A robust nonconforming $H^2$-element

Abstract:

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming $H^2$-element which is $H^1$-conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.