Arbitrary Quadrilateral Element And Anisotropic Cell Research

Since quadrilateral elements have some merits such as simple grids, smaller band width of stiffness matrix and economic computation and so on compared with triangle elements, and have better approximation to the boundary of the domain compared with rectanguler elements. It is very important to apply quadrilateral elements to kinds of problems from the points of theoretical and practical views.G. Acosta, R. Duran (SIAM. Numer. Anal. 2001) thought that RDP(N, ψ) seems to be the weakest mesh condition up to now and proved that RDP(N, ψ) is a sufficient condition such that the optimal interpolation error estamate under the norm of H₁ for Q₁ isoparametric element. One may ask whether or not the condition RDP(N, ψ) is also necessary? Acosta and Duran put it as an open problem. Z. Zhang conjectured that it is also necesarry condition. In this paper, we show that RDP(N, ψ) is not necessary by a conterexample.The convergence analysis of two new lower order nonconforming arbitrary convex quadrilateral elements for solving Stokes equations is studied in this paper. The approximation spaces to the pressure and velocity are piecewise constant and the degrees of freedom with integral values respectively. The two new elements not only have the simple construction and convenience in practice, but also have the advantage of better approximation to the boundary of the domain. we point out that one of two elements is a locking-free element and can be applied to the planar elasticity problem under rectangular grid; meanwhile, its approximate schemes with moving grid for solving nonstationary Stokes equations is presented in this paper and the optimal error estimates are obtained.It is well-known that the drgrees of freedom and the base of shape function space should match in the construction of finite element. On the one hand, the function of finite element space have countinuous with some kinds mean values, on the other hand, the drgrees of freedom are also the variable of the discrete equation, they should be selected simply and the total degrees of freedom should be as small as possible. As usual, it is diffcult to satify the above two requirements. In order to overcome the above difficulties, the " double set...