UTA Department of Mathematics

Special Mathematics Colloquium

Date/Time/Room: Monday (4/22/2002) at 1:30pm in 487 Pickard Hall

Speaker: Pavel Bochev, Computational Mathematics and Algorithms Department, Sandia National Laboratories

``Computational Electromagnetics and Differential Complexes''

Abstract: In 1966 Kane Yee from Lawrence-Livermore Lab proposed a finite difference scheme for the time-domain (FDTD) Maxwell's equations that quickly became a standard in computational electromagnetics (CEM). Almost three decades later, the field of computational electromagnetics abounds with discrete models of the Maxwell's equations based on finite-difference, finite volume, and finite element paradigms. Remarkably, successful CEM schemes share a common trait: on structured, Cartesian grids they all bear a striking similarity with the original Yee FDTD scheme! In contrast, attempts to extend approaches that worked well for the Laplace equation encountered annoying non-physical solutions (spurious modes) and less than perfect physical fidelity.

It turns out that the causes for the unprecedented success of some CEM schemes and the reasons for the failures of others can all be traced to the geometry of the Maxwell's equations that fits into the De Rham differential complex. On a more intuitive level, this means that the paradigmic operator for three-dimensional EM is the curl-curl rather than the div-grad (Laplace) operator.

We will show that successful schemes, such as Yee's original FDTD, FIT (Finite Integration Technique), Co-volume and edge element methods, all generate a discrete version of the De Rham complex which guarantees a physically correct approximation of the curl-curl operator. The schemes that fail to capture the structure of this complex lead in turn to curl-curl discretizations plagued by spurious modes.

To illustrate the importance of the De Rham structure in computational electromagnetics we will compare side by side two time domain simulations of magnetic diffusion. We will show how a node-centered discretization of a vector potential formulation leads to a severely distorted, unphysical solution, while an edge-face finite element method provides an excellent physical fidelity.

We will close by examining examples of differential complexes related to the Laplace and the Stokes equations which hint at new intriguing possibilities for robust and physically correct numerical methods.