Convergence Analysis of Incomplete Biquadratic Rectangular Element for Fourth-Order Singular Perturbation Problem on Anisotropic Meshes

Pingli Xie,Meng Hu

doi:10.1155/2014/234375

Abstract

The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.

Highlights

The elliptic perturbation problems, which are derived from the stationary formation of parabolic perturbation problems, such as the Cahn-Hilliard type equation, are very important in both theoretical research and applications
Let Π1h be the interpolation operator of the Lagrange bilinear rectangular element corresponding to the triangulation Th
We introduce the same mesh dependent semi–norm | ⋅ |m,h on space Vh + Hm: norm

Summary

Introduction

The elliptic perturbation problems, which are derived from the stationary formation of parabolic perturbation problems, such as the Cahn-Hilliard type equation, are very important in both theoretical research and applications. In [12], the convergence analysis of a nonconforming incomplete biquadratic rectangular plate element with the shape function space and the degrees of freedom P(K) = span{1, x, y, x2, xy, y2, x2y, xy2} and ΣK = {Vi, (∂V/∂n)(Bi) (i = 1, 2, 3, 4)}, respectively, is studied, where Vi is the function value at the vertex ai of element K, (∂V/∂n)(Bi) is the unit outer normal derivative value at the middle point Bi of the edge li of K, and n = (nx, ny) is the unit outer normal vector to li This element, similar to the famous triangular Morley element [3, 18, 19, 23], is a non C0 element, and its convergence order O(h) was given based on the Generalized Patch-Test. Numerical experiments are carried out in last section to confirm the theoretical analysis

Premilinaries

A Counterexample

Convergence Analysis in a Modified Discretization Form

Numerical Experiments