Multiwavelet based method of moments under discrete Sobolev-type norm

In this dissertation; the multiwavelet based method of moments (MBMM) under discrete Sobolev-type norm is developed. Owing to their higher order, compact support, symmetry/antisymmetry, continuity and smoothness, the multiscalet bases with a multiplicity of two demonstrate a robust ability in tracking unknown functions in electromagnetic (EM) integral equations. The discretization mesh in the MBMM is then much coarser than in the traditional method of moments (MoM) using pulse or triangle shape functions. By using the discrete Sobolev inner product, the MBMM changes the testing procedure in the MoM, thus changing the method of enforcing the boundary conditions for the integral equations. The new procedure samples the integral equations and their derivatives at observation points without numerical integrations, hence it is much easier to implement. This allows the MBMM to work efficiently for not only electrically large, but also medium and small problems. In contrast, some recently developed regular wavelet based algorithms and fast multipole methods are worth using only for electrically large problems due to the increased complexity of implementation. The accuracy of the new approach is guaranteed due to the extra constraint on the derivatives. Numerical examples of 2D scattering and guided-wave problems show the high convergence and precision of the new method. For 3D cases, the MBMM employs the directional derivative sampling in two orthogonal directions to realize the derivative tracking along all directions. The resulting nonsquare matrix equation is solved using the least square method (LSM). Therefore; the strengths appearing in the 2D cases are preserved in the 3D cases with a minor increase of the computational cost of the LSM. Numerical examples of 3D conducting sphere and rough surface scattering also demonstrate the high efficiency and easy implementation of the new algorithm.