{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T21:42:46Z","timestamp":1778708566590,"version":"3.51.4"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"270","license":[{"start":{"date-parts":[[2010,12,14]],"date-time":"2010-12-14T00:00:00Z","timestamp":1292284800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis normal d normal i normal v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">v<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\mathrm {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis normal d normal i normal v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">v<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\mathrm {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    operator with random weight function acting on a certain tensor product space of random fields with finite variance. We build on the Arnold-Falk-Winther multigrid algorithm presented in 1997 by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed. We extend the deterministic analysis to accommodate the modified\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis normal d normal i normal v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">v<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\mathrm {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    operator and establish spectral equivalence bounds with a new multigrid V-cycle operator that are independent of the spatial and stochastic discretization parameters. We implement multigrid within a block-diagonal preconditioner for the full saddle-point problem, derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned\n                    <sc>minres<\/sc>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02274-1","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"733-760","source":"Crossref","is-referenced-by-count":16,"title":["\ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) preconditioning for a mixed finite element formulation of the diffusion problem with random data"],"prefix":"10.1090","volume":"79","author":[{"given":"Howard","family":"Elman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Darran","family":"Furnival","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Catherine","family":"Powell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2009,12,14]]},"reference":[{"issue":"219","key":"1","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1090\/S0025-5718-97-00826-0","article-title":"Preconditioning in \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) and applications","volume":"66","author":"Arnold, Douglas N.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1137\/S0036142902418680","article-title":"Galerkin finite element approximations of stochastic elliptic partial differential equations","volume":"42","author":"Babu\u0161ka, Ivo","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","isbn-type":"print","volume-title":"Finite elements","author":"Braess, Dietrich","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0521011957","edition":"2"},{"key":"4","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"key":"5","series-title":"Springer Undergraduate Mathematics Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0533-6","volume-title":"Basic stochastic processes","author":"Brze\u017aniak, Zdzis\u0142aw","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/3540761756"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s00211-007-0086-x","article-title":"A fictitious domain approach to the numerical solution of PDEs in stochastic domains","volume":"107","author":"Canuto, Claudio","year":"2007","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"48","key":"7","doi-asserted-by":"publisher","first-page":"6359","DOI":"10.1016\/S0045-7825(01)00237-7","article-title":"Solution of stochastic partial differential equations using Galerkin finite element techniques","volume":"190","author":"Deb, Manas K.","year":"2001","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1093\/imanum\/drm006","article-title":"Solving the stochastic steady-state diffusion problem using multigrid","volume":"27","author":"Elman, Howard","year":"2007","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"1424","DOI":"10.1137\/070705817","article-title":"Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data","volume":"31","author":"Ernst, O. G.","year":"2008","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2-5","key":"10","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.cma.2004.04.008","article-title":"Finite elements for elliptic problems with stochastic coefficients","volume":"194","author":"Frauenfelder, Philipp","year":"2005","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"11","isbn-type":"print","volume-title":"Foundations of modern analysis","author":"Friedman, Avner","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0486640620"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic finite elements: a spectral approach","author":"Ghanem, Roger G.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387974563"},{"key":"13","first-page":"133","article-title":"Multigrid method for \ud835\udc07(\ud835\udc1d\ud835\udc22\ud835\udc2f) in three dimensions","volume":"6","author":"Hiptmair, R.","year":"1997","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"41-42","key":"14","doi-asserted-by":"publisher","first-page":"4723","DOI":"10.1016\/S0045-7825(03)00457-2","article-title":"A multigrid solver for two-dimensional stochastic diffusion equations","volume":"192","author":"Le Ma\u00eetre, O. P.","year":"2003","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/BF01385872","article-title":"Multilevel iterative methods for mixed finite element discretizations of elliptic problems","volume":"63","author":"Vassilevski, Panayot S.","year":"1992","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1137\/0712047","article-title":"Solutions of sparse indefinite systems of linear equations","volume":"12","author":"Paige, C. C.","year":"1975","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1093\/imanum\/dri017","article-title":"Parameter-free \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) preconditioning for a mixed finite element formulation of diffusion problems","volume":"25","author":"Powell, Catherine E.","year":"2005","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"718","DOI":"10.1137\/S0895479802404428","article-title":"Optimal preconditioning for Raviart-Thomas mixed formulation of second-order elliptic problems","volume":"25","author":"Powell, Catherine Elizabeth","year":"2003","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"19","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"key":"20","series-title":"Lecture Notes in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1170-9","volume-title":"Stochastic processes and orthogonal polynomials","volume":"146","author":"Schoutens, Wim","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/038795015X"},{"key":"21","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b137868","volume-title":"Domain decomposition methods---algorithms and theory","volume":"34","author":"Toselli, Andrea","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/3540206965"},{"key":"22","volume-title":"Topological vector spaces, distributions and kernels","author":"Tr\u00e8ves, Fran\u00e7ois","year":"1967"},{"key":"23","unstructured":"Dongbin Xiu. Generalized (Wiener-Askey) Polynomial Chaos. Ph.D. thesis, Brown University, 2004."},{"issue":"43","key":"24","doi-asserted-by":"publisher","first-page":"4927","DOI":"10.1016\/S0045-7825(02)00421-8","article-title":"Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos","volume":"191","author":"Xiu, Dongbin","year":"2002","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"5","key":"25","doi-asserted-by":"publisher","first-page":"1072","DOI":"10.1137\/0914065","article-title":"Multilevel iteration for mixed finite element systems with penalty","volume":"14","author":"Cai, Zhi Qiang","year":"1993","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02274-1\/S0025-5718-09-02274-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02274-1\/S0025-5718-09-02274-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:23:08Z","timestamp":1776788588000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02274-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,12,14]]},"references-count":25,"journal-issue":{"issue":"270","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["S0025-5718-09-02274-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02274-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,12,14]]}}}