Deflation Techniques for Computational Electromagnetism: Theoretical Considerations

This paper describes deflation of finite element (FE) matrices for electromagnetic fields. The condition of the FE matrices is improved by the matrix deflation which replaces small eigenvalues with zeros. It is known that convergence of linear solvers for FE equations can be improved by using the AV method as well as explicit and implicit error correction (EC) methods which have been derived from the multigrid method. These numerical results are theorized on the basis of the matrix deflation. In particular, augmented matrices appeared in the AV and implicit EC methods are shown to have good conditioning after preconditioning. These results suggest that the above methods are based on a common mathematical principle.