Comparison Results of Nonstandard P₂ Finite Element Methods for the Biharmonic Problem

Abstract

As modern variant of nonconforming schemes, discontinuous Galerkin finite element methods appear to be highly attractive for fourth-order elliptic PDEs. There exist various modifications and the most prominent versions with first-order convergence properties are the symmetric interior penalty DG method and the C0 interior penalty method which may compete with the classical Morley nonconforming FEM on triangles. Those schemes differ in their various jump and penalisation terms and also in the norms. This paper proves that the best-approximation errors of all the three schemes are equivalent in the sense that their minimal error in the respective norm and the optimal choice of a discrete approximation can be bounded from below and above by each other. The equivalence constants do only depend on the minimal angle of the triangulation and the penalisation parameter of the schemes; they are independent of any regularity requirement and hold for an arbitrarily coarse mesh.