Abstract
Multiscalets in the multiwavelet family are used as the basis and testing functions in Galerkin's method. Since the multiscalets are orthogonal to their translations under the Sobolev inner product, the resulting Galerkin's method behaves like a collocation method but possesses the ability of derivative tracking for unknown functions in solving integral equations. The former makes the method simple in implementation and the latter allows to use coarse meshes in discretization. These robust features have been demonstrated in solving two‐dimensional (2D) electromagnetic (EM) problems, but have not been exploited in three‐dimensional (3D) scenarios. For 3D problems, the unknown functions in the integral equations are dependent on two coordinate variables. In order to preserve the use of coarse meshes for 3D cases, we realize the omnidirectional derivative tracking by tracking the directional derivatives along two orthogonal directions, or equivalently tracking the gradient. This process yields a nonsquare matrix equation and we use the least‐squares method (LSM) to solve it. Numerical examples show that the multiscalet‐based Galerkin's method is also robust in solving for 3D EM integral equations with a minor cost increase from LSM. Copyright © 2007 John Wiley & Sons, Ltd.