Numerische Mathematik

, Volume 103, Issue 1, pp 155-169

First online:

The Morley element for fourth order elliptic equations in any dimensions

  • Wang MingAffiliated withLMAM, The School of Mathematical Sciences, Peking University
  • , Jinchao XuAffiliated withLMAM, The School of Mathematical Sciences, Peking UniversityDepartment of Mathematics, Pennsylvania State University Email author 

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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.


Nonconforming finite element Forth order elliptic equation Biharmonic Morley element