Utilization of wavelet concepts in finite elements for an efficient solution of Maxwell's equations

The principles of dilation and shift are two important properties that are attributed to wavelets. It is shown that inclusion of such properties in the choice of a basis in Galerkin's method can lead to a slow growth of the condition number of the system matrix obtained from the discretization of the differential form of Maxwell's equations. It is shown that for one‐dimensional problems the system matrix can be diagonalized. For two‐dimensional problems, however, the system matrix can be made mostly diagonal. This paper illustrates the application of the new type of “dilated” basis for a Galerkin's method (or equivalent, for example, finite element method) for the efficient solution of waveguide problems. Typical numerical results are presented to illustrate the concepts.