Error estimation of wavelet based modeling of electromagnetic waves in waveguides and resonators

Wavelets provide an effective toolbox for solving differential equations by representing the continuous functions by their wavelet expansion coefficients and the corresponding differential equations by discrete matrix equations. The wavelet basis functions are organized into resolution levels of different frequency terms at different locations, and the main advantage of the wavelet expansion representation is that the resolution level can be different at different locations, if the solution function contains higher frequency terms in one place and restricted to lower frequencies at other places. Wavelet based differential equation solving methods can be adaptive, it is possible to refine the solution locally, if the precision is not sufficient at some regions. In the present work a simple method for estimating the next resolution level wavelet coefficients is presented. Predicting the approximate value of these coefficients makes it possible to select the minimal set of wavelet basis functions for the next resolution level solution in a computationally economic way, or in the last resolution levels it can substitute the next level solution of the matrix equation.