Hybridized DG methods

This study is conducted in collaboration with Jay Gopalakrishnan (Department of Mathematics and Statistics, Portland State University, USA), Liang Li (School of Mathematical Sciences, University of Electronic Science and Technology of China), Nicole Olivares (Department of Mathematics and Statistics, Portland State University, USA) and Ronan Perrussel (Laplace Laboratory, UMR CNRS 5213).

Discontinuous Galerkin (DG) methods have been extensively considered for obtaining approximate solution of Maxwell's equations. Thanks to the discontinuity of the approximation, this kind of methods has many advantages, such as adaptivity to complex geometries through the use of unstructured possibly non-conforming meshes, easily obtained high order accuracy, hp-adaptivity and natural parallelism. However, despite these advantages, DG methods have one main drawback particularly sensitive for (psuedo)-stationary problems: the number of globally coupled degrees of freedom (DOFs) is much greater than the number of DOFs required by conforming finite element methods for the same accuracy. Consequently, DG methods are expensive in terms of both CPU time and memory consumption, especially for time-harmonic problems. Hybridization of DG methods is devoted to address this issue while keeping all the advantages of DG methods. The design of such a hybridizable discontinuous Galerkin (HDG) method for the discretization of the system of 3d time-harmonic Maxwell's equations is considered here.

HDG methods introduce an additional hybrid variable on the faces of the elements, on which the definition of the local (element-wise) solutions is based. A so-called conservativity condition is imposed on the numerical trace, whose definition involved the hybrid variable, at the interface between neighboring elements. As a result, HDG methods produce a linear system in terms of the DOFs of the additional hybrid variable only. In this way, the number of globally coupled DOFs is reduced. The local values of the electromagnetic fields can be obtained by solving local problems element-by-element. In this work, for the system of 3d time-harmonic Maxwell's equations, we propose a HDG formulation taking the tangential component of the magnetic field as the hybrid variable. We show that the reduced system of the hybrid variable has a wave-equation-like characterization, and the tangential components of the numerical traces for both electric and magnetic fields are single-valued. Moreover, numerical results seems to indicate that the approximate solutions for both E and H have optimal convergence orders.

Propagation of a plane wave in vacuum (2D case) : asymptotic convergence with non-uniform triangular meshes.

Related publications

L. Li, S. Lanteri and R. Perrussel
Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2d time-harmonic Maxwell’s equations
COMPEL, Vol. 2, No. 3, pp. 1112-1138 (2013)
Available as INRIA RR-7649 on Hyper Article Online

L. Li, S. Lanteri and R. Perrussel
A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equations
J. Comput. Phys., Vol. 256, pp. 563–581 (2014)
Available as INRIA RR-8251 on Hyper Article Online

L. Li, S. Lanteri and R. Perrussel
A class of locally well-posed hybridizable discontinuous Galerkin methods for the solution of time-harmonic Maxwell's equations
Comput. Phys. Comm., Vol. 192, pp. 23-31 (2015)

J. Gopalakrishnan, S. Lanteri, N. Olivares and R. Perrussel
Stabilization in relation to wavenumber in HDG methods
Available on arXiv (2015)