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Composite‐grid multigrid for diffusion on the sphere
Author(s) -
Adler James H.,
Lashuk Ilya,
MacLachlan Scott P.
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2115
Subject(s) - multigrid method , discretization , grid , mathematics , finite element method , partial differential equation , surface (topology) , laplace transform , coupling (piping) , algorithm , mathematical optimization , computer science , mathematical analysis , geometry , mechanical engineering , physics , engineering , thermodynamics
Summary Recently, there has been much interest in the solution of differential equations on surfaces and manifolds, driven by many applications whose dynamics take place on such domains. Although increasingly powerful algorithms have been developed in this field, many straightforward questions remain, particularly in the area of coupling advanced discretizations with efficient linear solvers. In this paper, we develop a structured refinement algorithm for octahedral triangulations of the surface of the sphere. We explain the composite‐grid finite‐element discretization of the Laplace–Beltrami operator on such triangulations and extend the fast adaptive composite‐grid scheme to provide an efficient solution of the resulting linear system. Supporting numerical examples are presented, including the recovery of second‐order accuracy in the case of a nonsmooth solution.