{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:42:19Z","timestamp":1776836539015,"version":"3.51.2"},"reference-count":63,"publisher":"American Mathematical Society (AMS)","issue":"340","license":[{"start":{"date-parts":[[2023,11,17]],"date-time":"2023-11-17T00:00:00Z","timestamp":1700179200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["456235063"],"award-info":[{"award-number":["456235063"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babu\u0161ka\u2019s theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babu\u0161ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzi\u2019s theory, in a combined abstract norm. The construction suggests to form the latter from individual\n                    <italic>fitted<\/italic>\n                    norms that are composed from proper seminorms.\n                  <\/p>\n                  <p>This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot\u2019s equations.<\/p>","DOI":"10.1090\/mcom\/3795","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T09:33:46Z","timestamp":1668677626000},"page":"607-634","source":"Crossref","is-referenced-by-count":14,"title":["A new practical framework for the stability analysis of perturbed saddle-point problems and applications"],"prefix":"10.1090","volume":"92","author":[{"given":"Qingguo","family":"Hong","sequence":"first","affiliation":[]},{"given":"Johannes","family":"Kraus","sequence":"additional","affiliation":[]},{"given":"Maria","family":"Lymbery","sequence":"additional","affiliation":[]},{"given":"Fadi","family":"Philo","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-319-93873-8_1","article-title":"Robust block preconditioners for Biot\u2019s model","author":"Adler, James H.","year":"2018","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319938738"},{"issue":"219","key":"2","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"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906210018","article-title":"Finite element exterior calculus, homological techniques, and applications","volume":"15","author":"Arnold, Douglas N.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0521868157","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/s002110100348","article-title":"Mixed finite elements for elasticity","volume":"92","author":"Arnold, Douglas N.","year":"2002","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"6","first-page":"1","article-title":"Survey lectures on the mathematical foundations of the finite element method","author":"Babu\u0161ka, Ivo","year":"1972"},{"key":"7","doi-asserted-by":"crossref","unstructured":"M. Bai, D. Elsworth, and J.-C. Roegiers, Multiporosity\/multipermeability approach to the simulation of naturally fractured reservoirs, Water Resources Res. 29 (1993), no. 6, 1621\u20131633.","DOI":"10.1029\/92WR02746"},{"key":"8","isbn-type":"print","first-page":"15","article-title":"Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems","author":"Battermann, A.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3764358351"},{"key":"9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492904000212","article-title":"Numerical solution of saddle point problems","volume":"14","author":"Benzi, Michele","year":"2005","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys. 12 (1941), no. 2, 155\u2013164.","DOI":"10.1063\/1.1712886"},{"key":"11","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1063\/1.1721956","article-title":"Theory of elasticity and consolidation for a porous anisotropic solid","volume":"26","author":"Biot, M. A.","year":"1955","journal-title":"J. Appl. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8979","issn-type":"print"},{"key":"12","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5","volume-title":"Mixed finite element methods and applications","volume":"44","author":"Boffi, Daniele","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642365188"},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"B961--B983","DOI":"10.1137\/20M1379708","article-title":"Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot\u2019s equations utilizing total pressure","volume":"43","author":"Boon, Wietse M.","year":"2021","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"6","key":"14","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1051\/m2an\/1996300607311","article-title":"Stability of saddle point problems with penalty","volume":"30","author":"Braess, Dietrich","year":"1996","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618635","volume-title":"Finite elements","author":"Braess, Dietrich","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521705189","edition":"3"},{"key":"16","first-page":"129","article-title":"On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers","volume":"8","author":"Brezzi, F.","year":"1974","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"key":"17","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"},{"issue":"4","key":"18","doi-asserted-by":"publisher","first-page":"943","DOI":"10.1088\/0266-5611\/18\/4\/301","article-title":"Iterative regularization of parameter identification problems by sequential quadratic programming methods","volume":"18","author":"Burger, Martin","year":"2002","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"113229","DOI":"10.1016\/j.cma.2020.113229","article-title":"Robust block preconditioners for poroelasticity","volume":"369","author":"Chen, Shuangshuang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"20","unstructured":"O. Coussy, Poromechanics, John Wiley & Sons, West Sussex, England, 2004."},{"key":"21","unstructured":"L. Demkowicz, Babu\u0161ka \u21d4 Brezzi??, Technical Report 08-06, ICE, The University of Texas at Austin, Texas Institute for Computational and Applied Mathematics, 2006."},{"issue":"1-2","key":"22","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/S0168-9274(02)00125-3","article-title":"Mathematical and numerical models for coupling surface and groundwater flows","volume":"43","author":"Discacciati, Marco","year":"2002","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"23","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","volume-title":"Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics","author":"Elman, Howard C.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780198528685"},{"key":"24","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements","volume":"159","author":"Ern, Alexandre","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387205748"},{"key":"25","isbn-type":"print","volume-title":"Practical optimization","author":"Gill, Philip E.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0122839501"},{"key":"26","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0063453","volume-title":"Finite element approximation of the Navier-Stokes equations","volume":"749","author":"Girault, V.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/3540095578"},{"key":"27","isbn-type":"print","first-page":"3","article-title":"Finite element methods for incompressible viscous flow","author":"Glowinski, Roland","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0444512241"},{"key":"28","doi-asserted-by":"crossref","unstructured":"L. Guo, Z. Li, J. Lyu, Y. Mei, J. Vardakis, D. Chen, C. Han, X. Lou, and Y. Ventikos, On the validation of a multiple-network poroelastic model using arterial spin labeling MRI data, Front. Comput. Neurosci. 13 (2019), PMID 31551742, PMCID PMC6733888.","DOI":"10.3389\/fncom.2019.00060"},{"key":"29","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Computer image processing and recognition","author":"Hall, Ernest L.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0123188504"},{"issue":"6","key":"30","doi-asserted-by":"publisher","first-page":"2483","DOI":"10.1137\/060660588","article-title":"Nodal auxiliary space preconditioning in \ud835\udc3b(\ud835\udc50\ud835\udc62\ud835\udc5f\ud835\udc59) and \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) spaces","volume":"45","author":"Hiptmair, Ralf","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"31","doi-asserted-by":"publisher","first-page":"2750","DOI":"10.1137\/14099810X","article-title":"Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem","volume":"54","author":"Hong, Qingguo","year":"2016","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1553\/etna_vol48s202","article-title":"Parameter-robust stability of classical three-field formulation of Biot\u2019s consolidation model","volume":"48","author":"Hong, Qingguo","year":"2018","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"4","key":"33","doi-asserted-by":"publisher","first-page":"e2242, 25","DOI":"10.1002\/nla.2242","article-title":"Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models","volume":"26","author":"Hong, Qingguo","year":"2019","journal-title":"Numer. Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1070-5325","issn-type":"print"},{"issue":"13","key":"34","doi-asserted-by":"publisher","first-page":"2523","DOI":"10.1142\/S0218202520500499","article-title":"Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot\u2019s consolidation and multiple-network poroelasticity models","volume":"30","author":"Hong, Qingguo","year":"2020","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"2","key":"35","doi-asserted-by":"publisher","first-page":"916","DOI":"10.1137\/19M1253988","article-title":"Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems","volume":"18","author":"Hong, Qingguo","year":"2020","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"1","key":"36","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s00211-015-0712-y","article-title":"A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations","volume":"132","author":"Hong, Qingguo","year":"2016","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"335","key":"37","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1090\/mcom\/3707","article-title":"An extended Galerkin analysis in finite element exterior calculus","volume":"91","author":"Hong, Qingguo","year":"2022","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"38","doi-asserted-by":"publisher","first-page":"A875--A898","DOI":"10.1137\/140974092","article-title":"Preconditioning heterogeneous \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) problems by additive Schur complement approximation and applications","volume":"38","author":"Kraus, Johannes","year":"2016","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"39","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1051\/m2an\/2019063","article-title":"Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity","volume":"54","author":"Kumar, Sarvesh","year":"2020","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"key":"40","series-title":"SpringerBriefs in Quantitative Finance","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-74101-7","volume-title":"Saddlepoint approximation methods in financial engineering","author":"Kwok, Yue Kuen","year":"2018","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319741000"},{"key":"41","first-page":"81","article-title":"Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation","volume":"59","author":"Lady\u017eenskaja, O. A.","year":"1976","journal-title":"Zap. Nau\\v{c}n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)"},{"key":"42","series-title":"Mathematics and its Applications, Vol. 2","volume-title":"The mathematical theory of viscous incompressible flow","author":"Ladyzhenskaya, O. A.","year":"1969"},{"issue":"1","key":"43","doi-asserted-by":"publisher","first-page":"A1--A24","DOI":"10.1137\/15M1029473","article-title":"Parameter-robust discretization and preconditioning of Biot\u2019s consolidation model","volume":"39","author":"Lee, Jeonghun J.","year":"2017","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"44","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1007\/s10915-016-0210-0","article-title":"Robust error analysis of coupled mixed methods for Biot\u2019s consolidation model","volume":"69","author":"Lee, Jeonghun J.","year":"2016","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"2","key":"45","doi-asserted-by":"publisher","first-page":"A722--A747","DOI":"10.1137\/18M1182395","article-title":"A mixed finite element method for nearly incompressible multiple-network poroelasticity","volume":"41","author":"Lee, J. J.","year":"2019","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"6","key":"46","doi-asserted-by":"publisher","first-page":"2029","DOI":"10.1137\/S1064827502418203","article-title":"Analysis of preconditioners for saddle-point problems","volume":"25","author":"Loghin, D.","year":"2004","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/nla.716","article-title":"Preconditioning discretizations of systems of partial differential equations","volume":"18","author":"Mardal, Kent-Andre","year":"2011","journal-title":"Numer. Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1070-5325","issn-type":"print"},{"key":"48","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite element methods for Maxwell's equations","author":"Monk, Peter","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0198508883"},{"issue":"3","key":"49","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0045-7825(92)90193-N","article-title":"Improved accuracy in finite element analysis of Biot\u2019s consolidation problem","volume":"95","author":"Murad, M\u00e1rcio A.","year":"1992","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"50","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1002\/nme.1620370407","article-title":"On stability and convergence of finite element approximations of Biot\u2019s consolidation problem","volume":"37","author":"Murad, M\u00e1rcio A.","year":"1994","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"51","unstructured":"J. Ne\u010das, Les m\u00e9thodes directes en th\u00e9orie des \u00e9quations elliptiques, Masson et Cie, \u00c9diteurs, Paris; Academia, \u00c9diteurs, Prague, 1967."},{"issue":"2","key":"52","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1137\/0719021","article-title":"Existence, uniqueness and approximation for generalized saddle point problems","volume":"19","author":"Nicolaides, R. A.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"53","doi-asserted-by":"publisher","first-page":"2951","DOI":"10.1137\/15M1050082","article-title":"Locking-free finite element methods for poroelasticity","volume":"54","author":"Oyarz\u00faa, Ricardo","year":"2016","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"54","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s10596-007-9045-y","article-title":"A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case","volume":"11","author":"Phillips, Phillip Joseph","year":"2007","journal-title":"Comput. Geosci.","ISSN":"https:\/\/id.crossref.org\/issn\/1420-0597","issn-type":"print"},{"issue":"2","key":"55","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s10596-007-9044-z","article-title":"A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case","volume":"11","author":"Phillips, Phillip Joseph","year":"2007","journal-title":"Comput. Geosci.","ISSN":"https:\/\/id.crossref.org\/issn\/1420-0597","issn-type":"print"},{"key":"56","isbn-type":"print","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-540-78841-6_1","article-title":"Introduction to model order reduction","author":"Schilders, Wil","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540788409"},{"key":"57","series-title":"Studies in Mathematics and its Applications","isbn-type":"print","volume-title":"Navier-Stokes equations","volume":"2","author":"Temam, Roger","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0444875581","edition":"3"},{"key":"58","unstructured":"K. Terzaghi, Erdbaumechanik auf bodenphysikalischer Grundlage, F. Deuticke, 1925."},{"key":"59","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":"60","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1017\/S0022112010004428","article-title":"Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus","volume":"667","author":"Tully, B.","year":"2011","journal-title":"J. Fluid Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1120","issn-type":"print"},{"issue":"1","key":"61","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/(SICI)1099-1506(199601\/02)3:1<1::AID-NLA67>3.3.CO;2-5","article-title":"Preconditioning mixed finite element saddle-point elliptic problems","volume":"3","author":"Vassilevski, Panayot S.","year":"1996","journal-title":"Numer. Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1070-5325","issn-type":"print"},{"issue":"4","key":"62","doi-asserted-by":"publisher","first-page":"1189","DOI":"10.1002\/num.21865","article-title":"Convergence analysis of a new mixed finite element method for Biot\u2019s consolidation model","volume":"30","author":"Yi, Son-Young","year":"2014","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"2","key":"63","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1137\/100814767","article-title":"Nonstandard norms and robust estimates for saddle point problems","volume":"32","author":"Zulehner, Walter","year":"2011","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-340\/S0025-5718-2022-03795-9\/mcom3795_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-340\/S0025-5718-2022-03795-9\/S0025-5718-2022-03795-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:49:08Z","timestamp":1776833348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-340\/S0025-5718-2022-03795-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,17]]},"references-count":63,"journal-issue":{"issue":"340","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["S0025-5718-2022-03795-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3795","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":[[2022,11,17]]}}}