{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T08:17:17Z","timestamp":1769069837427,"version":"3.49.0"},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["Project-ID 255510958 \u2013 SPP 1748"],"award-info":[{"award-number":["Project-ID 255510958 \u2013 SPP 1748"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11625101"],"award-info":[{"award-number":["11625101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,7,1]]},"abstract":"Abstract<\/jats:title>\n The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces \n \n \n \n C<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {C^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> at interior vertices, while solely global \n \n \n \n C<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {C^{1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\ncontinuity is required for the conformity in \n \n \n \n H<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {H^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Since the Argyris finite element functions are not<\/jats:italic>\n \n \n \n \n C<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {C^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> at the midpoints of edges in general, the bisection\nof an edge for mesh-refinement leads to non-nestedness: the standard Argyris finite element space \n \n \n \n \n A<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udcaf<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A^{\\prime}(\\mathcal{T})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> associated to a triangulation \n \n \n \ud835\udcaf<\/m:mi>\n <\/m:math>\n \n {\\mathcal{T}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with a refinement\n\n \n \n \n \ud835\udcaf<\/m:mi>\n ^<\/m:mo>\n <\/m:mover>\n <\/m:math>\n \n {\\widehat{\\mathcal{T}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is not<\/jats:italic> a subspace of the standard Argyris finite element space \n \n \n \n \n A<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \ud835\udcaf<\/m:mi>\n ^<\/m:mo>\n <\/m:mover>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A^{\\prime}(\\widehat{\\mathcal{T}})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> associated to the refined triangulation \n \n \n \n \ud835\udcaf<\/m:mi>\n ^<\/m:mo>\n <\/m:mover>\n <\/m:math>\n \n {\\widehat{\\mathcal{T}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThis paper suggests an extension \n \n \n \n A<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udcaf<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A(\\mathcal{T})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of \n \n \n \n \n A<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udcaf<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A^{\\prime}(\\mathcal{T})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> that allows for nestedness \n \n \n \n \n A<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udcaf<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2282<\/m:mo>\n \n A<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \ud835\udcaf<\/m:mi>\n ^<\/m:mo>\n <\/m:mover>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A(\\mathcal{T})\\subset A(\\widehat{\\mathcal{T}})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> under mesh-refinement.\nThe extended Argyris finite element space \n \n \n \n A<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udcaf<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {A(\\mathcal{T})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is called hierarchical, but is still based on the concept of the Argyris finite element as a triple\n\n \n \n \n (<\/m:mo>\n T<\/m:mi>\n ,<\/m:mo>\n \n \n P<\/m:mi>\n 5<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n T<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n \n (<\/m:mo>\n \n \u039b<\/m:mi>\n 1<\/m:mn>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mi>\n ,<\/m:mo>\n \n \u039b<\/m:mi>\n 21<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {(T,P_{5}(T),(\\Lambda_{1},\\dots,\\Lambda_{21}))}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in the sense of Ciarlet. The other main results of this paper\nare the optimal convergence rates of an adaptive mesh-refinement algorithm via the abstract framework of the axioms of adaptivity\nand uniform convergence of a local multigrid V-cycle algorithm for the effective solution of the discrete system.<\/jats:p>","DOI":"10.1515\/cmam-2021-0083","type":"journal-article","created":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T02:00:26Z","timestamp":1622512826000},"page":"529-556","source":"Crossref","is-referenced-by-count":10,"title":["Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms"],"prefix":"10.1515","volume":"21","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[{"name":"Humboldt-Universit\u00e4t zu Berlin , 10099 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Hu","sequence":"additional","affiliation":[{"name":"LMAM and School of Mathematical Sciences , Peking University , Beijing 100871 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,6,1]]},"reference":[{"key":"2023033111373288687_j_cmam-2021-0083_ref_001","doi-asserted-by":"crossref","unstructured":"C. Bacuta, J. H. Bramble and J. E. Pasciak,\nShift theorems for the biharmonic Dirichlet problem,\nRecent Progress in Computational and Applied PDEs,\nKluwer\/Plenum, New York (2002), 1\u201326.","DOI":"10.1007\/978-1-4615-0113-8_1"},{"key":"2023033111373288687_j_cmam-2021-0083_ref_002","doi-asserted-by":"crossref","unstructured":"P. Binev, W. Dahmen and R. DeVore,\nAdaptive finite element methods with convergence rates,\nNumer. Math. 97 (2004), no. 2, 219\u2013268.","DOI":"10.1007\/s00211-003-0492-7"},{"key":"2023033111373288687_j_cmam-2021-0083_ref_003","doi-asserted-by":"crossref","unstructured":"A. Bonito and R. H. Nochetto,\nQuasi-optimal convergence rate of an adaptive discontinuous Galerkin method,\nSIAM J. Numer. 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