Overlapping Schwarz preconditioner for the mixed formulation of plane elasticity

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.