中文摘要：

平均技巧对于数值处理偏微分方程的自适应有限元方法是一个普遍的方法，能通过一个简单的后处理提供有效的后验误差估计。在自适应网格上，平均技巧可产生一个梯度的逼近，这一逼近比有限元解的梯度逼近精度更高。考虑光滑系数问题及间断系数的问题，给出了包括线性问题和非线性问题的大量数值实验。

英文摘要：

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of partial differential equations since they provide the efficient posteriori error estimates by a simple postprocessing. It is shown that the averaging techniques can yield higher accuracy approximations to the gradient of the solution than that of the finite element solution on adaptive finite element grids. Both smooth coefficients problems and large jump coefficients problems will be considered. Some numerical experiments including linear problems and nonlinear prob- lems will be reported.