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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Ludmil Zikatanov, Pennsylvania State University
Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell's Equations


This work is on the numerical approximation of incoming solutions to Maxwell's equations with maximally dissipative boundary conditions, whose energy decays exponentially with time. We use the standard Nedelec-Raviart-Thomas elements and a Crank-Nicholson scheme to approximate such solutions. We prove that with divergence free initial conditions, the fully discrete approximation to the electric field is weakly divergence-free for all time. We show numerically that the finite-element solution approximates well the asymptotically disappearing solutions constructed analytically when the mesh size becomes small. This is a joint work with James Adler (Tufts University) and Vesselin Petkov (University of Bordeaux).