. Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire, 6:363–395, 1989.CrossRefGoogle Scholar
[17]Chang, K.-C. and Ding, W.-Y.. A result on the global existence for heat flows of harmonic maps from D2 into S2. In Nematics (Orsay, 1990), volume 332 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci, pages 37–47. Kluwer Acad. Publ., Dordrecht, 1991.Google Scholar
[18]Chang, K.-C., Ding, W.-Y., and Ye, R.. Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Diff. Geom., 36:507–515, 1992.Google Scholar
[19]Chen, S. H.. A sufficient condition for blowup solutions of nonlinear heat equations. J. Math. Anal. Appl., 293:227–236, 2004.CrossRefGoogle Scholar
[20]Chen, Y. and Ding, W.-Y.. Blow-up and global existence for heat flows of hamronic maps. Invent. Math., 99:567–578, 1990.CrossRefGoogle Scholar
[21]Coron, J.-M. and Ghidaglia, J.-M.. Explosion en temps fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris Sér. I Math., 308:339–344, 1989.Google Scholar
[22]Simone, A. De and Podio-Guidugli, P.. On the continuum model of deformable ferromagnetic solids. Arch. Ration. Mech. Anal., 136:201–233, 1996.Google Scholar
[23]Evans, J. D., Galaktionov, V. A., and Williams, J. F.. Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation. SIAM J. Math. Anal., 38:64–102 (electronic), 2006.CrossRefGoogle Scholar
[24]Freire, A.. Uniqueness of the harmonic map flow from surfaces to general targets. Comment. Math. Helv., 70:310–338, 1995.CrossRefGoogle Scholar
[25]Friedman, A. and Mcleod, B.. Blow up of positive solutions of semilinear heat equations. Indiana Univ. Math. J., 34:425–447, 1985.CrossRefGoogle Scholar
[26]Guan, M., Gustafson, S., and Tsai, T.-P.. Global existence and blow-up for harmonic map heat flow. J. Diff. Eq., 246:1–20, 2009.CrossRefGoogle Scholar
[27]Huang, W., Ma, J., and Russell, R. D.. A study of MMPDE moving mesh methods for the numerical simulation of blowup in reaction diffusion equations. J. Comput. Phys., 227:6532–6552, 2008.CrossRefGoogle Scholar
[28]Huang, W., Ren, Y., and Russell, R. D.. Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys., 113:279–290, 1994.CrossRefGoogle Scholar
[29]Huang, W., Ren, Y., and Russell, R. D.. Moving mesh partial differential equations (MMPDEs) based upon the equidistribution principle. SIAM J. Numer. Anal., 31:709–730, 1994.CrossRefGoogle Scholar
[30]Huang, W. and Russell, R. D.. Adaptive Moving Mesh Methods. Springer, New York, 2011. Applied Mathematical Sciences Series, Vol. 174.CrossRefGoogle Scholar
[31]Ma, J., Jiang, Y., and Xiang, K.. On a moving mesh method for solving partial integro-differential equations. J. Comput. Math., 27:713–728, 2009.CrossRefGoogle Scholar
[32]Struwe, M.. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv., 60:558–581, 1985.CrossRefGoogle Scholar
[33]Tang, B., Sapiro, G., and Casselles, V.. Color image enhancement via chromaticity diffusion. IEEE Trans. Image Proc., 10:701–707, 2001.CrossRefGoogle ScholarPubMed
[34]Dam, A. van and Zegeling, P. A.. Balanced monitoring of flow phenomena in moving mesh methods. Comm. Comput. Phys., 7:138–170, 2010.Google Scholar
[35]Berg, J. B. van den, Hulshof, J., and King, J. R.. Formal asymptotics of bubbling in the harmonic map heat flow. SIAM J. Appl. Math., 63:1682–1717, 2003.Google Scholar
[36]Hout., R. van derFlow alignment in nematic liquid crystals in flows with cylindrical symmetry. Diff. Int. Eq., 14:189–211, 2001.Google Scholar
[37]Schans., M. van derHarmonic map heat flow. Technical report, Mathematical Institute Leiden, 2006. Master Thesis.Google Scholar
[38]Vese, L. A. and Osher, S. J.. Numerical methods for p-harmonic flows and applications to image processing. SIAM J. Numer. Anal., 40:2085–2104, 2002.CrossRefGoogle Scholar