{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T18:50:14Z","timestamp":1649184614696},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Methods Appl. Math."],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem\nby means of  the <jats:italic>hp<\/jats:italic>-version of the finite element method.\nThe system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method.\nThe development of an efficient preconditioner  requires three ingredients, a preconditioner related to the components of the velocity modes,\na preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair\nwhich satisfies the discrete inf-sup-condition.\nThe last condition is also important in order to obtain a stable discretization scheme.\nThe preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation.\nMoreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix\nif the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can\nbe solved in almost optimal complexity.\nThis yields  quasioptimal <jats:italic>hp<\/jats:italic>-FEM solvers for the Stokes problems and the linear elasticity problems.\nThe latter are robust with respect to the contraction ratio \u03bd.\nThe efficiency of the presented solver is shown in several numerical examples.<\/jats:p>","DOI":"10.2478\/cmam-2012-0030","type":"journal-article","created":{"date-parts":[[2013,2,13]],"date-time":"2013-02-13T12:23:49Z","timestamp":1360758229000},"page":"369-390","source":"Crossref","is-referenced-by-count":0,"title":["Schwarz Type Solvers for -FEM Discretizations of Mixed Problems"],"prefix":"10.2478","volume":"12","author":[{"given":"Sven","family":"Beuchler","sequence":"first","affiliation":[{"name":"1Institute for Numerical Simulation, Wegelerstra\u00dfe 6, 53115 Bonn, Germany."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Purrucker","sequence":"additional","affiliation":[{"name":"1Institute for Numerical Simulation, Wegelerstra\u00dfe 6, 53115 Bonn, Germany."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/12\/4\/article-p369.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.2478\/cmam-2012-0030\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T14:22:33Z","timestamp":1619101353000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/12\/4\/article-p369.xml"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"references-count":0,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.2478\/cmam-2012-0030","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012]]}}}