Iterative methods for stabilized discrete convection-diffusion problems

Yin-Tzer Shih

Department of Mathematics, University of Maryland, College Park, MD 20742

Howard C. Elman

Department of Computer Science, University of Maryland, College Park, MD 20742


In this paper, we study the computational cost of solving the convection-diffusion equation using various stabilization strategies and iteration solution algorithms. The choice of stabilization strategy influences the properties of the discrete solution and also the choice of solution algorithm. The stabilizations considered here are based on streamline diffusion, crosswind diffusion and shock-capturing. Most notably, shock-capturing discretizations lead to nonlinear algebraic systems (even for linear problems) and require nonlinear algorithms. We compare various preconditioned Krylov subspace methods including Newton-Krylov methods for the nonlinear problems, as well as several preconditioners based on relaxation and incomplete factorization. We find that although enhanced stabilization based on shock-capturing requires fewer degrees of freedom than linear stabilizations to achieve comparable accuracy, the nonlinear algebraic systems are more costly to solve than those derived from a judicious combination of streamline diffusion and crosswind diffusion. Solution algorithms based on GMRES with incomplete block-matrix factorization preconditioners developed by Axelsson are robust and efficient.