# On Skew-symmetric Preconditioning for Strongly Non-symmetric Linear Systems

L.A. Krukier, Mikhail A. Bochev

To solve iteratively linear system $Au=b$ with large sparse strongly non-symmetric matrix $A$ we propose preconditioning $\hat A \hat u = \hat b$, $\hat A=(I+\tau L_1)^{-1} A (I+\tau U_1)^{-1},\; \tau>0$ where respectively lower and upper triangular matrices $L_1$ and $U_1$ are so that $L_1+U_1=1/2(A-A^*)$. Such preconditioning technique may be treated as a variant of ILU-factorization, and we call it MSSILU --- Modified Skew-Symmetric ILU. We investigate and optimize (with respect to $\tau$) convergence of preconditioned Richardson method (RM) of the following special form: ${\hat x}^{m+1}=(I-\tau \hat A){\hat x}^m+\tau {\hat b},\; m\geq 0$, where $\tau$ is the same as in $\hat A$. For this method we give an estimate for rate of convergence in relevant Euclidean norm for the case of positivereal matrix $A$. Numerical experiments have included solving linear systems arising from 5-point FD approximation of convection--diffusion equation with dominated convection by MSSILU+RM, MSSILU+GMRES(2) and MSSILU+GMRES(10).