Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations
The Stokes equation is fundamental in fluid mechanics. We used bivariate Bernstein polynomial bases to construct the function space for mixed finite element methods to solve the 2D Stokes equation. Our results show that the numerical accuracy and convergence order using bicubic and lower-order Lagra...
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241706 |
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| author | Lanyin Sun Siya Wen |
| author_facet | Lanyin Sun Siya Wen |
| author_sort | Lanyin Sun |
| collection | DOAJ |
| description | The Stokes equation is fundamental in fluid mechanics. We used bivariate Bernstein polynomial bases to construct the function space for mixed finite element methods to solve the 2D Stokes equation. Our results show that the numerical accuracy and convergence order using bicubic and lower-order Lagrange interpolation polynomials are comparable to those achieved with Bernstein polynomial bases. However, high-order Lagrange interpolation functions often suffer from the Runge's phenomenon, which limits their effectiveness. By employing high-order Bernstein polynomial bases, we have significantly improved the numerical solutions, effectively mitigating the Runge phenomenon. This approach highlights the advantages of Bernstein polynomial bases in achieving stable and accurate solutions for the 2D Stokes equation. |
| format | Article |
| id | doaj-art-90f5903f40af4928bb760c55cb6020e9 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-90f5903f40af4928bb760c55cb6020e92025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912359783600010.3934/math.20241706Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equationsLanyin Sun0Siya Wen1School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, ChinaSchool of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, ChinaThe Stokes equation is fundamental in fluid mechanics. We used bivariate Bernstein polynomial bases to construct the function space for mixed finite element methods to solve the 2D Stokes equation. Our results show that the numerical accuracy and convergence order using bicubic and lower-order Lagrange interpolation polynomials are comparable to those achieved with Bernstein polynomial bases. However, high-order Lagrange interpolation functions often suffer from the Runge's phenomenon, which limits their effectiveness. By employing high-order Bernstein polynomial bases, we have significantly improved the numerical solutions, effectively mitigating the Runge phenomenon. This approach highlights the advantages of Bernstein polynomial bases in achieving stable and accurate solutions for the 2D Stokes equation.https://www.aimspress.com/article/doi/10.3934/math.20241706mixed finite element methodbernstein polynomial basis2d stokes equations |
| spellingShingle | Lanyin Sun Siya Wen Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations AIMS Mathematics mixed finite element method bernstein polynomial basis 2d stokes equations |
| title | Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations |
| title_full | Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations |
| title_fullStr | Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations |
| title_full_unstemmed | Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations |
| title_short | Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations |
| title_sort | applications of mixed finite element method based on bernstein polynomials in numerical solution of stokes equations |
| topic | mixed finite element method bernstein polynomial basis 2d stokes equations |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241706 |
| work_keys_str_mv | AT lanyinsun applicationsofmixedfiniteelementmethodbasedonbernsteinpolynomialsinnumericalsolutionofstokesequations AT siyawen applicationsofmixedfiniteelementmethodbasedonbernsteinpolynomialsinnumericalsolutionofstokesequations |