Some Narrow Element And Anisotropic Element Analysis

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...