GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Scaling Matrices to Prescribed Row and Column Maxima
SIAM Journal on Matrix Analysis and Applications
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Some New Bounds on the Condition Numbers of Optimally Scaled Matrices
Journal of the ACM (JACM)
Journal of Computational Physics
SIAM Journal on Scientific Computing
Fine tuning interface relaxation methods for elliptic differential equations
Applied Numerical Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
ACM Transactions on Mathematical Software (TOMS)
A non-standard finite element method based on boundary integral operators
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
Linear systems with large differences between the coefficients, called ''discontinuous coefficients'', often arise when physical phenomena in heterogeneous media are modeled by partial differential equations (PDEs). Such problems are usually solved by domain decomposition techniques, but these can be difficult to implement when subdomain boundaries are complicated or the grid is unstructured. It is known that for such systems, diagonal scaling can sometimes improve the eigenvalue distribution and the convergence properties of some algorithm/preconditioner combinations. However, there seems to be no study outlining both the usefulness and limitations of this approach. It is shown that L"2-scaling of the equations is a generally useful preconditioner for such problems when the system matrices are nonsymmetric, but only when the off-diagonal elements are small to moderate. Tests were carried out on several nonsymmetric linear systems with discontinuous coefficients derived from convection-diffusion elliptic PDEs with small to moderate convection terms. It is shown that L"2-scaling improved the eigenvalue distribution of the system matrix by reducing their concentration around the origin very significantly. Furthermore, such scaling improved the convergence properties of restarted GMRES and Bi-CGSTAB, with and without the ILU(0) preconditioner. Since ILU(0) is theoretically oblivious to diagonal scaling, these results indicate that L"2-scaling also improves the runtime numerical stability.