SPARSE-MATRIX APPROXIMATION TO AN INTEGRAL-EQUATION OF SCATTERING

An integral equation formulation of scattering of waves obeying Helmholtz's equation is considered in two dimensions, along with a standard discretization that generates a full matrix equation. A transformation involving banded matrices is introduced that (for most large problems) replaces this full matrix by one which, although full, has nearly all of its elements reduced in size by many orders of magnitude. Sample calculations show that in many cases these small elements may be approximated by zero with little loss in accuracy, thereby generating a sparse matrix equation. This transformation is effective whenever much of the scatter(s) is somewhat smooth, although isolated discontinuities are allowed. It is speculated that with minor changes this transformation will also be effective for three-dimensional problems and for a variety of integral equations of wave motion.