The concept of the M -decomposition was introduced by Cockburn et al. in Math. Comp. vol. 86 (2017), pp. 1609-1641 to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the M -decomposition to construct HDG methods for the Maxwell's equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an M -decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Unlike the elliptic case, we obtain a superconvergent rate for the curl of the solution, not the solution, and this is confirmed by our numerical experiments. of problems posed in two dimensions. To see how this is possible, consider the usual time harmonic Maxwell system for the electric field E (a complex valued vector function):
The rigorous coupled-wave approach was used to compute the plane-wave absorptance of a thin-film tandem solar cell with a metallic surface-relief grating as its back reflector. The absorptance is a function of the angle of incidence and the polarization state of incident light; the free-space wavelength; and the period, duty cycle, the corrugation height, and the shape of the unit cell of the surface-relief grating. The solar cell was assumed to be made of hydrogenated amorphous-silicon alloys and the back reflector of bulk aluminum. The incidence and the grating planes were taken to be identical. The AM1.5 solar irradiance spectrum was used for computations in the 400-1100 nm wavelength range. Inspection of parametric plots of the solar-spectrum-integrated (SSI) absorption efficiency and numerical optimization using the differential evolution algorithm were employed to determine the optimal surface-relief grating. For direct insolation, the SSI absorption efficiency is maximizable by appropriate choices of the period, the duty cycle, and the corrugation height, regardless of the shape of the corrugation in each unit cell of the grating. A similar conclusion also holds for diffuse insolation, but the maximum efficiency for diffuse insolation is about 20% smaller than for direct insolation. Although a tin-doped indium-oxide layer at the front and an aluminum-doped zinc-oxide layer between the semiconductor material and the backing metallic layer change the optimal depth of the periodic corrugations, the optimal period of the corrugations does not significantly change.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.