Use of Coifman intervallic wavelets in 2-D and 3-D scattering problems

The method of moments (MOM) has been used to solve antenna and scattering problems for several decades, due both to its flexibility in handling complex geometric structures and to its numerical accuracy. However, for electrically large problems, the MOM often becomes incapable of achieving solutions due to its requirements for vast amounts of local memory and processor cycles. To overcome this difficulty, orthonormal wavelets have been introduced, which create very sparse moment matrices that can be evaluated by iterative techniques. Nevertheless, the traditional orthonormal wavelets have demonstrated several limitations. The use of intervallic wavelets is presented; they form an orthonormal basis and preserve the same multi-resolution analysis as other unbounded wavelets. In contrast to periodic wavelets, endpoint values are not restricted if the unknown function is expanded in terms of intervallic wavelets. Very sparse impedance matrices have been obtained with this method. Zero elements of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. The majority of matrix elements are evaluated directly, without performing numerical integration procedures such as Gaussian quadrature. The construction of intervallic wavelets is presented. Numerical examples of 2-D and 3-D scattering problems are discussed, and the relative error of this method is studied analytically.