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

摘要

To solve iteratively linear system $Au=b$ with large sparse strongly non-symmetric matrix $A$ we propose preconditioning $hat A hat u = hat b$, $hat A=(I+au L_1)^{-1} A (I+au U_1)^{-1},; au>0$ 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. udWe investigate and optimize (with respect to $au$) convergence of preconditioned Richardson method (RM) of the following special form: ${hat x}^{m+1}=(I-au hat A){hat x}^m+au {hat b},; mgeq 0$, where $au $ 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 $A$. udNumerical 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).ud