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A matrix-free Discontinuous Galerkin method for the time dependent Maxwell equations in unbounded domains

A Discontinuous Galerkin (DG) Finite Element Method (FEM) approach for the 3D time dependent Maxwell equations in unbounded domains is presented. The method (implemented in the FEM library NGsolve) is based on the covariant transformation of a modal orthogonal polynomial basis, originally defined on a reference simplex. The approach leads to an explicit time stepping scheme for which the mass matrix to be inverted is at most $d\!\times\!d$ block-diagonal (in $d\!=\!2,3$ spatial dimensions) while the matrix which discretizes the curl operators on the right-hand side of the system is a small reference matrix, independent from geometric properties of mesh elements. Furthermore, we show that the introduced optimizations are preserved when unbounded domains are also included in the formulation through a complex-stretching based approach.