Crouzeix--Raviart--Marini 实现各向异性网格上可计算的先验 $L^2$ 误差界

中文摘要 AI 辅助

我们考虑二维和三维 Dirichlet Poisson 问题的协调 $\u0050one$ 和最低阶 Crouzeix--Raviart (CR) 逼近。在没有全局 $H^2$ 正则性假设的情况下，我们在各向异性单纯形网格上获得了可计算的先验 $L^2$ 误差界。对于分片常数数据，Marini 关系从标量 CR 解给出了一个均衡的最低阶 Raviart--Thomas (RT) 通量。所得 Marini 常数的平方是仅涉及标量协调和 CR 矩阵的问题的最大广义特征值。与逐单元 Poincaré 常数结合，它控制了协调和 CR 误差。CR $L^2$ 估计使用了离散 CR--协调间隙算子的正性，而不是非协调 Aubin--Nitsche 论证。我们还推导了 Marini 缺陷的精确分解，将其分解为 CR--协调能量间隙和一个没有纵横比因子的显式几何项。在我们这里考虑的代数分级 L 形网格族上，$q>3/2$ 产生 $\kappa_{M,h}=O(h)$ 和可计算的 $O(h^2)$ $L^2$ 算子界。

英文摘要

We consider conforming $\Pone$ and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3/2$ yields $κ_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.

发表机构

• Team FEM

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