{"title":"论二维 P4+ 三角形和三维 P6+ 四面体无发散有限元的收敛性","authors":"Shangyou Zhang","doi":"10.1002/num.23088","DOIUrl":null,"url":null,"abstract":"We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0003\" display=\"inline\" location=\"graphic/num23088-math-0003.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>P</mi>\n</mrow>\n<mrow>\n<mi>k</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {P}_k $$</annotation>\n</semantics></math>-<math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0004\" display=\"inline\" location=\"graphic/num23088-math-0004.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msubsup>\n<mrow>\n<mi>P</mi>\n</mrow>\n<mrow>\n<mi>k</mi>\n<mo form=\"prefix\">−</mo>\n<mn>1</mn>\n</mrow>\n<mrow>\n<mtext>disc</mtext>\n</mrow>\n</msubsup>\n</mrow>\n$$ {P}_{k-1}^{\\mathrm{disc}} $$</annotation>\n</semantics></math> mixed finite element method for <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0005\" display=\"inline\" location=\"graphic/num23088-math-0005.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>k</mi>\n<mo>≥</mo>\n<mn>4</mn>\n</mrow>\n$$ k\\ge 4 $$</annotation>\n</semantics></math> on 2D triangular grids or <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0006\" display=\"inline\" location=\"graphic/num23088-math-0006.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>k</mi>\n<mo>≥</mo>\n<mn>6</mn>\n</mrow>\n$$ k\\ge 6 $$</annotation>\n</semantics></math> on tetrahedral grids, even in the case the inf-sup condition fails. By a simple <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0007\" display=\"inline\" location=\"graphic/num23088-math-0007.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msup>\n<mrow>\n<mi>L</mi>\n</mrow>\n<mrow>\n<mn>2</mn>\n</mrow>\n</msup>\n</mrow>\n$$ {L}^2 $$</annotation>\n</semantics></math>-projection of the discrete <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0008\" display=\"inline\" location=\"graphic/num23088-math-0008.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>P</mi>\n</mrow>\n<mrow>\n<mi>k</mi>\n<mo form=\"prefix\">−</mo>\n<mn>1</mn>\n</mrow>\n</msub>\n</mrow>\n$$ {P}_{k-1} $$</annotation>\n</semantics></math> pressure to the space of continuous <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0009\" display=\"inline\" location=\"graphic/num23088-math-0009.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>P</mi>\n</mrow>\n<mrow>\n<mi>k</mi>\n<mo form=\"prefix\">−</mo>\n<mn>1</mn>\n</mrow>\n</msub>\n</mrow>\n$$ {P}_{k-1} $$</annotation>\n</semantics></math> polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"65 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements\",\"authors\":\"Shangyou Zhang\",\"doi\":\"10.1002/num.23088\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0003.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<msub>\\n<mrow>\\n<mi>P</mi>\\n</mrow>\\n<mrow>\\n<mi>k</mi>\\n</mrow>\\n</msub>\\n</mrow>\\n$$ {P}_k $$</annotation>\\n</semantics></math>-<math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0004.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<msubsup>\\n<mrow>\\n<mi>P</mi>\\n</mrow>\\n<mrow>\\n<mi>k</mi>\\n<mo form=\\\"prefix\\\">−</mo>\\n<mn>1</mn>\\n</mrow>\\n<mrow>\\n<mtext>disc</mtext>\\n</mrow>\\n</msubsup>\\n</mrow>\\n$$ {P}_{k-1}^{\\\\mathrm{disc}} $$</annotation>\\n</semantics></math> mixed finite element method for <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0005\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0005.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<mi>k</mi>\\n<mo>≥</mo>\\n<mn>4</mn>\\n</mrow>\\n$$ k\\\\ge 4 $$</annotation>\\n</semantics></math> on 2D triangular grids or <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0006\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0006.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<mi>k</mi>\\n<mo>≥</mo>\\n<mn>6</mn>\\n</mrow>\\n$$ k\\\\ge 6 $$</annotation>\\n</semantics></math> on tetrahedral grids, even in the case the inf-sup condition fails. By a simple <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0007\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0007.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<msup>\\n<mrow>\\n<mi>L</mi>\\n</mrow>\\n<mrow>\\n<mn>2</mn>\\n</mrow>\\n</msup>\\n</mrow>\\n$$ {L}^2 $$</annotation>\\n</semantics></math>-projection of the discrete <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0008\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0008.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<msub>\\n<mrow>\\n<mi>P</mi>\\n</mrow>\\n<mrow>\\n<mi>k</mi>\\n<mo form=\\\"prefix\\\">−</mo>\\n<mn>1</mn>\\n</mrow>\\n</msub>\\n</mrow>\\n$$ {P}_{k-1} $$</annotation>\\n</semantics></math> pressure to the space of continuous <math altimg=\\\"urn:x-wiley:num:media:num23088:num23088-math-0009\\\" display=\\\"inline\\\" location=\\\"graphic/num23088-math-0009.png\\\" overflow=\\\"scroll\\\">\\n<semantics>\\n<mrow>\\n<msub>\\n<mrow>\\n<mi>P</mi>\\n</mrow>\\n<mrow>\\n<mi>k</mi>\\n<mo form=\\\"prefix\\\">−</mo>\\n<mn>1</mn>\\n</mrow>\\n</msub>\\n</mrow>\\n$$ {P}_{k-1} $$</annotation>\\n</semantics></math> polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.\",\"PeriodicalId\":19443,\"journal\":{\"name\":\"Numerical Methods for Partial Differential Equations\",\"volume\":\"65 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-01-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerical Methods for Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/num.23088\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Methods for Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/num.23088","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements
We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the - mixed finite element method for on 2D triangular grids or on tetrahedral grids, even in the case the inf-sup condition fails. By a simple -projection of the discrete pressure to the space of continuous polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.
期刊介绍:
An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.