{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:02:45Z","timestamp":1776844965386,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2003,3,22]],"date-time":"2003-03-22T00:00:00Z","timestamp":1048291200000},"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 consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript p\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W Subscript p Superscript 1\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">W^1_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is that the given space of functions on the reference element contain all polynomial functions of total degree at most\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01439-4","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"909-922","source":"Crossref","is-referenced-by-count":186,"title":["Approximation by quadrilateral finite elements"],"prefix":"10.1090","volume":"71","author":[{"given":"Douglas","family":"Arnold","sequence":"first","affiliation":[]},{"given":"Daniele","family":"Boffi","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Falk","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,3,22]]},"reference":[{"key":"1","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"key":"2","first-page":"281","article-title":"On cosine operator functions in Banach spaces","volume":"36","author":"Nagy, B.","year":"1974","journal-title":"Acta Sci. Math. (Szeged)","ISSN":"https:\/\/id.crossref.org\/issn\/0001-6969","issn-type":"print"},{"key":"3","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 153","volume-title":"Geometric measure theory","author":"Federer, Herbert","year":"1969"},{"key":"4","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite element methods for Navier-Stokes equations","volume":"5","author":"Girault, Vivette","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540157964"},{"key":"5","doi-asserted-by":"crossref","unstructured":"F. Kikuchi, M. Okabe, and H. Fujio, Modification of the 8-node serendipity element, Comp. Methods Appl. Mech. Engrg. 179 (1999), 91\u2013109.","DOI":"10.1016\/S0045-7825(99)00031-6"},{"key":"6","doi-asserted-by":"crossref","unstructured":"R. H. McNeal and R. L. Harder, Eight nodes or nine?, Int. J. Numer. Methods Engrg. 33 (1992), 1049\u20131058.","DOI":"10.1002\/nme.1620330510"},{"issue":"2","key":"7","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":"8","doi-asserted-by":"crossref","unstructured":"P. Sharpov and Y. Iordanov, Numerical solution of Stokes equations with pressure and filtration boundary conditions, J. Comp. Phys. 112 (1994), 12\u201323.","DOI":"10.1006\/jcph.1994.1078"},{"key":"9","doi-asserted-by":"crossref","unstructured":"G. Strang and G. Fix, A Fourier analysis of the finite element variational method, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E. II Ciclo, 1971, pp. 793\u2013840.","DOI":"10.1007\/978-3-642-10984-3_7"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/s002110000104","article-title":"Interpolation error estimates of a modified 8-node serendipity finite element","volume":"85","author":"Zhang, Jing","year":"2000","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"11","unstructured":"O. C. Zienkiewicz and R. L. Taylor, The finite element method, fourth edition, volume 1: Basic formulation and linear problems, McGraw-Hill, London, 1989."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01439-4\/S0025-5718-02-01439-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01439-4\/S0025-5718-02-01439-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:07Z","timestamp":1776726007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01439-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3,22]]},"references-count":11,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-02-01439-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01439-4","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":[[2002,3,22]]}}}