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Solving the Laplace‐Beltrami equation on S 2 using spherical triangles
Author(s) -
Mu Jun
Publication year - 1996
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199609)12:5<627::aid-num6>3.0.co;2-m
Subject(s) - mathematics , laplace's equation , galerkin method , discretization , finite element method , quadrature (astronomy) , mathematical analysis , numerical analysis , conjugate gradient method , projection (relational algebra) , partial differential equation , algebraic equation , domain (mathematical analysis) , geometry , algorithm , physics , nonlinear system , quantum mechanics , electrical engineering , thermodynamics , engineering
The geometry of the domain − S 2 , causes difficulty in solving the Laplace‐Beltrami Equation, for example, in discretization for the differential equation. To overcome this problem, we study a numerical method, which is based on the finite element approximation with a hierarchical refinement of icosahedron for the grid. We construct a geometrically intrinsic base vector field for the Galerkin approximation. In this way, no artificial poles are introduced, and the numerical grids are distributed more evenly. We use radial projection to map the curved triangle onto a flat one, so that existing quadrature schemes can be applied for the numerical integration. The resulting system of linear algebraic equations is solved by using a conjugate gradient method. © 1996 John Wiley & Sons, Inc.