双调和方程的最优内罚间断伽辽金格式

中文摘要 AI 辅助

本文针对平面双调和方程，提出了一种使用次数为 $k=3$ 或 $4$ 的分片多项式的最优内罚间断伽辽金（IPDG）格式。在标准 IPDG 方法中，大的罚参数迫使离散解进入一个过度约束的空间，严重降低精度——这一现象称为数值锁死。为克服此问题，我们的方法仅强制顶点连续性，并将函数及其法向导数的跳跃分别投影到 $\mathcal{P}^{k-3}$ 和 $\mathcal{P}^{k-2}$ 上。因此，当罚参数趋于无穷时，离散解被强制位于约束子空间 $V_{h,\infty}^k$ 中，我们将其识别为最优非协调有限元空间 $B_h^k$。这一内在联系从根本上消除了数值锁死。我们证明了在网格依赖的能量范数下 $\mathcal{O}(h^{k-1})$ 的最优误差估计，以及在 $L^2$ 范数下 $\mathcal{O}(h^{k+1})$ 的最优误差估计。在凸域和 L 形域上的数值实验证实，所提出的格式是稳健的，并且即使对于极大的罚参数也完全无锁死。

英文摘要

This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree $k=3$ or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy---phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto $\mathcal{P}^{k-3}$ and $\mathcal{P}^{k-2}$, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace $V_{h,\infty}^k$, which we identify as the optimal nonconforming finite element space $B_h^k$. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of $\mathcal{O}(h^{k-1})$ in a mesh-dependent energy norm and $\mathcal{O}(h^{k+1})$ in the $L^2$ norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.

发表机构

• School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
• Beijing Key Laboratory on MCAACI, Beijing Institute of Technology（北京理工大学MCAACI北京市重点实验室）
• Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
• School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）

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