High order geometric methods with splines: fast solution with explicit time-stepping for Maxwell equations

We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the AmpereMaxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.