Block preconditioners for finite element discretization of incompressible flow with thermal convection

SUMMARYWe derive block preconditioners for a finite element discretization of incompressible flow coupled toheat transport by the Boussinesq approximation. Our techniques rely on effectively approximating theSchur complement obtained by eliminating the fluid variables to obtain an equation for temperature alone.Additionally, the method utilizes existing block-structured preconditioners and multilevel methods for theNavier—Stokes equations and scalar convection-diffusion. We find that the preconditioner remains robustand scalable even when the subsolves are applied quite inexactly. Copyright c 0000 John Wiley & Sons,Ltd. Received ... KEY WORDS: Finite element, incompressible flow, Benard convection, multiphysics, block precondi-´tioner. 1. INTRODUCTIONIn this paper, we extend block preconditioners for inf-sup stable finite element discretizations of theincompressible Navier–Stokes equations,  u+ uru+ rp= fru= 0;(1)posed on some domain ˆR d for d= 2;3 and equipped with appropriate boundary conditions, toa particular non-dimensionalization of a coupled fluid-thermal problem, Benard convection:´ u+ uru+ rp= RaPrgT^ru= 01Pr T+ urT= 0;(2)again posed on some domain along with boundary conditions. The fluid velocity and pressureare uand p, respectively, in both equations, and the temperature is T. The in (1) is the fluidviscosity. The Rayleigh number Rameasures the ratio of energy from buoyant forces to viscous

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