High-performance adaptive finite element solution of nonlinear multiscale Maxwell equations

Nonlinear multiscale Maxwell equations arise in applications where the linear susceptibility assumption does not hold and the material contains tiny space inhomogeneities (e.g. cracks, wires, narrow slots) that are difficult to deal with because of their size. Applications include integrated optics (photonic cristals with defects, soliton propagation) and geophysics (measurements of oil-production fields, flaw detection in oil-and-gas winning equipment). Nonlinearities are also frequently caused by frequency dispersion and high-intensity signals. When solving the Maxwell equations numerically, the finite element method is an excellent tool to handle complicated geometries, inhomogeneities, material nonlinearity and frequency dispersion. The so-called Whitney (or edge) finite elements are intrinsically well suited for the Maxwell equations, because they automatically satisfy important properties of the equations, such as zero divergence of the magnetic flux and the proper treatment of discontinuities in the electric and magnetic fields at material interfaces. The hp-adaptivity (which is crucial for achieving a high computational efficiency for problems with localized structures and singularities) is, however, not straightforward for Whitney elements, and is presently an active research area. The aim of this project is to develop new efficient adaptive finite element techniques for the numerical solution of the nonlinear multiscale Maxwell equations. A complementary approach will be taken where the Dutch research team will focus on discontinuous Galerkin (DG) vector finite element methods suitable for hp-adaptation and the Russian team will concentrate on a new vector finite element method which can handle subscale inhomogeneities with the help of the finite superelement approach. The combined DG-vector finite element method will be then extended to the solution of the nonlinear Maxwell equations within the inexact Newton-Krylov method framework. The new finite element discretizations will result in matrices whose properties are not well studied and special preconditioning techniques for the solution of linear systems and eigenproblems, based on the multigrid approach, will be designed. Keywords: finite elements, finite superelement method, discontinuous Galerkin method, edge finite elements, hp-adaptivity, Krylov subspace methods, multigrid preconditioners, high-performance parallel computing.