四阶椭圆方程的非协调有限元方法

【作者】 谢萍丽；

【导师】 程晋；

【作者基本信息】 复旦大学 ， 计算数学， 2009， 博士

【摘要】 在论文中,我们主要讨论了四阶椭圆问题的一些非协调有限元逼近。由于技术上的困难,我们通常采用非协调有限元来逼近四阶问题。但是,并不是所有的板元对四阶奇异摄动问题都关于摄动参数一致收敛,而且大多数的讨论都是在网格满足正则性条件或拟一致假设的前提下进行的。本文主要是在不同的网格剖分下讨论了一些非协调板元应用到四阶板弯曲问题和四阶奇异摄动问题时的收敛性。第一章,我们给出了一些基本空间的定义,记号和有限元方法的一些基础知识以及其相关的性质。第二章,我们给出了两个Morley型的非C~0非协调板元在各向异性网格下对四阶板弯曲问题的逼近,利用Poincare不等式得到了插值误差的一个显式估计,通过引入特殊的插值算子得到了相容误差的估计,从而证明了收敛性结果,得到了能量模和零模意义下和正则网格下相同的收敛速度,相应的数值例子再次验证了我们的理论分析。第三章,我们首先通过一个反例说明了第二章中的第一个Morley型的非C~0非协调板元逼近像Poisson方程这样的二阶问题时并不收敛,这就意味着该单元在通常的有限元离散格式下逼近四阶奇异摄动问题时关于奇异摄动参数不是一致收敛的。接着,我们证明了它在一个修正的有限元离散格式下是收敛的,并且在各向异性网格下也得到了关于摄动参数一致收敛的估计结果,并进行了相关的数值试验。最后,我们证明了无论网格是否满足正则性条件或拟一致假设,第二章中的第二个Morley型非协调板元逼近四阶奇异摄动问题时关于奇异摄动参数都是一致收敛的。由于该单元也是非C~0的,这就表明对四阶奇异摄动问题的收敛性并不要求有限元空间一定是H~1的子空间。最后的数值例子再次表明了该单元的有效性。第四章,在本章中,我们首先证明了一类C~0型非协调板元逼近四阶奇异摄动问题时的一个一般收敛性结果,并给出了边界层情形时的误差分析,得到了只依赖于右端项的误差估计。接着,我们分析了一个双参数三角形元的性质,利用前面的结论得到了相应的收敛性结果,最后,我们也进行了相关的数值试验。更多还原

【Abstract】 In this thesis,several nonconforming finite element methods for plate problems and fourth order elliptic singular perturbation problems are discussed.For the reason of technique difficulties,nonconforming plate elements are always employed to solve plate bending problems.Many successful plate elements have been constructed,however, they are not always convergent for fourth order singular perturbation problems.On the other hand,most of the discussions are based on regular condition or quasi-uniform assumption of triangulations.Here,we focus on the convergence analysis of different kinds of nonconforming finite element approximations to fourth order elliptic problems with different mesh fashions.In Chapter 1,we introduce some definitions and notations of basic spaces,present the fundamental knowledge and relational properties of finite element method,such as some important notions,inequalities and useful theorems.In Chapter 2,the approximation of plate bending problem by employing two Morley-type non-C~0 nonconforming plate elements under anisotropic meshes are discussed, the optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches.Some numerical tests are given to confirm the theoretical analysis.In Chapter 3,a counterexample is given to show that the first element in Chapter 2 diverges for the reduced second order equations,which means that it is not convergent for fourth order elliptic singular perturbation problems uniformly.As an alternative,the convergence in the energy norm uniformly with respect to the perturbation parameter is derived when a modified finite element approximation is employed.Numerical experiments are carried out to confirm our theoretical results.Moreover,by employing some new tricks,we obtain the convergence results of the second element in Chapter 2 even when the regular condition or quasi-uniform assumption on the triangulation is not satisfied.Since this element is also non-C~0,this means that the convergence does not require the finite element space being a subspace of H~1(Ω).Similar estimates are also presented for other two different approximation formulations.Numerical results are also given at the end of this chapter.In Chapter 4,a general convergence theorem for C~0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem is presented.The error estimates only related to the righthand term of the equation are also derived when there exist boundary layers.Then the properties of a nine parameter triangular element constructed by double set method is studied,the corresponding convergence results are obtained by the former theorem,numerical experiments are carried out to check our analysis results.更多还原