一个𝐶1-𝑃7三角形上的贝尔有限元

IF 1 4区 数学 Q3 MATHEMATICS, APPLIED Computational Methods in Applied Mathematics Pub Date : 2023-11-22 DOI:10.1515/cmam-2023-0068
Xuejun Xu, Shangyou Zhang
{"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":null,"pages":null},"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\":null,\"pages\":null},\"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}
引用次数: 0

摘要

我们在每个三角形的三条边上,通过限制c1c ^{1} - p7p_ {7} Bell有限元的法向导数从p6p_{6}多项式到p5p_{5}多项式,以及它的二阶法向导数从p5p_{5}多项式到p4p_{4}多项式,构造了c1c ^{1} - p7p_ {7} Bell有限元。在一个三角形上,有限元空间包含p6p_{6}多项式空间。我们证明了该方法在l2 ^{2}范数下收敛于7阶。通过消除c1c ^{1} - p7p_ {7} Argyris有限元边缘上的所有自由度,新单元的整体自由度从27 ^ V 27V渐近地大幅降低到12 ^ V 12V,其中三角形网格中的顶点数。当c1 C^{1} - p6p_ {6} Argyris有限元的整体自由度为19 ^ V 19V时,新单元同样精确,但更经济。数值试验表明,新单元比现有单元精度更高,且具有更少的全局未知量。
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A 𝐶1-𝑃7 Bell Finite Element on Triangle
We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in L 2 L^{2} -norm. By eliminating all degrees of freedom on edges of C 1 C^{1} - P 7 P_{7} Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27 V 27V to 12 V 12V asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C 1 C^{1} - P 6 P_{6} Argyris finite element is 19 V 19V , 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.
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2.40
自引率
7.70%
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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.
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Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations Computational Methods in Applied Mathematics (CMAM 2022 Conference, Part 2)
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