Incremental Unknowns for convection-diffusion equations

Abstract

When iterative methods (e.g. MR, GCR, Orthomin(k)) are used to approximate the solution of a nonsymmetric linear system $AU=b$, where $A$ is a $N\times N$ matrix with positive symmetric part, i.e. $M={1 \over 2 }(A+A^t)$ is positive definite, the convergence rates depend on the number $\nu(A)= {\lambda_{max}(A^tA)}^{1/2} / \lambda_{min}(M)$ which is the conterpart of condition number of $A$ when $A$ is symmetric positive definite. In this paper, we use Incremental Unknowns (IU) in conjunction with the above iterative methods to approximate the solution of the nonsymmetric linear system generated by the discretization of a convection-diffusion equation. We show theoretically that, with the use of IU, $\nu$ is of order $O((log (h))^2)$ instead of $O({1 \over h^2})$ which is the order of $\nu$ when nodal unknowns are used; here $h$ is the mesh size for the finite differences. With the use of (2.3) in Theorem 2.1, we can show that \underbar {at most} $O((log (h))^4)$ iterations are needed to attain a good solution. Numerical results are also presented. They show that actually, $O(|log (h)|)$ iterations are needed with the use of IU. The Incremental Unknown methods are efficient and easy to implement.