A finite-element, parameter stepping solution procedure for the steady-state Boussinesq equations

摘要

In this paper we present a solution procedure for buoyancy-induced flows, based on a finite-element discretization and a continuation strategy. At each stage of the continuation procedure, the nonlinear discrete equations are linearized using the Newton-Raphson procedure and the resulting system of equations solved by a preconditioned biconjugate gradient algorithm. The solution procedure is tested on the buoyancy-driven cavity problem, and results obtained for a range of Rayleigh numbers. We prove that under appropriate conditions, the continuation strategy will converge to the true solution. The proof requires the assumption of the existence and uniqueness of solutions at each Rayleigh number; however, we finally prove the existence of weak solutions of the underlying continuous problem and give a sufficient condition for uniqueness of the solution.