New error estimates of the Morley element for the plate bending problems

Abstract

In this paper, we derive some new error estimates of a nonconforming Morley element for the plate bending problems. Our aim is fourfold, first, it shows that the consistency error of the Morley element can be controlled by its corresponding approximation error up to an arbitrarily high order term, especially which can be of arbitrary order provided that f on the right hand side is piecewise smooth enough, thus the total energy norm is dominated alone by the approximation error. Second, as a byproduct, we derive the error estimate under the regularity assumption u∈H2+s(Ω) with s∈(0,1], which fills the gap in the a priori error estimate of the Morley element with low regularity for s∈(0,12]. Third, based on this new error bound, a robust convergence is proved even if the solution is exactly in H2(Ω). Finally, by a slight modification of the variational formulation, a new error estimate is also derived for the case f∈H−1(Ω). The key role is played by some quasi-interpolation operators and the weak continuous properties of Morley finite element space. The main ideas and results presented here are motivated by Mao and Shi (2010) and can be viewed as its extension to fourth order problems.