A 3D curvilinear discontinuous Galerkin time-domain solver for nanoscale light–matter interactions

Abstract

International audienc

High order HDG method and domain decomposition solvers for frequency‐domain electromagnetics

“…The corresponding mesh consists of 1 645 874 elements and 3 521 251 faces. We, however, note that taking full benefit of higher order interpolation would also require an appropriate treatment of curved boundaries as it has been demonstrated in Li et al 5 for a similar HDG method in the two-dimensional case, or in Viquerat and Scheid 19 in the framework of a DG method for the three-dimensional time-domain Maxwell equations. In Figure 4, we show physical solutions obtained with the HDG-P method for increasing values of the polynomial interpolation order .…”

Simulation of three-dimensional nanoscale light interaction with spatially dispersive metals using a high order curvilinear DGTD method

“…Furthermore, the explicit time integration of the obtained semi-discrete scheme is now performed by a Low Storage Runge-Kutta (LSRK) scheme. Finally, we extend the use of curvilinear elements from a previous work [33] in order to guarantee the high order nature of our scheme even if complex geometries are of concern. 9 3.1.…”

“…Although this approach presents the advantage of proposing simple transformations of the FE matrices from the reference element to the physical element, in the case of high-order methods as depicted in Figure 2, it represents a serious hindrance, since it limits the accuracy of the spatial discretization to second order. In [33], we proposed an implementation of curvilinear (isoparametric) elements in the DGTD framework for Maxwell's equations, along with local dispersion models for metals. In this case, no additional work was required for local dispersion models, since they only consist of additional ODEs to the Maxwell PDE system.…”

“…The implementation of curvilinear elements in the DG framework requires the numerical integration and storage of the FE matrices on the curvilinear cells only (the method for linear cells remaining the same). As presented in [33], the fully-discrete scheme can be written in terms of (i) the mass matrix ( (U * i φ ik ) · n (with its analoguous scalar counterpart) in addition to those appearing in Maxwell's equations…”

“…This work must be done for each curvilinear face, leading to 9 integrals per face for the Maxwell terms, and to 4 additional integrals per face for the hydrodynamic terms. The remaining of the method is identical to what is described in [33], and will not be detailed here.…”