Integral equation solvers for real world applications - some challenge problems

This paper presents some real world applications and problems for integral equation solvers (IES). Integral equation solvers are, in general, more complex to implement compared to differential equation solvers (DESs). This is due to the need for the Green's function method, which generally involves the evaluation of singular integrals. Moreover, due to the dense matrix system, acceleration solution methods have to be invoked before IESs are competitive with differential equation solvers. Also, linearity of the media has to be assumed before frequency-domain and Green's function techniques can be used. In contrast to DESs, the advantage of IESs, lies in the smaller number of unknowns and favorable scaling properties for memory and CPU requirements. DESs are simple to implement, but usually exhibit worse scaling properties when applied to surface scattering problems. The presence of grid-dispersion error worsens their scaling properties for large scale computing. On the other hand, DESs in the time domain can easily account for nonlinear phenomena. Hence, for an area replete with nonlinear physics, such as computational mechanics or computational fluid dynamics, DESs outrank integral equation solvers in popularity. The advantages of IESs in EM make them popular for solving a number of scattering problems. This is especially so when they have been accelerated with fast algorithms.