Arnoldi-Chebyshev approach for convection-diffusion computations

In a convection-diffusion equation, if the convection term is very dominant, the linear system of equations which result from either finite-differencing or finite elementing will not have a strictly diagonally dominant coefficient matrix. So, if one tries a conventional iteration method (Jacobi or Gauss-Seidel) to solve the linear system of equations, the iteration matrix may not satisfy the spectral radius condition for convergence, and hence, no converging solution may be obtained. The problem can be overcome, under certain conditions, if one uses a two-step iteration procedure involving the spectral enveloping ellipse for the iteration matrix. In this paper, we present such a two-step method that combines an Arnoldi-Chebyshev approach for convection-diffusion computations.