{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T04:00:37Z","timestamp":1740110437568,"version":"3.37.3"},"reference-count":43,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004735","name":"Hunan Provincial Natural Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004735","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Applied Mathematics and Computation"],"published-print":{"date-parts":[[2021,8]]},"DOI":"10.1016\/j.amc.2021.126091","type":"journal-article","created":{"date-parts":[[2021,2,28]],"date-time":"2021-02-28T03:46:43Z","timestamp":1614484003000},"page":"126091","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":4,"special_numbering":"C","title":["An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients"],"prefix":"10.1016","volume":"402","author":[{"given":"Kejia","family":"Pan","sequence":"first","affiliation":[]},{"given":"Hai-Wei","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Yufeng","family":"Xu","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"year":"1999","series-title":"Fractional Differential Equations","author":"Podlubny","key":"10.1016\/j.amc.2021.126091_bib0001"},{"key":"10.1016\/j.amc.2021.126091_bib0002","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1007\/s10915-018-0869-5","article-title":"Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach","volume":"79","author":"Hao","year":"2019","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0003","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1137\/17M1116222","article-title":"Boundary problems for the fractional and tempered fractional operators","volume":"16","author":"Deng","year":"2018","journal-title":"Multiscale Model. Simul."},{"year":"2015","series-title":"Numerical Methods for Fractional Calculus","author":"Li","key":"10.1016\/j.amc.2021.126091_bib0004"},{"year":"2015","series-title":"Finite Difference Methods of Fractional Differential Equations","author":"Sun","key":"10.1016\/j.amc.2021.126091_bib0005"},{"key":"10.1016\/j.amc.2021.126091_bib0006","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.cam.2003.09.028","article-title":"Numerical solution of the space fractional Fokker-Planck equation","volume":"166","author":"Liu","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0007","doi-asserted-by":"crossref","first-page":"4038","DOI":"10.1016\/j.jcp.2009.02.011","article-title":"Finite difference approximations for a fractional advection diffusion problem","volume":"228","author":"Sousa","year":"2009","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0008","doi-asserted-by":"crossref","first-page":"8095","DOI":"10.1016\/j.jcp.2010.07.011","article-title":"A direct O(Nlog2N) finite difference method for fractional diffusion equations","volume":"229","author":"Wang","year":"2010","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0009","doi-asserted-by":"crossref","first-page":"A2444","DOI":"10.1137\/12086491X","article-title":"A fast finite difference method for two-dimensional space-fractional diffusion equations","volume":"34","author":"Wang","year":"2012","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0010","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/j.jcp.2014.10.053","article-title":"A fourth-order approximation of fractional derivatives with its applications","volume":"281","author":"Hao","year":"2015","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0011","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.jcp.2015.07.001","article-title":"Fast numerical solution for fractional diffusion equations by exponential quadrature rule","volume":"299","author":"Zhang","year":"2015","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0012","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1016\/j.jcp.2011.10.005","article-title":"Multigrid method for fractional diffusion equations","volume":"231","author":"Pang","year":"2012","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0013","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/j.jcp.2015.08.052","article-title":"Multigrid methods for space fractional partial differential equations","volume":"302","author":"Jiang","year":"2015","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0014","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.cma.2017.08.019","article-title":"Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver","volume":"327","author":"Ainsworth","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.amc.2021.126091_bib0015","doi-asserted-by":"crossref","first-page":"A4007","DOI":"10.1137\/17M115164X","article-title":"Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations","volume":"40","author":"Donatelli","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0016","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.jcp.2017.02.008","article-title":"A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations","volume":"336","author":"Lin","year":"2017","journal-title":"J. Comput. Phys."},{"issue":"e2178","key":"10.1016\/j.amc.2021.126091_bib0017","first-page":"1","article-title":"Solving time-periodic fractional diffusion equations via diagonalization technique and multigrid","volume":"25","author":"Wu","year":"2018","journal-title":"Numer. Linear Algebr. Appl."},{"key":"10.1016\/j.amc.2021.126091_bib0018","doi-asserted-by":"crossref","first-page":"1034","DOI":"10.1007\/s10915-017-0480-1","article-title":"Uniform convergence of v-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation","volume":"74","author":"Chen","year":"2018","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0019","doi-asserted-by":"crossref","first-page":"3471","DOI":"10.1016\/j.camwa.2019.05.017","article-title":"Parallel-in-time multigrid for space-time finite element approximations of two-dimensional space-fractional diffusion equations","volume":"78","author":"Yue","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2021.126091_bib0020","unstructured":"X.Q. Yue, K.J. Pan, et\u00a0al., A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems, arXiv:1906.06829."},{"key":"10.1016\/j.amc.2021.126091_bib0021","doi-asserted-by":"crossref","first-page":"A948","DOI":"10.1137\/18M1191488","article-title":"Multigrid methods for discrete fractional sobolev spaces","volume":"41","author":"Baerland","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0022","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.jcp.2013.02.025","article-title":"A circulant preconditioner for fractional diffusion equations","volume":"242","author":"Lei","year":"2013","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0023","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.jcp.2013.07.040","article-title":"Preconditioned iterative methods for fractional diffusion equation","volume":"256","author":"Lin","year":"2014","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0024","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1007\/s11425-008-0119-7","article-title":"Analysis of extrapolation cascadic multigrid method (EXCMG)","volume":"51","author":"Chen","year":"2008","journal-title":"Sci. China Ser. A"},{"key":"10.1016\/j.amc.2021.126091_bib0025","doi-asserted-by":"crossref","first-page":"684","DOI":"10.4208\/jcm.1110-m11si05","article-title":"On extrapolation cascadic multigrid method","volume":"29","author":"Chen","year":"2011","journal-title":"J. Comput. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0026","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1007\/s10915-016-0275-9","article-title":"An extrapolation cascadic multigrid method combined with a fourth-order compact scheme for 3d poisson equation","volume":"70","author":"Pan","year":"2017","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0027","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s002110050234","article-title":"The cascadic multigrid method for elliptic problems","volume":"75","author":"Bornemann","year":"1996","journal-title":"Numer. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0028","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.cam.2018.10.018","article-title":"An economical cascadic multigrid method for the weak Galerkin finite element approximation of second order elliptic problems","volume":"362","author":"Shao","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0029","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s11075-019-00690-1","article-title":"A type of cascadic multigrid method for coupled semilinear elliptic equations","volume":"82","author":"Xu","year":"2020","journal-title":"Numer. Algorithms"},{"key":"10.1016\/j.amc.2021.126091_bib0030","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.jcp.2017.04.069","article-title":"A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems","volume":"344","author":"Pan","year":"2017","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.amc.2021.126091_bib0031","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/j.camwa.2017.05.023","article-title":"On the convergence of an extrapolation cascadic multigrid method for elliptic problems","volume":"74","author":"Hu","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2021.126091_bib0032","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1016\/j.camwa.2018.12.024","article-title":"An EXCMG accelerated multiscale multigrid computation for 3d poisson equation","volume":"77","author":"Dai","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2021.126091_bib0033","article-title":"An efficient extrapolation full multigrid method for elliptic problems in two and three dimensions","author":"Li","year":"2020","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0034","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1137\/15M1033952","article-title":"A fractional laplace equation: regularity of solutions and finite element approximations","volume":"55","author":"Acosta","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.amc.2021.126091_bib0035","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0898-1221(95)00228-6","article-title":"Some estimates of the rate of convergence for the cascadic conjugate-gradient method","volume":"31","author":"Shaidurov","year":"1996","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.amc.2021.126091_bib0036","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/PL00005416","article-title":"A cascadic multigrid algorithm for semilinear elliptic problems","volume":"86","author":"Timmermann","year":"2000","journal-title":"Numer. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0037","first-page":"551","article-title":"Cascadic multigrid for parabolic problems","volume":"18","author":"Shi","year":"2000","journal-title":"J. Comput. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0038","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1007\/s11425-008-0112-1","article-title":"Cascadic multigrid methods for parabolic problems","volume":"51","author":"Du","year":"2008","journal-title":"Sci. China Ser. A"},{"key":"10.1016\/j.amc.2021.126091_bib0039","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-019-0769-9","article-title":"A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains","volume":"38","author":"Du","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0040","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.apnum.2018.12.002","article-title":"On a second order scheme for space fractional diffusion equations with variable coefficients","volume":"137","author":"Vong","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"10.1016\/j.amc.2021.126091_bib0041","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1007\/s10915-017-0388-9","article-title":"Fast iterative method with a second-order implicit difference scheme for time-space fractional convection-diffusion equation","volume":"72","author":"Gu","year":"2017","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0042","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-020-01193-1","article-title":"A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation","volume":"83","author":"Zhao","year":"2020","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.amc.2021.126091_bib0043","article-title":"Numerical correction of finite difference solution for two-dimensional space-fractional diffusion equations with boundary singularity","author":"Hao","year":"2020","journal-title":"Numer. Algorithms"}],"container-title":["Applied Mathematics and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0096300321001399?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0096300321001399?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T04:38:54Z","timestamp":1672807134000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0096300321001399"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8]]},"references-count":43,"alternative-id":["S0096300321001399"],"URL":"https:\/\/doi.org\/10.1016\/j.amc.2021.126091","relation":{},"ISSN":["0096-3003"],"issn-type":[{"type":"print","value":"0096-3003"}],"subject":[],"published":{"date-parts":[[2021,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients","name":"articletitle","label":"Article Title"},{"value":"Applied Mathematics and Computation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.amc.2021.126091","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2021 Elsevier Inc. All rights reserved.","name":"copyright","label":"Copyright"}],"article-number":"126091"}}