用于冯·卡门障碍问题的二次 $C^0$ 内点罚方法
A Quadratic $C^0$ Interior Penalty Method for the von Kármán Obstacle Problem
AI总结：
本文针对冯·卡门板位移障碍问题，提出二次 $C^0$ 内点罚方法，证明其适定性并推导误差估计，通过数值实验验证方法有效性，明确罚参数与障碍尺寸阈值的影响。
AI中文摘要：
本文针对冯·卡门板的位移障碍问题，提出并分析了一种二次 $C^0$ 内点罚方法。离散空间由拉格朗日 $P_2$ 有限元构成，障碍约束施加在顶点处。冯·卡门括号的三线性形式通过边上的项进行修正，使其在离散能量范数下有界。本文证明了离散问题的适定性，即存在离散解，且在数据满足小性条件时解唯一。本文量化了离散能量范数的索伯列夫常数和弗里德里希斯常数，明确了其对网格尺寸的依赖关系。主要结果是离散能量范数下的误差估计，阶数为 $\u039f(h^\u03b1)$，其中 $1/2<\u03b1\u22641$ 是多边形域上双调和算子的椭圆正则性指数。在正方形域和L形域上的数值实验验证了预测的收敛阶。其中一个算例的重合集具有正测度，另一个算例的重合集内部为空。实验还展示了罚参数对收敛阶的影响，并确定了障碍尺寸的阈值，超过该阈值后，迭代求解器在细网格上会失效。
英文摘要：
This article proposes and analyses a quadratic $C^0$ interior penalty method for the displacement obstacle problem of the von Kármán plate. The discrete space consists of Lagrange $P_2$ finite elements and the obstacle constraint is imposed at the vertices. The trilinear form of the von Kármán bracket is modified by terms on the edges so that it is bounded in the discrete energy norm. The well-posedness of the discrete problem, namely the existence of a discrete solution and its uniqueness under a smallness condition on the data, is established. The Sobolev and Friedrichs constants of the discrete energy norm are quantified with an explicit dependence on the mesh size. The main result is an error estimate of order $\mathcal{O}(h^α)$ in the discrete energy norm, where $1/2<α\le1$ is the index of elliptic regularity of the biharmonic operator on the polygonal domain. Numerical experiments on a square and on an L-shaped domain confirm the predicted rates. The coincidence set has positive measure in one example and empty interior in another. The experiments also show how the penalty parameter affects the rates and identify a threshold in the size of the obstacle beyond which the iterative solver fails on fine meshes.
