Von Kármán障碍问题的Morley有限元法
Morley Finite Element Method for the von Kármán Obstacle Problem
摘要：
本文研究了带有凸障碍物约束的薄板大变形问题的vonKármán方程，将其转化为一个耦合系统的半线性四阶障碍物问题，并给出了该问题的Morley有限元逼近.第一部分建立了von Kármán障碍问题的适定性，并讨论了解在先验和后验小性条件下的唯一性。文章的第二部分讨论了Frehse从1971年以来的正则性结果，并将其与多边形区域上解的正则性结合起来。文章的第三部分给出了Morley有限元逼近小数据条件下von Kármán障碍问题的最优收敛速度的先验误差估计。文章最后用数值结果说明了最佳收敛速度对数据小性假设的要求。
英文摘要：
This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.
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