Nodal high-order methods on unstructured grids

We present a convergent high-order accurate scheme for the solution of linear conservation laws in geometrically complex domains. As our main example we include a detailed development and analysis of a scheme for the time-domain solution of Maxwell's equations in a three-dimensional domain. The fully unstructured spatial discretization is made possible by the use of a high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles and tetrahedra, while the equations themselves are satisfied in a discontinuous Galerkin form with the boundary conditions being enforced weakly through a penalty term. Accuracy, stability, and convergence of the semidiscrete approximation to Maxwell's equations is established rigorously and bounds on the growth of the global divergence error are provided. Concerns related to efficient implementations are discussed in detail. This sets the stage for the presentation of examples, verifying the theoretical results, and illustrating the versatility, flexibility, and robustness when solving two- and three-dimensional benchmark problems in computational electromagnetics. Pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated.