{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:39:43Z","timestamp":1776839983875,"version":"3.51.2"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"354","license":[{"start":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T00:00:00Z","timestamp":1756944000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002848","name":"Comisi\u00c3\u00b3n Nacional de Investigaci\u00c3\u00b3n Cient\u00c3\u00adfica y Tecnol\u00c3\u00b3gica","doi-asserted-by":"publisher","award":["1230013"],"award-info":[{"award-number":["1230013"]}],"id":[{"id":"10.13039\/501100002848","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The classical continuous mixed formulation of linear elasticity with pointwise symmetric stresses allows for a conforming finite element discretization with piecewise polynomials of degree at least three. Symmetric stress approximations of lower polynomial order are only possible when their\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d i v\">\n                        <mml:semantics>\n                          <mml:mi>div<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {div}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conformity is weakened to the continuity of normal-normal components. In two dimensions, this condition is meant pointwise along edges for piecewise polynomials, but a corresponding characterization for general piecewise\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis d i v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>div<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\operatorname {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    tensors has been elusive.\n                  <\/p>\n                  <p>\n                    We introduce such a space and establish a continuous mixed formulation of linear planar elasticity with pointwise symmetric stresses that have, in a distributional sense, continuous normal-normal components across the edges of a shape-regular triangulation. The displacement is split into an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    field and a tangential trace on the skeleton of the mesh. The well-posedness of the new mixed formulation follows with a duality lemma relating the normal-normal continuous stresses with the tangential traces of displacements.\n                  <\/p>\n                  <p>\n                    For this new formulation we present a lowest-order conforming discretization. Stresses are approximated by piecewise quadratic symmetric tensors, whereas displacements are discretized by piecewise linear polynomials. The tangential displacement trace acts as a Lagrange multiplier and guarantees global\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d i v\">\n                        <mml:semantics>\n                          <mml:mi>div<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {div}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conformity in the limit as the mesh-size tends to zero. We prove locking-free, quasi-optimal convergence of our scheme and illustrate this with numerical examples.\n                  <\/p>","DOI":"10.1090\/mcom\/4017","type":"journal-article","created":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T14:01:15Z","timestamp":1725458475000},"page":"1571-1602","source":"Crossref","is-referenced-by-count":2,"title":["Normal-normal continuous symmetric stresses in mixed finite element elasticity"],"prefix":"10.1090","volume":"94","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[]},{"given":"Norbert","family":"Heuer","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,9,4]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/s10915-004-4807-3","article-title":"A mixed finite element method for elasticity in three dimensions","volume":"25","author":"Adams, Scot","year":"2005","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"9","key":"2","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1002\/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B","article-title":"Vector potentials in three-dimensional non-smooth domains","volume":"21","author":"Amrouche, C.","year":"1998","journal-title":"Math. Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"issue":"263","key":"3","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1090\/S0025-5718-08-02071-1","article-title":"Finite elements for symmetric tensors in three dimensions","volume":"77","author":"Arnold, Douglas N.","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","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"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00211-004-0548-3","article-title":"Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis","volume":"99","author":"Bartels, S.","year":"2004","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"6","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":"31","key":"7","doi-asserted-by":"publisher","first-page":"3391","DOI":"10.1016\/S0045-7825(02)00254-2","article-title":"The discontinuous Petrov-Galerkin method for elliptic problems","volume":"191","author":"Bottasso, Carlo L.","year":"2002","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"494","DOI":"10.1016\/j.camwa.2016.05.004","article-title":"Breaking spaces and forms for the DPG method and applications including Maxwell equations","volume":"72","author":"Carstensen, C.","year":"2016","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"9","unstructured":"Carsten Carstensen and Norbert Heuer, A fractional-order trace-dev-div inequality,  arXiv:2403.01291, 2024."},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1137\/S0036142995285873","article-title":"Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem","volume":"35","author":"Cessenat, Olivier","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1002\/num.20640","article-title":"A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions","volume":"27","author":"Demkowicz, L.","year":"2011","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"169","key":"12","doi-asserted-by":"publisher","first-page":"39","DOI":"10.2307\/2007791","article-title":"Global estimates for mixed methods for second order elliptic equations","volume":"44","author":"Douglas, Jim, Jr.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"13","doi-asserted-by":"publisher","first-page":"1705","DOI":"10.1016\/j.camwa.2017.11.029","article-title":"Superconvergence in a DPG method for an ultra-weak formulation","volume":"75","author":"F\u00fchrer, Thomas","year":"2018","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"14","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1016\/j.cma.2018.08.041","article-title":"Fully discrete DPG methods for the Kirchhoff-Love plate bending model","volume":"343","author":"F\u00fchrer, Thomas","year":"2019","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"318","key":"15","doi-asserted-by":"publisher","first-page":"1587","DOI":"10.1090\/mcom\/3381","article-title":"An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation","volume":"88","author":"F\u00fchrer, Thomas","year":"2019","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"T. F\u00fchrer, N. Heuer, and A. H. Niemi, A DPG method for shallow shells, Numer. Math. 152 (2022), no. 1, 76\u201399.","DOI":"10.1007\/s00211-022-01308-w"},{"key":"17","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/BF00286498","article-title":"Singularit\u00e9s en elasticit\u00e9","volume":"107","author":"Grisvard, P.","year":"1989","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"19","unstructured":"K\u00e5re Hellan, Analysis of elastic plates in flexure by a simplified finite element method, Acta Polytech. Scand. Civ. Eng. Build. Constr. Ser. 46 (1967), 1."},{"key":"20","doi-asserted-by":"crossref","unstructured":"Leonard R. Herrmann, Finite-element bending analysis for plates, J. Eng. Mech. Div. 93 (1967), no. 5, 13\u201326.","DOI":"10.1061\/JMCEA3.0000891"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"283","DOI":"10.4208\/jcm.1412-m2014-0071","article-title":"Finite element approximations of symmetric tensors on simplicial grids in \u211d\u207f: the higher order case","volume":"33","author":"Hu, Jun","year":"2015","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"9","key":"22","doi-asserted-by":"publisher","first-page":"1649","DOI":"10.1142\/S0218202516500408","article-title":"Finite element approximations of symmetric tensors on simplicial grids in \u211d\u207f: the lower order case","volume":"26","author":"Hu, Jun","year":"2016","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"23","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF01436186","article-title":"On the convergence of a mixed finite-element method for plate bending problems","volume":"21","author":"Johnson, Claes","year":"1973","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"24","isbn-type":"print","volume-title":"Strongly elliptic systems and boundary integral equations","author":"McLean, William","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0521663326"},{"issue":"8","key":"25","doi-asserted-by":"publisher","first-page":"1761","DOI":"10.1142\/S0218202511005568","article-title":"Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity","volume":"21","author":"Pechstein, Astrid","year":"2011","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"2","key":"26","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1002\/nme.3319","article-title":"Anisotropic mixed finite elements for elasticity","volume":"90","author":"Pechstein, A.","year":"2012","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s00211-017-0933-3","article-title":"An analysis of the TDNNS method using natural norms","volume":"139","author":"Pechstein, Astrid S.","year":"2018","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1137\/17M1118427","article-title":"A decomposition result for Kirchhoff plate bending problems and a new discretization approach","volume":"56","author":"Rafetseder, Katharina","year":"2018","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"29","doi-asserted-by":"publisher","first-page":"4130","DOI":"10.1137\/08073901X","article-title":"Efficient assembly of \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) and \ud835\udc3b(\ud835\udc50\ud835\udc62\ud835\udc5f\ud835\udc59) conforming finite elements","volume":"31","author":"Rognes, Marie E.","year":"2009","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1023\/A:1007639413619","article-title":"Corner singularities and regularity of weak solutions for the two-dimensional Lam\u00e9 equations on domains with angular corners","volume":"60","author":"R\u00f6ssle, Andreas","year":"2000","journal-title":"J. Elasticity","ISSN":"https:\/\/id.crossref.org\/issn\/0374-3535","issn-type":"print"},{"issue":"36","key":"31","first-page":"21","article-title":"The regularity of boundary value problems for the Lam\u00e9 equations in a polygonal domain","author":"S\u00e4ndig, Anna-Margarete","year":"1989","journal-title":"Rostock. Math. Kolloq.","ISSN":"https:\/\/id.crossref.org\/issn\/0138-3248","issn-type":"print"},{"issue":"190","key":"32","doi-asserted-by":"publisher","first-page":"483","DOI":"10.2307\/2008497","article-title":"Finite element interpolation of nonsmooth functions satisfying boundary conditions","volume":"54","author":"Scott, L. Ridgway","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"33","series-title":"Lecture Notes of the Unione Matematica Italiana","isbn-type":"print","volume-title":"An introduction to Sobolev spaces and interpolation spaces","volume":"3","author":"Tartar, Luc","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540714828"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04017-6\/S0025-5718-2024-04017-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:42:21Z","timestamp":1776836541000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04017-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,4]]},"references-count":33,"journal-issue":{"issue":"354","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["S0025-5718-2024-04017-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4017","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":[[2024,9,4]]}}}