一类Kirchhoff 板弯问题自适应高阶混合元方法及理论分析

本文针对Kirchhoff 板弯问题提出了一个基于高阶Hellan-Herrmann-Johnson (简记为H-H-J)方法的自适应有限元算法, 分析了它的收敛性和计算复杂度. 证明了算法在执行过程中, 相应的拟能量误差会以几何级数单调衰减, 从而得到收敛性. 利用此单调下降性质, 进一步给出了算法的计算复杂度. 推导过程中的一个关键步骤是建立基于平衡方程的单元误差表示(error indicator) 与平衡方程右端载荷震荡项(data oscillation) 的局部等价关系.

A class of high order adaptive mixed element methods for Kirchhoff plate bending problems

In this paper,we deal with convergence and complexity of an adaptive algorithm for Kirchhoff bending plate problems.The algorithm is based on high order Hellan-Herrmann-Johnson methods（k 2,where k denotes the polynomial degree of the discrete moment-field space）.We derive a contraction property for the scaled sum of the energy-norm error,the error indicators and the data oscillation involving a given transverse load in two consecutive adaptive loops.Then a complexity estimate in terms of the number of degrees of freedom is developed.The key ingredient in the analysis is a local equivalence of the data oscillation and the element error indicator arising from the equilibrium equation.