Preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and Stokes problems Original Research Article

Mixed spectral methods applied to elasticity and Stokes problems in three dimensions produce large symmetric indefinite systems of equations. These systems are sparse and more ill-conditioned than those produced by h-version finite element methods. These stiffness matrices have the structure of saddle point problems with a penalty term. In this paper, preconditioned iterative methods of conjugate residual type are introduced for these systems. The convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n. Several numerical experiments on three dimensional model problems confirm the results obtained.