| There are many problems in engineering and scientific disciplines that can be described by convection-diffusion equations. In a convection-diffusion equation, if the convection term is very dominant, the linear system of equations that result from either finite-differencing or finite elementing will not have a 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. We present such a two step method that combines an Arnoldi-Chebyshev approach for convection-diffusion computations, that generate faster and better solutions. We also study a domain decomposition method for solving convection diffusion problems. Next, we present a finite difference singular perturbation technique for solving problems with boundary layers. |
