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In this paper we develop a leap-frog-type finite element method for modeling the electromagnetic wave propagation in metamaterials. The metamaterial model equations are represented by integrodifferential Maxwell’s equations, which are quite challenging to analyze and solve in that we have to solve a coupled problem with different partial differential equations given in different material regions. Our method is based on a mixed finite element method using edge elements, which can easily handle the tangential continuity of the electric field. Stability analysis and optimal error estimate are carried out for the proposed scheme. The scheme is implemented and confirmed to obey the proved optimal convergence rate by using a smooth analytical solution. Then the scheme is extended to model wave propagation in heterogeneous media composed of metamaterials and free space, and extensive numerical results (using a rectangular edge element, a triangular edge element, and mixed edge elements) demonstrate the effectiveness of our algorithm for modeling the exotic backward wave propagation phenomenon in metamaterials. To the best of our knowledge, this is the first paper with an exhaustive simulation of backward wave propagation in metamaterials using time-domain finite element methods.

This is the note for a series of lectures that the author gave at the Centre de Recerca Matem´atica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N= Norm (T), and W= N/T is the Weyl group. If w W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.