Iterative subspace correction methods for thin elastic structures and Korn's type inequality in subspaces

Abstract

We present a methodology for the efficient solution of three–dimensional problems for thin elastic structures, such as shells, plates, rods, arches and beams, based on the synergy of two fundamental concepts: the subspace correction and the Korn's type inequality in subspaces . The former provides the theoretical background which enables the development of modern iterative methods for large–scale problems, such as domain decomposition and multilevel methods, which are sine qua non for high–performance scientific and engineering computing, and the latter is responsible for the design of iterative methods for thin elastic structures with convergence which is uniform with respect to the thickness. The subspace correction methods are based on the decomposition of the space where the solution is sought into the sum of subspaces. In this paper we show that using the Korn's type inequality in subspaces we can introduce subspace decompositions for which the convergence rate of the corresponding subspace correction methods is independent of both the thickness and the discretization parameters.