The Morley element for fourth order elliptic equations in any dimensions
- 315 Downloads
In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general n-dimensional Morley element consists of all quadratic polynomials defined on each n-simplex with degrees of freedom given by the integral average of the normal derivative on each (n-1)-subsimplex and the integral average of the function value on each (n-2)-subsimplex. Explicit expressions of nodal basis functions are also obtained for this element on general n-simplicial grids. Convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation.
KeywordsNonconforming finite element Forth order elliptic equation Biharmonic Morley element
Unable to display preview. Download preview PDF.
- 1.Bazeley, G.P., Cheung, Y.K., Irons, B.M., Zienkiewicz, O.C.: Triangular elements in plate bending — conforming and nonconforming solutions, in Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson A. F. Base, Ohio, 1965, 547–576Google Scholar
- 2.Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland. Amsterdam, New York, 1978Google Scholar
- 3.Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numer. R-1, 9–53 (1985)Google Scholar
- 4.Morley, L.S.D: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, 149–169 (1968)Google Scholar
- 6.Shi Zhong-ci: On the error estimates of Morley element. Numerica Mathematica Sinica 12,2 113–118 (1990)Google Scholar
- 7.Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, 1973Google Scholar
- 8.Wang Ming: On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J Numer Anal 39,2 363–384 (2002)Google Scholar
- 9.Wang Ming and Jinchao Xu, Some tetrahedron nonconforming elements for fourth order elliptic equations, Math Comp, acceptedGoogle Scholar