Property G and Large Interior Eigenproblems. |
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| Accurate eigensolutions of the frequency domain Maxwell's equations are important in accelerator simulations. Edge-based finite element discretizations lead to generalized eigenproblems, K x = lambda M x. Typically the eigenvalues of interest are tightly clustered. The Shift-Invert Lanczos (SIL) method is theoretically ideal for our eigenproblems. However, it requires accurate solutions of the shifted linear systems, which are large and ill-conditioned. We develop a hybrid method based on Inexact SIL and Inexact JOCC iterations. Its convergence properties are comparable to those of Exact SIL, and it is highly parallelizable. By studying the SIL systems, we have discovered an important property of certain linear systems and solvers: Property G. When Property G is present, reducing the residual by a modest amount gives a sufficiently accurate solution (far better than expected from the condition number). The software implementation confirms desirable parallel performance, and has already proved valuable to accelerator designs. |
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algweb@cerfacs.fr Last Update: Apr 4, 2003 |