通过交换插值算子实现连续压力Stokes离散化的压力鲁棒性

中文摘要 AI 辅助

不可压缩Stokes方程采用连续压力逼近的常见有限元离散化——MINI单元、Taylor-Hood单元或稳定化等阶单元——不具备压力鲁棒性：速度误差受到压力最佳逼近误差乘以逆粘性系数的污染。已有的补救方法是将动量方程中的测试函数替换为保散度重构算子；对于连续压力逼近，该构造较为繁琐，需要求解局部问题（Lederer, Linke, Merdon & Schöberl, SIAM J. Numer. Anal., 2017）。我们提出一种更简单的替代方案：用适当阶的Nédélec插值算子对力进行插值，而非重构测试函数。利用连接Nédélec和Lagrange插值算子的交换图性质，我们证明所得方法对任何$H^1$-相容的连续压力离散化都是压力鲁棒的。我们给出了MINI单元的完整分析，随后对任意次数$k$的稳定化等阶$P^kP^k$单元进行分析，证明插值引起的一致性误差是压力鲁棒的，并以最优阶收敛。我们通过多个数值算例验证了该理论。

英文摘要

Common finite element discretizations for the incompressible Stokes equations with a continuous pressure approximation -- the MINI element, the Taylor--Hood element, or stabilized equal-order elements -- are not pressure-robust: the velocity error is polluted by the pressure best-approximation error, multiplied by the inverse viscosity. The established remedy replaces the test function in the momentum equation by a divergence-preserving reconstruction operator; for continuous pressure approximations this construction is cumbersome, requiring the solution of local problems (Lederer, Linke, Merdon & Schöberl, SIAM J. Numer. Anal., 2017). We propose a simpler alternative: interpolating the force by a Nédélec interpolation operator of suitable order instead of reconstructing the test function. Exploiting the commuting diagram property linking Nédélec and Lagrange interpolation operators, we show that the resulting method is pressure-robust for any $H^1$-conforming, continuous-pressure discretization. We give the complete analysis for the MINI element, and then for stabilized equal-order $P^kP^k$ elements of arbitrary degree $k$, proving that the interpolation-induced consistency error is pressure-robust and converges of optimal order. We validate the theory by several numerical examples.

发表机构

• University of Hamburg（汉堡大学）
• TU Wien（维也纳工业大学）

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