CS Distinguished Lecture

SPEAKER: Tom Manteuffel, University of Colorado at Boulder

TITLE: First-Order System LL* (FOSLL*) for Maxwell's Equations in 3D with Edge Singularities

DATE: Monday, April 30, 2007
TIME: 4:00 P.M.
PLACE: 1404 Siebel Center
201 N. Goodwin Ave.

ABSTRACT

The L²-norm version of first--order system least squares (FOSLS) attempts to reformulate a given system of partial differential equations as a first-order system so that applying an L-norm least-squares principle yields a functional whose bilinear part is H¹-elliptic. This means that the minimization process amounts to solving a loosely coupled system of elliptic scalar equations. An unfortunate limitation of the L²-norm FOSLS approach is that this product H¹ equivalence generally requires sufficient smoothness of the original problem. Inverse-norm FOSLS overcomes this limitation, but at a substantial loss of real efficiency. The FOSLL* approach described in this talk is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations.

This talk provides a theoretical foundation for the FOSLL* methodology and application to the eddy current form of Maxwell's equations. It is shown that singularities due to discontinuous coefficients are easily treated. However, singularities due to reentrant edges require a further modification. A partially weighted norm is used only on the slack equations. The solution retains optimal order accuracy and the resulting linear systems are easily solved by multigrid methods.

Comparison is made to the curl/curl formulation, which requires Nedelec finite elements, and the weighted regularization approach, which requires a finite element space with a C subspace. The FOSLL* approach uses standard H conforming finite element spaces, is shown to have equal or better accuracy, obtained at a smaller cost. Numerical examples are presented that support the theory.