Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

Abstract

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are (i) valid in the pre-asymptotic regime; (ii) solely depend on the geometry and the polynomial order; and (iii) are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order (p = 1, 2, 3, 4) hierarchic H (curl)-conforming polynomial basis functions.

Discontinuous Galerkin methods in nanophotonics

“…Busch, M. König, and J. Niegemann: Discontinuous Galerkin methods in nanophotonics It should be noted that in this article we want to solve the very same first-order system as in the time-domain. In particular, we will not derive a wave equation for one of the electromagnetic fields as, for example, done in [53]. Therefore, our approach will not lead to optimal performance, but to maximum consistency with an existing time-domain code.…”

“…Furthermore, we are challenged by the non-symmetric nature of the system matrix H 0 , which leads to issues regarding preconditioning and solving the related system of linear equations. Symmetric DG discretisations of the second-order wave equation, as discussed in [53], for example, seem preferable from an efficiency point of view.…”

“…Expressed in other terms, this means that the nodal basis contains superfluous degrees of freedom. To eliminate this redundant information one can construct a modal, vectorial basis which locally satisfies the divergence condition, i. e., in an element-wise fashion [53,54,71]. However, since the locally divergence-free basis functions are not invariant under affine transformations, such an implementation dramatically increases the memory consumption.…”

Free and Forced Vibration Analysis of Airtight Cylindrical Vessels with Doubly Curved Shells of Revolution by Using Jacobi-Ritz Method

“…The selection of an admissible displacement function is an important factor in guaranteeing a stable convergence and accuracy of the solution. In the handling of the continuity conditions, the use of spring stiffness, which can be seen as penalty parameters, makes the selection of admissible functions very flexible [37][38][39][40][41][42][43][44][45][46][47]. Subscript is omitted here for the sake of brevity.…”

“…From the mathematical point of view, unlike the domain decomposition method which uses the Lagrange multipliers and the weight residual least squares method [35], when the penalty parameters is defined as a very high value the solution of the proposed method may not converge [38][39][40]. And it is necessary to emphasize that the research will focus on the elastic deformation rather than rigid deformation.…”