Locally divergence-free discontinuous Galerkin methods for the Maxwell equations

By Bernardo Cockburn, Fengyan Li, Chi-Wang Shu. Journal of Computational Physics, 2004

318 citations recorded by the source corpus.

In this paper, we develop the locally divergence-free discontinuous Galerkin method for numerically solving the Maxwell equations. The distinctive feature of the method is the use of approximate solutions that are exactly divergence-free inside each element. As a consequence, this method has a smaller computational cost than that of the discontinuous Galerkin method with standard piecewise polynomial spaces. We show that, in spite of this fact, it produces approximations of the same accuracy. We also show that this method is more efficient than the discontinuous Galerkin method using globally divergence-free piecewise polynomial bases. Finally, a post-processing technique is used to recover (2k + 1)th order of accuracy when piecewise polynomials of degree k are used.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/649c8d0b686262fc38bf955d6f14ea559223e79c

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Citations in the corpus

High Order Strong Stability Preserving Time Discretizations (2009, background): “…[2], radiation hydrodynamics [64], two-species plasma flow [54], atmospheric transport [10], large-eddy simulation [69], Maxwell's equations [11], semiconductor devices [7], lithotripsy [88], geometrical optics [12], Schrodinger equations [9] [43], and combined with a range of spatial…”
Discontinuous Galerkin methods in nanophotonics (2011, methodology): “To reduce the number of spurious modes, one could use a locally divergence-free basis on each element [78].”
An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling (2010, methodology): “It has been applied to a wide range of applications such as electromagnetics (Cockburn et al. 2004), aeroacoustics (Toulopoulos & Ekaterinaris 2006) and plasma physics (Jacobs & Hesthaven 2006), just to cite a few examples.”
Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions (2004, unknown): “There has some recent work on using divergence free basis function in each individual elements, thus eliminating one of the sources of the divergence error [22].”
Time-Domain Finite-Difference and Finite-Element Methods for Maxwell Equations in Complex Media (2008, methodology): “Galerkin method [37], the finite integration technique (FIT)”
Convergence and stability of a discontinuous galerkin time-domain method for the 3D heterogeneous maxwell equations on unstructured meshes (2005, background): “Much work has recently been done in that direction, including divergence-free basis functions [2, 6].”
A unified approach for a posteriori high-order curved mesh generation using solid mechanics (2016, methodology): “In particular, the high-order discontinuous Galerkin method has become popular in this area due to its ability to propagate waves over long periods of time with a reduced computational cost compared to alternative low-order methods [13,15,38,40,42].”