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Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill\u2010conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two\u2010sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three\u2010by\u2010three block linear system are proved to be positive\u2010definite, based on which we propose an efficient block Gauss\u2013Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner31<\/jats:sup> as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill\u2010conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent.<\/jats:p>","DOI":"10.1002\/nla.2372","type":"journal-article","created":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T09:21:32Z","timestamp":1616145692000},"update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An efficient positive\u2010definite block\u2010preconditioned finite volume solver for two\u2010sided fractional diffusion equations on composite mesh"],"prefix":"10.1002","volume":"28","author":[{"given":"Pingfei","family":"Dai","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences Zhejiang University Hangzhou Zhejiang China"},{"name":"School of Science Hangzhou Normal University Hangzhou Zhejiang China"},{"name":"Department of Mathematics University of South Carolina Columbia South Carolina 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