Iterative methods for stabilized discrete convection-diffusion problems

In this paper we study the computational cost of solving theconvection-diffusion equation using various discretization strategiesand iteration solution algorithms. The choice of discretizationinfluences the properties of the discrete solution and alsothe choice of solution algorithm. The discretizations consideredhere are stabilized low-order finite element schemes using streamlinediffusion, crosswind diffusion and shock-capturing. The latter,shock-capturing discretizations lead to nonlinear algebraicsystems and require nonlinear algorithms. We compare variouspreconditioned Krylov subspace methods including Newton-Krylovmethods for nonlinear problems, as well as several preconditionersbased on relaxation and incomplete factorization. We find thatalthough enhanced stabilization based on shock-capturing requiresfewer degrees of freedom than linear stabilizations to achievecomparable accuracy, the nonlinear algebraic systems are morecostly to solve than those derived from a judicious combinationof streamline diffusion and crosswind diffusion. Solution algorithmsbased on GMRES with incomplete block-matrix factorization preconditioningare robust and efficient.