Scalable solutions to integral-equation and finite-element simulations

Abstract
When developing numerical methods, or applying them to the simulation and design of engineering components, it inevitably becomes necessary to examine the scaling of the method with a problem's electrical size. The scaling results from the original mathematical development; for example, a dense system of equations in the solution of integral equations, as well as the specific numerical implementation. Scaling of the numerical implementation depends upon many factors; for example, direct or iterative methods for solution of the linear system, as well as the computer architecture used in the simulation. In this paper, scalability will be divided into two components—scalability of the numerical algorithm specifically on parallel computer systems and algorithm or sequential scalability. The sequential imple- mentation and scaling is initially presented, with the parallel implementation following. This progression is meant to illustrate the differences in using current parallel platforms and sequential machines and the resulting savings. Time to solution (wall-clock time) for differing problem sizes are the key parameters plotted or tabulated. Sequential and parallel scalability of time harmonic surface integral equation forms and the finite-element solution to the partial differential equations are considered in detail.