一维问题有限元逼近的强超收敛性

【作者】 赵庆华；

【导师】 朱起定；

【作者基本信息】 湖南师范大学 ， 计算数学， 2002， 硕士

【摘要】 本文研究一维问题(包括两点边值问题，一维抛物问题及抛物型积分-微分方程)有限元逼近的强超收敛性。除引言和准备知识外，主要有三个部分。 首先对于一类两点边值问题，利用投影型插值算子获得了函数及导数的内部强超收敛点。这些点是与解u有关的，但我们可用含u_h的一个简单公式来确定它们的位置。在此基础上，我们继续讨论了超收敛单元片应力恢复(SPR)技巧与有限元校正的问题。用比较简明的方法证明了关于SPR技巧的一些结果，更重要的是经过校正获得了整体强超收敛性。 其次，我们将上述两点边值问题的结论推广到了一类抛物方程，获得了单元内部函数及导数的强超收敛点并经过校正获得了整体强超收敛性。 最后，我们对一类抛物型积分-微分方程获得了真解与Ritz-Volterra投影之间的强超逼近估计，并在此基础上得到了一个强超收敛二择一定理。更多还原

【Abstract】 In this thesis, we study ultraconvergence of finite element approximation in one dimensional case (include two-point boundary value problem, one dimensional parabolic equation and parabolic type integro-differential equation ). Besides introduction and preliminary, there are three main parts.In the first part, we consider a class of two-point boundary value problems. We obtained interior ultraconvergent points for function and derivative using projection interpolation. These points depend on the solution u, but we can locate them by a simple formular with uh. Furthermore, we discuss SPR technique and finite element correction, we proved some results of SPR technique and obtained global ultraconvergence by correction.In the second part, we extend the above results of two-point boundary value problem to a class of parabolic equations. We obtained interior ultraconvergent points for function and derivative. We also obtained global ultraconvergence by correction.In the third part, for a class of parabolic integro-differential equations we obtained ultra-approximation between u and its Ritz-Volterra projection which leads to a ultraconvergence alternating theorem.更多还原