In this talk, we discuss the construction and analysis of higher-order time integration schemes for the full discretization of linear Maxwell equations on locally refined spatial grids with discontinuous Galerkin methods. Roughly speaking, we decompose the spatial domain into two subdomains, which we refer to as stiff and nonstiff subdomains, respectively. In the stiff subdomain, we gather all mesh elements which have a small diameter or a small material parameter (leading to stiff degrees of freedom). All other mesh elements are assigned to the nonstiff subdomain. We assume that the degrees of freedom in the stiff subdomain is considerably smaller than in the nonstiff subdomain.

The locally implicit scheme is based on a higher-order implicit method, e.g., an algebraically stable Runge–Kutta method like a Gauss collocation method. Our main contribution is to propose a preconditioned Krylov subspace method for solving the linear systems arising in each time step. The preconditioner is designed in such a way that its convergence only depends on the nonstiff subdomain but not on the stiff one. We will sketch a proof of this result using approximation theory in the complex plane using Faber polynomials. Finally, we verify our theoretical findings by numerical experiments.

This approach is applicable to any implicit scheme and also works for exponential integrators, where the action of the matrix exponential is approximated by rational Krylov subspace methods. It is even applicable to nonlinear problems, where such linear systems arise within the Newton iterations.

This is joint work with Jonas Köhler and Pratik Kumbhar