and T.Warburton. High-order/spectral methods on unstructured grids. I. Time-domain solution of Maxwell’s equations

Abstract. We present an ab initio development of a convergent high-order accurate scheme for the solution of linear conservation laws in geometrically complex domains. As our main example we present a detailed development and analysis of a scheme suitable 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 dened on the triangles and tetrahedra. Careful choices of the unstructured nodal grid points ensure high-order/spectral accuracy, while the equations themselves are satised in a discontinuous Galerkin form with the boundary conditions being enforced weakly through a penalty term. Accuracy, stability, and convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Concerns related to eÆcient implementations are discussed in detail. This sets the stage for the presentation of examples, verifying the theoretical results, as well as illustrating the versatility, exibility, and robustness when solving two- and three-dimensional benchmarks in computa-tional electromagnetics. Pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated. Subject classication. Applied Mathematic