  We propose a general algorithm to enumerate all solutions of a
zero-dimensional polynomial system with respect to a given cost function. The
algorithm is developed and is used to study a polynomial system obtained by
discretizing the steady cavity flow problem in two dimensions. The key
technique on which our algorithm is based is to solve polynomial optimization
problems via sparse semidefinite programming relaxations (SDPR), which has been
adopted successfully to solve reaction-diffusion boundary value problems
recently. The cost function to be minimized is derived from discretizing the
fluid's kinetic energy. The enumeration algorithm's solutions are shown to
converge to the minimal kinetic energy solutions for SDPR of increasing order.
We demonstrate the algorithm with SDPR of first and second order on polynomial
systems for different scenarios of the cavity flow problem and succeed in
deriving the $k$ smallest kinetic energy solutions. The question whether these
solutions converge to solutions of the steady cavity flow problem is discussed,
and we pose a conjecture for the minimal energy solution for increasing
Reynolds number.
