A new Korn's type inequality for thin domains and its application to iterative methods

Abstract

We present a new Korn's type inequality which estimates the ratio of the energy norm to the Sobolev norm in a given subspace in terms of the angle it forms with an explicitly extracted finite dimensional subspace. This inequality provides crucial information for improving the convergence of various iterative algorithms for elasticity problems in thin domains. This is demonstrated in the case of the semi-discrete iterative algorithm EDRA (see E.E. Ovtchinnikov and L.S. Xanthis, Effective dimensional reduction for elliptic problems, C.R. Acad. Sci. Paris, Series I, 320: 879–884, 1995). We show both theoretically and numerically that the convergence of the modified EDRA is independent of the thickness of the domain and of the semi-discretisation parameters.