# Meeting Details

Title: First-Order System LL*(FOSLL*) for Maxwell's equations in 3D with edge singularities Department of Mathematics Colloquium Thomas Manteuffel, University of Colorado Many important applications involve the solution of the eddy current approximation to Maxwell's equations. For example, mathematical models of magnetically confined plasma in tokomak geometry lead to magnetohydrodynamic equations (MHD) in which Maxwell's equations play a major role. Sandia's Z-pinch reactor provides another important application. In this talk we focus on a first-order system least-squares (FOSLS) formulation of the eddy current approximation to Maxwell's equations. The $L2$--norm version of FOSLS yields a least-squares functional whose bilinear part is $H1$--elliptic. This means that the minimization process amounts to solving a loosely coupled system of scalar elliptic 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 $C1$ subspace. The FOSLL$^*$ approach uses standard $H1$ 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.