The discrete Korn's type inequality in subspaces and iterative methods for thin elastic structures

Abstract

When iterative methods are applied to thin elastic structures, such as shells, plates, rods, arches and beams, they suffer from slow convergence with diminishing thickness. To overcome this difficulty the authors have introduced the fundamental concept of the Korn's type inequality in subspaces (see Proc. R. Soc. Lond. A 453 (1997) 2003–2016). Here, we present the discrete Korn's type inequality in subspaces which, when applied to discrete methods, such as the finite element method, enables the design of iterative algorithms for thin elastic structures with convergence rate independent of both the thickness and the discretisation parameters. As a paradigm we consider the steepest descent method with the subspace correction preconditioning and present p-version finite element results for a model shell problem which show that a simple modification of this method based on the discrete Korn's type inequality in subspaces yields a radical improvement in the convergence.