论二维 P4+ 三角形和三维 P6+ 四面体无发散有限元的收敛性

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Numerical Methods for Partial Differential Equations Pub Date : 2024-01-21 DOI:10.1002/num.23088

Shangyou Zhang

引用次数: 0

摘要

我们的研究表明，用 Pk$$ {P}_k $$-Pk-1disc$$ {P}_{k-1}^{mathrm{disc}} 混合有限元法求解稳态斯托克斯方程时，离散速度解以最优阶收敛。$$ 混合有限元法在二维三角形网格上计算 k≥4$ k\ge 4 $$ 或在四面体网格上计算 k≥6$ k\ge 6 $$，即使在 inf-sup 条件失效的情况下也是如此。通过将离散的 Pk-1$$ {P}_{k-1} $ $ 压力简单地投影到连续的 Pk-1$$ {P}_{k-1} $ 多项式空间的 L2$$ {L}^2 $$投影，我们证明了这种后处理压力解也能以最优阶收敛。二维和三维数值测试都验证了这一理论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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

P_{k} $$ {P}_k $$

P_{k - 1}^{disc} $$ {P}_{k-1}^{\mathrm{disc}} $$

mixed finite element method for

k \geq 4 $$ k\ge 4 $$

on 2D triangular grids or

k \geq 6 $$ k\ge 6 $$

on tetrahedral grids, even in the case the inf-sup condition fails. By a simple

L^{2} $$ {L}^2 $$

-projection of the discrete

P_{k - 1} $$ {P}_{k-1} $$

pressure to the space of continuous

P_{k - 1} $$ {P}_{k-1} $$

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.

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来源期刊

Numerical Methods for Partial Differential Equations 数学-应用数学

CiteScore

7.20

自引率

2.60%

发文量

审稿时长

9 months

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.