一些窄边元和各向异性元分析

【作者】 李清善；

【导师】 陈绍春；

【作者基本信息】 郑州大学 ， 基础数学， 2003， 博士

【摘要】 本文主要研究了二维空间中的二阶问题和Stokes问题及三维空间中的二阶问题，研究了窄边四边形元的双线性插值及二维和三维空间中各向异性元的误差估计，构造出一些用于二阶问题和Stokes问题的窄边单元和各向异性单元。本文结果均是在不要求区域剖分满足正则性条件下获得的。 传统有限元的研究，要求对区域的剖分满足正则性条件。随着有限元应用领域的不断扩大，正则性条件已成为影响有限元应用的制约条件。Zenisek等人给出了窄边四边形上的双线性插值定理，但是，我们发现起关键作用的特殊区域上的Poincaré不等式的证明是不正确的，而且所给出的误差估计式中的常数也不是最优的。我们给出了Poincaré不等式和迹不等式的改进形式，并给出了正确的证明。通过证明过程的精细估计，我们给出了优化的插值定理，定理中的常数比原来定理中的常数小得多(约为原来常数的1／2～1／5)。 研究了著名的类Wilson非协调元。通过对Wilson元的非协调部分增加一个高阶项，构造出不满足正则性条件的窄边类Wilson非协调元，得到了窄边类Wilson元的插值误差估计；同时通过特殊的技巧，给出了窄边类Wilson元的收敛性分析，证明了二阶问题的误差也能达到最优收敛阶。用类似于类Wilson非协调元的方法还研究了五参数窄边四边形非协调有限元法，证明了该元具有特殊的收敛性质(即相容误差比插值误差高一阶)。把窄边类Wilson非协调元和五参数窄边四边形非协调元应用于定常Stokes方程，研究了试探函数不要求满足不可压条件的协调有限元解法和非协调有限元解法。通过对Stokes方程离散格式的构造，我们给出了它们相应的误差分析和最优误差估计。 将二维类Wilson非协调元推广到三维各向异性Wilson砖元和各向异性类Wilson六面体非协调元。讨论了空间单元上的各向异性插值，分别给出了平行六面体元、砖元和六面体元上的各向异性插值误差估计。对于各向异性插值误差，参考元上的插值误差估计起着重要作用，我们给出一个一般的插值误差估计的判定定理，该定理应用于各向异性单元，易于操作，易于验证。验证了Wilson砖元具有各向异性特征，给出该元的各向异性插值误差，分析了各向异性Wilson砖元对二阶问题收敛，而且证明了二阶边值问题误差估计具有各向异性特征，即误差估计可以从单元的不同边长分别受益。利用平面窄边类Wilson元的特殊性质，我们给出了空间各向异性类Wilson六面体非协调元的构造，证明了各向异性类Wilson六面体元也具有某些特殊的性质。由此证明了二阶问题的各向异性类Wilson六面体元的解也具有各向异性特征。对于九参数砖非协调元和九参数六面体非协调元我们也给出了类似的结论. 对于各向异性三角形元和各向异性四面体元，我们分别讨论了它们的各向异性插值，将它们应用于定常stokes方程，我们研究了有限元空间不满足散度条件的各向异性非协调有限元解法，同样得到了逼近解的误差分析和最优收敛阶.更多还原

【Abstract】 The second-order problem and Stokes problem for two dimensional spaces and the second-order problem for three dimensional space are investigated in this paper. The bilinear interpolation theorem of narrow quadrilateral and error estimates of anisotropic elements are obtained. Some narrow quadrilateral elements and anisotropic elements which are applied to second-order problem and Stokes problem are produced. Our results are derived without satisfying the regular condition.The researches of classical finite elements need the meshes of triangulation satisfying the regular condition. However, with the continuous enlargement of application range of finite elements, this condition has been restrictive facter of affacting use of finite elements. Zenisek and Vanmaele considered anisotropic, convex, quadrilateral, isoparametric finite elements with bilinear shape function. They derived the interpolation error estimates and treated the constants in the estimates carefully. But we find that the Poincare inequality on the special domain which plays the key role in the proof of error estimates is wrong, and the constants of error estimates are also not optimal. We give the improved forms of Poincare inequality and trace inequalities, and give their correct proofs. Through the sharper estimate for the proof process, we present the optimized interpolation theorems, and the constants of error estimate are much less than original ones (it is about 1/2?/5 of the original ones).The famous quasi-Wilson nonconforming element is studied. By adding a high-order item to the part of nonconforming, we construct the narrow quasi-Wilson nonconforming element which needn’t satisfy the regular condition, and give the interpalation error estimate of the element. Meanwhile, by using a special technique, we give the convergent analysis of narrow quasi-Wilson element and prove that the error of second-order problem also can reach the optimal convergent order. Using the method analogous to quasi-Wilson element, we study the nonconforming finite element of five parameters narrow quadrilateral, and prove that this element has a special property, that is, consistency error is one order higher than that of interpolation error. Applying the narrow quasi-Wilson element and five parameters quadrilateral element to the stationary Stokes equation, we derive a finite element method which eliminates the div v ?0 restriction on the trial functions, and give their corresponding error analysis and optimal error estimate.The two dimensional quasi-Wilson nonconforming element is generalized to three dimensional anisotropic Wilson’s brick nonconforming element and anisotropic quasi-Wilson hexahedral nonconforming element. We discuss the anisotropic interpolation error estimates of parallelepiped element, of brick element and of hexahedral element, respectively. For anisotropic interpolation, the error estimate on references plays an important role. We give a general decision theorem about anisotropic interpolation. This theorem is easier to be operated and applied than Apel’s. It is verified that Wilson’s Brick element has anisotropic characteristic and the anisotropic interpolation error is obtained. We analyse the convergence of quasi-Wilson’s brick element for second-order problem, and prove that error estimate of second-order boundary value problem has anisotropic behaviour. Based on the idea of two dimensional narrow quasi-Wilson element, we give a construction of three dimensional anisotropic quasi-Wilson hexahedral nonconforming element, prove that this element also has some special properties, and that the solution of second-order problem aslo has anisotropic characteristic. For nine-parameters brick nonconforming element and nine parameters hexahedral nonconforming element, we also give the similar conclusions.We considered anisotropic triangular elements and anisotropic tetrahedral elements, and give their anisotropic interpolation error estimates, respectively. Applying the triangular element to the stationary Stokes equation, we o更多还原