Moving Mesh Methods for Singular Problems on a Sphere Using Perturbed Harmonic Mappings

Yana Di, Ruo Li, Tao Tang, Pingwen Zhang

Research output: Contribution to journalJournal articlepeer-review

8 Citations (Scopus)
24 Downloads (Pure)


This work is concerned with developing moving mesh strategies for solving problems defined on a sphere. To construct mappings between the physical domain and the logical domain, it has been demonstrated that harmonic mapping approaches are useful for a general class of solution domains. However, it is known that the curvature of the sphere is positive, which makes the harmonic mapping on a sphere not unique. To fix the uniqueness issue, we follow Sacks and Uhlenbeck [Ann. of Math. (2), 113 (1981), pp. 1–24] to use a perturbed harmonic mapping in mesh generation. A detailed moving mesh strategy including mesh redistribution and solution updating on a sphere will be presented. The moving mesh scheme based on the perturbed harmonic mapping is then applied to the moving steep front problem and the Fokker–Planck equations with high potential intensities on a sphere. The numerical experiments show that with a moderate number of grid points our proposed moving mesh algorithm can accurately resolve detailed features of singular problems on a sphere.

Original languageEnglish
Pages (from-to)1490-1508
Number of pages19
JournalSIAM Journal on Scientific Computing
Issue number4
Publication statusPublished - 15 Sept 2006

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Harmonic mapping
  • Moving mesh methods
  • Perturbed harmonic mapping
  • Singularity
  • Spherical domain


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