TY - JOUR
T1 - Moving Mesh Methods for Singular Problems on a Sphere Using Perturbed Harmonic Mappings
AU - Di, Yana
AU - Li, Ruo
AU - Tang, Tao
AU - Zhang, Pingwen
N1 - Funding information:
LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China ([email protected], [email protected], [email protected]). The research of the first two authors was supported in part by the Joint Applied Mathematics Research Institute between Peking University and Hong Kong Baptist University. The research of the fourth author was partially supported by the special funds for Major State Research Projects (2005CB1704) and the National Science Foundation of China for Distinguished Young Scholars (10225103).
Mathematics Department, Hong Kong Baptist University, Kowloon Tong, Hong Kong (ttang@ math.hkbu.edu.hk). The research of this author was supported by CERG grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, NSAF grant 10476032 of the National Science Foundation of China, and by the International Research Team on Complex System of Chinese Academy of Sciences.
Publisher copyright:
Copyright © 2006 Society for Industrial and Applied Mathematics
PY - 2006/9/15
Y1 - 2006/9/15
N2 - 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.
AB - 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.
KW - Harmonic mapping
KW - Moving mesh methods
KW - Perturbed harmonic mapping
KW - Singularity
KW - Spherical domain
UR - http://www.scopus.com/inward/record.url?scp=34547223991&partnerID=8YFLogxK
U2 - 10.1137/050642514
DO - 10.1137/050642514
M3 - Journal article
AN - SCOPUS:34547223991
SN - 1064-8275
VL - 28
SP - 1490
EP - 1508
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 4
ER -