Hierarchical Conforming Finite Element Methods for the Biharmonic Equation

Abstract

The paper deals with hierarchical bases in spaces of conforming $C^1 $ elements in connection with the approximate solution of the biharmonic equation \[ \Delta ^2 u = f\quad {\text{in }}\Omega ,\qquad u = \frac{{\partial u}}{{\partial n}} = 0\quad {\text{on }}\partial \Omega \] on a plane polygonal domain $\Omega $. Two different composite finite elements are studied: piecewise quadratic Powell–Sabin elements and piecewise cubic elements of Clough–Tocher type. The main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization (the so-called hierarchical multilevel method) will yield suboptimal convergence rates in comparison with standard multigrid schemes.