A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains
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A fast finite difference method is developed for solving space-fractional diffusion equations with variable coefficient in convex domains using a volume penalization approach. The resulting coefficient matrix can be written as the discretized matrix from the extended rectangular domain plus a diagonal matrix with jumping entries due to the penalization parameter. An efficient preconditioner is constructed based on the combination of two approximate inverse circulant matrices. The preconditioned BiCGSTAB method, with the proposed preconditioner, is implemented for solving the resulting linear system. Numerical results are carried out to demonstrate the utility of the proposed algorithm.
KeywordsAnomalous diffusion Finite difference method Space-fractional diffusion equation Circulant preconditioner Penalization
Mathematics Subject Classification35R05 65F08 65F10 65M06
The research is supported in part by the National Science Foundation under Grant DMS-1216923, the OSD/ARO MURI Grant W911NF-15-1-0562, the National Natural Science Foundation of China under Grants 11831010, 11471194, 11571115 and 11371229, Taishan research project of Shandong Province, the research grant 0118/2018/A3 from FDCT of Macao, and MYRG2018-00015-FST from University of Macau.
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