A preconditioner for the equations of linear elasticity discretized by the PEERS element

Block diagonal preconditioners for the mixed finite element formulation of planar linear elasticity using the PEERS element by Arnold, Brezzi and Douglas [1] are studied. This mixed finite element approach treats the stress tensor, the displacement field and the anti-symmetric part of the strain tensor as independent variables. The resulting linear system of equations is symmetric but indefinite due to its saddle point structure. The validity of an inf-sup condition with respect to the H(div; )2 L2( )2 L2( ) norm suggests an optimal block diagonal preconditioner for the MINRES iterative method. Equivalent preconditioning of the blocks associated with stresses, displacements and anti-symmetric part leads to convergence rates of the block diagonal preconditioner uniformly bounded with respect to the refinement level and the Lamé parameter which determines the compressibility of the material. While the L2( ) blocks can simply be preconditioned by its diagonal parts, a multilevel technique is employed for the H(div; ) block. The smoothing iteration for this technique is based on the discrete Helmholtz decomposition of the underlying finite element space similar to the approach by Hiptmair [13]. Numerical tests for a benchmark problem of planar linear elasticity illustrate the efficiency of this preconditioner, in particular, its level independence and uniform convergence in the incompressible limit. Copyright c 2004 John Wiley & Sons, Ltd.