{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T13:33:45Z","timestamp":1770816825964,"version":"3.50.1"},"reference-count":37,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11071041"],"award-info":[{"award-number":["11071041"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["N_CUHK437\/16"],"award-info":[{"award-number":["N_CUHK437\/16"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We first derive some explicit bounds on the spectra of generalized\nnon-symmetric singular or nonsingular saddle point matrices. Then we propose two new nonsingular preconditioners for solving generalized singular saddle point problems, and show that GMRES determines a solution without breakdown when applied to the resulting preconditioned systems with any initial guess. Furthermore, the detailed spectral properties of the\npreconditioned systems are analyzed. The nonsingular preconditioners are also applied to solve the singular\nfinite element saddle point systems\narising from the discretization of the Stokes problems to test their performance.<\/jats:p>","DOI":"10.1515\/cmam-2017-0006","type":"journal-article","created":{"date-parts":[[2017,6,2]],"date-time":"2017-06-02T13:21:55Z","timestamp":1496409715000},"page":"237-256","source":"Crossref","is-referenced-by-count":2,"title":["Spectral Analysis, Properties and Nonsingular Preconditioners for Singular Saddle Point Problems"],"prefix":"10.1515","volume":"18","author":[{"given":"Na","family":"Huang","sequence":"first","affiliation":[{"name":"Institute of Computational Mathematics and Scientific\/Engineering Computing , Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing , 100190 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chang-Feng","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science & FJKLMAA , Fujian Normal University , Fuzhou , 350007 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Zou","sequence":"additional","affiliation":[{"name":"Department of Mathematics , The Chinese University of Hong Kong , Shatin, N.T. , Hong Kong , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,5,31]]},"reference":[{"key":"2023033109580948067_j_cmam-2017-0006_ref_001_w2aab3b7b1b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"O.  Axelsson,\nUnified analysis of preconditioning methods for saddle point matrices,\nNumer. 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