The Global Stress Superconvergence Of The Three-dimensional Wilson Element

The Wilson finite element, known as Wilson's brick in the three-dimensional finite element applications, is widely used in computational mechanics and structural engineering. The corresponding finite element space consists of piecewise quadratic functions which are required to be continuous only at the vertices of elements. This reduce continuity property determines the nonconforming character of the Wilson element in solving second-order elliptic boundary value problems.For the two-dimensional case, the paper [4] obtained the super-convergence and asymptotic expanson of the Wilson element. The papers [22], [23] discussed the global stress superconvergence of Wilson's element for the second-order elliptic boundary value problems and eigenvalue problems, and improved the accuracy order of the completely quadratic interpolation for the Wilson element to O(h²) (in the meaning of || ? ||_1,2). Numerical experiments of [10], [14], [30] indicated that the computed eigenvalues by the Wilson element approximate exact one's form below.For 3-D case, because of the number of unknown is O(h^-3) ( h is the diameter of the mesh), the amount of storage and calculation is very cumbersome which is difficult to accomplish in general computers. This is called "3-D trouble". Therefore, papers about research of 3-D Wilson's element are very few and there are no according numerical experiments in present.This paper discusses superconvergence of 3-D Wilson's element. We prove the global stress superconvergence of completely and incompletely quadratic interpolation in 3-D Wilson's element theoretically and give some examples to support our conclusion. Numerical experiments show that eigenvalues by 3-D Wilson's element approximate exact eigenvalues from the lower direction.