Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2d time‐harmonic Maxwell's equations

Abstract

PurposeThis work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.Design/methodology/approachThe proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.FindingsThe approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L2 norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.Originality/valueThe work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.