Mass-lumped high-order cell methods for the time-dependent Maxwell's equations

Abstract

Our goal is the efficient numerical approximation of solutions to the time-domain linear Max-well system. Methods like the classical finite-difference time-domain method rely on the approximation of the electric and the magnetic field on two interlaced (Cartesian) grids respectively. Our method expands this idea to general triangular/tetrahedral meshes by defining a dual grid using the barycentric subdivision of the primal mesh. The resulting primal and dual ements are both composed of the same set of quadrilateral (in 2d) and hexahedral (in 3d) cells. We use Lagrangian polynomial basis functions with respect to tensor product integration points on the unit square/cube which are mapped to the physical cells and suitably combined to obtain differing, element-wise conforming bases on the dual and primal mesh respectively. This approach provides generically computable, stable high order bases. Using lumped mass matrices with respect to the defining points of the basis functions leads to block diagonal mass matrices with block-size independent of the polynomial degree. Thus the combination of the spatial discretization with explicit time-stepping schemes yields an efficient method with very little memory requirement.