一类四阶抛物积分微分方程混合元方法的超收敛分析

Superconvergence Analysis of a Mixed Finite Method for a Class of Fourth Order Parabolic Integro-Differential Equations

本文的主要目的是利用双线性元Q_(11)及Q_(01)×Q_(10)元研究一类非线性四阶抛物积分微分方程的混合有限元方法.一方面,利用上述两种元的高精度结果以及对时间t的导数转移技巧,在半离散格式下,导出原始变量u和中间变量w=-?u在H^1-模意义下及流量p(向量)=-?u在(L^2)~2-模意义下具有O(h^2)阶的超逼近性质.进一步地,借助插值后处理技术,得到上述变量的整体超收敛结果.另一方面,建立一个新的向后Euler全离散格式.通过采取新的分裂技术,得到u和w在H^1-模意义下及p在(L^2)~2-模意义下具有O(h^2+?t)阶的超逼近和超收敛结果.这里,h和?t分别表示空间剖分参数和时间步长.最后,给出一个数值算例,计算结果验证了理论分析的正确性.

The purpose of this paper is to study a mixed finite element method for a class of fourth order parabolic integro-differential equations with bilinear Q11element and Q01×Q10element.On one hand,the superclose properties of order O（h-2）for original u and intermediate variable w in H-1-norms and the flux p in（L^2）-2-norm are obtained for semi-discrete scheme through the known high accuracy results of above two elements and derivative transforming technique with respect to time variable.Furthermore,the global superconvergence results are deduced through interpolated postprocessing approach.On the other hand,a new backward Euler fully-discrete scheme is developed.The superclose and superconvegence results of order O（h-2＋ t）for u and w in H-1-norms and p in（L^2）-2-norms are proved respectively through a new splitting approach.Here,h and？t is the subdivision parameter for the space and time step,respectively.Finally,numerical results are provided to confirm the theoretical analysis.