On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-01-21 DOI:10.1002/num.23088

Shangyou Zhang

引用次数: 0

Abstract

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.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

论二维 P4+ 三角形和三维 P6+ 四面体无发散有限元的收敛性

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。

求助全文

约1分钟内获得全文去求助

来源期刊