Incremental Unknowns for convection-diffusion equations
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.
Min Chen (chen@math.purdue.edu)