{"title":"一个𝐶1-𝑃7三角形上的贝尔有限元","authors":"Xuejun Xu, Shangyou Zhang","doi":"10.1515/cmam-2023-0068","DOIUrl":null,"url":null,"abstract":"We construct a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0001.png\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0002.png\" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Bell finite element by restricting its normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0003.png\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0004.png\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, and its second normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0004.png\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0006.png\" /> <jats:tex-math>P_{4}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0003.png\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial space. We show the method converges at order 7 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0008.png\" /> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm. By eliminating all degrees of freedom on edges of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0001.png\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0002.png\" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Argyris finite element, the global degrees of freedom of the new element are reduced substantially from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>27</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0011.png\" /> <jats:tex-math>27V</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>12</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0012.png\" /> <jats:tex-math>12V</jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0001.png\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0003.png\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Argyris finite element is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>19</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0068_ineq_0015.png\" /> <jats:tex-math>19V</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"128 2","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A 𝐶1-𝑃7 Bell Finite Element on Triangle\",\"authors\":\"Xuejun Xu, Shangyou Zhang\",\"doi\":\"10.1515/cmam-2023-0068\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We construct a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0001.png\\\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0002.png\\\" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Bell finite element by restricting its normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0003.png\\\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0004.png\\\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, and its second normal derivative from a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0004.png\\\" /> <jats:tex-math>P_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0006.png\\\" /> <jats:tex-math>P_{4}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0003.png\\\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> polynomial space. We show the method converges at order 7 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0008.png\\\" /> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm. By eliminating all degrees of freedom on edges of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0001.png\\\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0002.png\\\" /> <jats:tex-math>P_{7}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Argyris finite element, the global degrees of freedom of the new element are reduced substantially from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mn>27</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0011.png\\\" /> <jats:tex-math>27V</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mn>12</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0012.png\\\" /> <jats:tex-math>12V</jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0001.png\\\" /> <jats:tex-math>C^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>P</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0003.png\\\" /> <jats:tex-math>P_{6}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Argyris finite element is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mn>19</m:mn> <m:mo></m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2023-0068_ineq_0015.png\\\" /> <jats:tex-math>19V</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.\",\"PeriodicalId\":48751,\"journal\":{\"name\":\"Computational Methods in Applied Mathematics\",\"volume\":\"128 2\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-11-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/cmam-2023-0068\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/cmam-2023-0068","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
We construct a C1C^{1}-P7P_{7} Bell finite element by restricting its normal derivative from a P6P_{6} polynomial to a P5P_{5} polynomial, and its second normal derivative from a P5P_{5} polynomial to a P4P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P6P_{6} polynomial space. We show the method converges at order 7 in L2L^{2}-norm. By eliminating all degrees of freedom on edges of C1C^{1}-P7P_{7} Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27V27V to 12V12V asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C1C^{1}-P6P_{6} Argyris finite element is 19V19V, the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.
期刊介绍:
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.