一类Kirchhoff板弯问题自适应高阶混合元方法及理论分析

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

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.

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.