{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T19:54:10Z","timestamp":1780343650631,"version":"3.54.1"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T00:00:00Z","timestamp":1645747200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article we construct parallel solvers analyze the efficiency and accuracy of general parallel solvers for three dimensional parabolic problems with the fractional power of elliptic operators. The proposed discrete method are targeted for general non-constant elliptic operators, the second motivation for the usage of such schemes arises when non-uniform space meshes are essential. Parallel solvers are required to solve the obtained large size systems of linear equations. The detailed scalability analysis is done in order to compare the efficiency of prposed parallel algorithms. Results of computational experiments are presented and analyzed.<\/jats:p>","DOI":"10.3390\/axioms11030098","type":"journal-article","created":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T10:00:40Z","timestamp":1645783240000},"page":"98","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3262-3048","authenticated-orcid":false,"given":"Raimondas","family":"\u010ciegis","sequence":"first","affiliation":[{"name":"Department of Mathematical Modelling, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul\u0117tekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ignas","family":"Dap\u0161ys","sequence":"additional","affiliation":[{"name":"Department of Mathematical Modelling, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul\u0117tekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Remigijus","family":"\u010ciegis","sequence":"additional","affiliation":[{"name":"Kaunas Faculty, Institute of Social Sciences and Applied Informatics, Vilnius University, Muitin\u0117s St 8, LT-44280 Kaunas, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1029\/2000WR900409","article-title":"Subordinated advection-dispersion equation for contaminant transport","volume":"37","author":"Baeumer","year":"2001","journal-title":"Water Resour. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","article-title":"The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics","volume":"37","author":"Metzler","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"20140352","DOI":"10.1098\/rsif.2014.0352","article-title":"Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization","volume":"11","author":"Kay","year":"2014","journal-title":"J. R. Soc. Interface"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.cam.2016.06.020","article-title":"Volume constrained 2-phase segmentation method utilizing a linear system solver based on the best uniform polynomial approximation of x\u22121\/2","volume":"310","author":"Harizanov","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.cam.2017.09.007","article-title":"A second-order operator splitting Fourier spectral method for fractional-in-space reaction\u2013diffusion equations","volume":"333","author":"Lee","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"\u010ciegis, R., \u010ciegis, R., and Dap\u0161ys, I. (2021). A Comparison of discrete schemes for numerical solution of parabolic problems with fractional power elliptic operators. 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Evolution equations in Hilbert spaces via the lacunae method. arXiv.","DOI":"10.3390\/fractalfract6050229"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2083","DOI":"10.1090\/S0025-5718-2015-02937-8","article-title":"Numerical approximation of fractional powers of elliptic operators","volume":"84","author":"Bonito","year":"2015","journal-title":"Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1093\/imanum\/drz054","article-title":"hp-FEM for the fractional heat equation","volume":"41","author":"Melenk","year":"2021","journal-title":"IMA J. Numer. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1137\/14096308X","article-title":"A PDE approach to space-time fractiobal parabolic problems","volume":"54","author":"Nochetto","year":"2016","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"109141","DOI":"10.1016\/j.jcp.2019.109141","article-title":"A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations","volume":"405","author":"Zhang","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"A1494","DOI":"10.1137\/16M1106122","article-title":"The AAA algorithm for rational approximation","volume":"40","author":"Nakatsukasa","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e2167","DOI":"10.1002\/nla.2167","article-title":"Optimal solvers for linear systems with fractional powers of sparse SPD matrices","volume":"25","author":"Harizanov","year":"2018","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/j.apnum.2021.03.006","article-title":"Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operator","volume":"165","author":"Vabishchevich","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"584","DOI":"10.3846\/mma.2020.12139","article-title":"A three-level parallelisation scheme and application to the Nelder-Mead algorithm","volume":"25","author":"Bugajev","year":"2020","journal-title":"Math. Model. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"A2288","DOI":"10.1137\/110835347","article-title":"Aggregation-based algebraic multigrid for convection-diffusion equations","volume":"34","author":"Notay","year":"2012","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"e4216","DOI":"10.1002\/cpe.4216","article-title":"Parallel solvers for fractional power diffusion problems","volume":"29","author":"Margenov","year":"2017","journal-title":"Concurr. Comput. Pract. Exp."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1109\/TSP.2006.882087","article-title":"A modified split-radix FFT with fewer arithmetic operations","volume":"55","author":"Johnson","year":"2007","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/98\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:27:10Z","timestamp":1760135230000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/98"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,25]]},"references-count":21,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["axioms11030098"],"URL":"https:\/\/doi.org\/10.3390\/axioms11030098","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,25]]}}}