An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

Abstract

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.