Applied Mathematics Colloquium: Dr. Lise-Marie Imbert-Gerard

Title: Variable coefficients and numerical methods for electromagnetic waves

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.

In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.