Overlapping Schwarz preconditioner for the mixed formulation of plane elasticity

Abstract

Recently a stable pair of finite element spaces for the mixed formulation of the plane elasticity system has been developed by Arnold and Winther. Here we construct a two-level overlapping Schwarz preconditioner for the resulting discrete system. Essentially, this reduces to finding an efficient preconditioner for the form ( ⋅ , ⋅ ) + ( div ⋅ , div ⋅ ) in the symmetric tensor space H ( div , Ω ) . The main difficulty comes from the well known complexity of building preconditioners for the div operator. We solve it by taking a decomposition similar to the Helmholz decomposition. Both additive and multiplicative preconditioners are studied, and the conditioner numbers are shown to be uniform with respect to the mesh size.