Quasi-least-squares finite element method for steady flow and heat transfer with system rotation

Two quasi-least-squares finite element schemes based on L2 inner product are proposed to solve a steady Navier–Stokes equations, coupled to the energy equation by the Boussinesq approximation and augmented by a Coriolis forcing term to account for system rotation. The resulting nonlinear systems are linearized around a characteristic state, resulting in linearized least-squares models that yield algebraic systems with symmetric positive definite coefficient matrices. Existence of solutions are investigated and a priori error estimates are obtained. The performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess.