An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

Abstract

This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (P3) and possesses \({\cal O}({h^2})\) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with \({\cal O}({h^2})\) accuracy in the broken H2 norm. Besides, a discrete strengthened Miranda-Talenti estimate (∇ 2h ·, ∇ 2h ·) = (∆h ·, ∆h ·), which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet’s triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.