Field-of-values analysis of preconditioned linearized Rayleigh-Benard convection problems

In this paper we use the notion of field-of-values (FOV) equivalence of matrices to study a class of block-triangular preconditioners for the fixed-point linearization of the Rayleigh–Bénard convection problem discretized with inf–sup stable finite element spaces. First, sufficient conditions on the nondimensional parameters of the problem are determined in order to establish the FOV-equivalence between the system matrix and the preconditioners. Four upper triangular block preconditioners belonging to the general proposed class are then considered. Numerical experiments show that the Generalized Minimal Residual (GMRES) convergence is robust with respect to the mesh size for these preconditioned systems. We also compare the performance of the different preconditioners in terms of computational time.