{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:47:20Z","timestamp":1776836840426,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming\n                    <italic>rotated<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">Q_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper W\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">W<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {W}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle with any number of smoothing steps and close-to-optimal condition number estimates for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper V\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">V<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {V}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper W\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">W<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {W}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycles with a sufficiently large number of smoothing steps and variable\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper V\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">V<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {V}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming\n                    <italic>rotated<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">Q_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00920-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:53Z","timestamp":1027707233000},"page":"667-693","source":"Crossref","is-referenced-by-count":19,"title":["Multigrid and multilevel methods for nonconforming \ud835\udc44\u2081 elements"],"prefix":"10.1090","volume":"67","author":[{"given":"Zhangxin","family":"Chen","sequence":"first","affiliation":[]},{"given":"Peter","family":"Oswald","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"211","key":"1","doi-asserted-by":"publisher","first-page":"943","DOI":"10.2307\/2153478","article-title":"On the implementation of mixed methods as nonconforming methods for second-order elliptic problems","volume":"64","author":"Arbogast, Todd","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"153","key":"2","doi-asserted-by":"publisher","first-page":"35","DOI":"10.2307\/2007724","article-title":"An optimal order process for solving finite element equations","volume":"36","author":"Bank, Randolph E.","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1137\/0727056","article-title":"Multigrid methods for nonconforming finite element methods","volume":"27","author":"Braess, D.","year":"1990","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"4","series-title":"Pitman Research Notes in Mathematics Series","isbn-type":"print","volume-title":"Multigrid methods","volume":"294","author":"Bramble, James H.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0582234352"},{"issue":"191","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008789","article-title":"Parallel multilevel preconditioners","volume":"55","author":"Bramble, James H.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"193","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008527","article-title":"The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms","volume":"56","author":"Bramble, James H.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"185","key":"7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008649","article-title":"An optimal-order multigrid method for \ud835\udc431 nonconforming finite elements","volume":"52","author":"Brenner, Susanne C.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","isbn-type":"print","first-page":"54","article-title":"Multigrid methods for nonconforming finite elements","author":"Brenner, Susanne C.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712483"},{"key":"9","unstructured":"S. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity, Preprint, 1995, submitted."},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1051\/m2an\/1993270100091","article-title":"Analysis of mixed methods using conforming and nonconforming finite element methods","volume":"27","author":"Chen, Zhangxin","year":"1993","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"8","key":"11","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/0898-1221(93)90173-S","article-title":"Projection finite element methods for semiconductor device equations","volume":"25","author":"Chen, Zhangxin","year":"1993","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"12","unstructured":"Zhangxin Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems, East-West J. Numer. Math. 4 (1996), 1\u201333."},{"key":"13","doi-asserted-by":"crossref","unstructured":"Zhangxin Chen, R. E. Ewing, Y. Kuznetsov, R. Lazarov, and S. Maliassov, Multilevel preconditioners for mixed methods for second order elliptic problems, J. Numer. Lin. Alg. Appl. 30 (1996), 427\u2013453.","DOI":"10.1002\/(SICI)1099-1506(199609\/10)3:5<427::AID-NLA92>3.0.CO;2-I"},{"issue":"214","key":"14","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1090\/S0025-5718-96-00703-X","article-title":"Domain decomposition algorithms for mixed methods for second-order elliptic problems","volume":"65","author":"Chen, Zhangxin","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","unstructured":"Zhangxin Chen, D. Y. Kwak, and Y. J. Yon, Multigrid algorithms for nonconforming and mixed methods for nonsymmetric and indefinite problems, IMA Preprint Series #1277, 1994, SIAM J. Scientific Computing, 1998 to appear."},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s002110050115","article-title":"On the abstract theory of additive and multiplicative Schwarz algorithms","volume":"70","author":"Griebel, M.","year":"1995","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"215","key":"17","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1090\/S0025-5718-96-00735-1","article-title":"Analysis of a class of nonconforming finite elements for crystalline microstructures","volume":"65","author":"Klou\u010dek, Petr","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/BF01135254","article-title":"The computation of the dynamics of the martensitic transformation","volume":"6","author":"Klou\u010dek, P.","year":"1994","journal-title":"Contin. Mech. Thermodyn.","ISSN":"https:\/\/id.crossref.org\/issn\/0935-1175","issn-type":"print"},{"key":"19","unstructured":"C. Lee, A nonconforming multigrid method using conforming subspaces, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, N. Melson et al., eds., NASA Conference Publication, vol. 3224, 1993, pp.317\u2013330."},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/BF01396226","article-title":"On a hierarchical basis multilevel method with nonconforming \ud835\udc431 elements","volume":"62","author":"Oswald, P.","year":"1992","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"21","series-title":"Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-91215-2","volume-title":"Multilevel finite element approximation","author":"Oswald, Peter","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3519027194"},{"issue":"215","key":"22","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1090\/S0025-5718-96-00717-X","article-title":"Preconditioners for nonconforming discretizations","volume":"65","author":"Oswald, Peter","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"23","doi-asserted-by":"crossref","unstructured":"P. Oswald, Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations, Appl. Numer. Math. 23 (1997), 139\u2013158.","DOI":"10.1016\/S0168-9274(96)00065-7"},{"issue":"2","key":"24","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1002\/num.1690080202","article-title":"Simple nonconforming quadrilateral Stokes element","volume":"8","author":"Rannacher, R.","year":"1992","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"25","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"issue":"3","key":"26","first-page":"229","article-title":"Multigrid techniques for a divergence-free finite element discretization","volume":"2","author":"Turek, S.","year":"1994","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"3","key":"27","first-page":"238","article-title":"The \ud835\udc4a-cycle multigrid method for finite elements with nonnested spaces","volume":"23","author":"Wang, Ming","year":"1994","journal-title":"Adv. in Math. (China)","ISSN":"https:\/\/id.crossref.org\/issn\/1000-0917","issn-type":"print"},{"key":"28","isbn-type":"print","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1017\/S0962492900002385","article-title":"Old and new convergence proofs for multigrid methods","author":"Yserentant, Harry","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0521443563"},{"key":"29","isbn-type":"print","first-page":"174","article-title":"Convergence estimates for some multigrid algorithms","author":"Xu, Jinchao","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/089871253X"},{"issue":"4","key":"30","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1137\/1034116","article-title":"Iterative methods by space decomposition and subspace correction","volume":"34","author":"Xu, Jinchao","year":"1992","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00920-X\/S0025-5718-98-00920-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00920-X\/S0025-5718-98-00920-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:44:45Z","timestamp":1776721485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00920-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":30,"journal-issue":{"issue":"222","published-print":{"date-parts":[[1998,4]]}},"alternative-id":["S0025-5718-98-00920-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00920-x","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":[[1998]]}}}