Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations

Lanyin Sun; Siya Wen; Lanyin Sun; Siya Wen

doi:10.3934/math.20241706

AIMS Mathematics

2024, Volume 9, Issue 12: 35978-36000. doi: 10.3934/math.20241706

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Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations

Lanyin Sun ^,,
Siya Wen

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

Received: 22 September 2024 Revised: 19 November 2024 Accepted: 13 December 2024 Published: 25 December 2024
MSC : 65D17, 65M60

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.
- mixed finite element method,
- Bernstein polynomial basis,
- 2D Stokes equations
Citation: Lanyin Sun, Siya Wen. Applications of mixed finite element method based on Bernstein polynomials in numerical solution of Stokes equations[J]. AIMS Mathematics, 2024, 9(12): 35978-36000. doi: 10.3934/math.20241706

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Abstract

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.

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