Conservation Properties of the Discontinuous Galerkin Method for Maxwell Equations

Gjonaj, Erion; Lau, Thomas; Weiland, Thomas

doi:10.1109/iceaa.2007.4387311

Cited by 8 publications

(7 citation statements)

References 6 publications

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“…Thus, spurious numerical solutions exist which are not compatible with (4). This statement is, in fact, part of a general result obtained by Gjonaj et al [2007] showing that any DG-FEM will violate charge conservation independently from the type of the approximation spaces used in the formulation.…”

Section: Dg-femmentioning

confidence: 63%

“…A major issue related with DG-FEM, however, is the unphysical charge distribution induced by discretization which gives rise to spurious numerical solutions. In the work by Gjonaj et al [2007] it was shown on topological grounds that, except for tensor product formulations on Cartesian grids, any other DG-FEM will violate charge conservation independently from the type of the approximation spaces used. The appearance of unphysical solutions related to the violation of charge conservation has consequences on both the numerical accuracy and on the stability of time domain simulations.…”

Section: Introductionmentioning

confidence: 99%

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A projection penalization approach for the high‐order DG‐FEM in the time domain

Gjonaj

Weiland

2011

Radio Science

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[1] We investigate the Discontinuous Galerkin-Finite Element Method (DG-FEM) for the solution of Maxwell's equations in the time domain. The charge conservation laws are known to be violated by this method which consequently gives rise to spurious numerical solutions. It is, however, possible to introduce additional constrains in the formulation which impose charge conservation either exactly or approximately. In this work, a new approach based on a topological orthogonal projector into a high-order H(curl)-conforming approximation space is proposed. Using this approach we derive a constrained DG-FEM formulation which is strictly free of spurious modes. Furthermore, we construct a time domain penalization scheme which allows to separate the spurious modes related to unphysical charges from the physical solutions while maintaining accuracy and energy conservation.Citation: Gjonaj, E., and T. Weiland (2011), A projection penalization approach for the high-order DG-FEM in the time domain, Radio Sci., 46, RS0E10,

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Section: Dg-femmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

A projection penalization approach for the high‐order DG‐FEM in the time domain

Gjonaj

Weiland

2011

Radio Science

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“…There exists a large amount of literature on algorithms for solving this problem, coming as well from the mathematics as from the physics community, and a recent survey is available in [19]. However, despite some insightful studies on the proper discrete Gauss laws that should be preserved by a DG-PIC scheme [36,48], we believe that the problem of designing a stable charge-conserving coupling for non-conforming solvers remained an open one. In this context the non-conforming methods obtained with our structure-preserving approach may be seen as general and satisfactory solution.…”

Section: Compatible Maxwell Solvers With Particles Imentioning

confidence: 99%

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

Pinto¹,

Sonnendrücker²

2017

The SMAI Journal of Computational Mathematics

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Abstract. This article is the first of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific chargeconserving schemes are provided for the 2D case. In this article we study two schemes which include a strong discretization of the Ampere law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by a Raviart-Thomas finite element interpolation for the current source, thanks to its commuting diagram properties. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.Math. classification. 35Q61, 65M12, 65M60, 65M75.

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“…However, DG formulations exhibit spurious mode solutions due to violation of charge (and possibly energy) conservation [6].…”

Section: Introductionmentioning

confidence: 99%

GPU Accelerated Time-Domain Discrete Geometric Approach Method for Maxwell’s Equations on Tetrahedral Grids

et al. 2018

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A recently introduced time domain method for the numerical solution of Maxwell's equations on unstructured grids is reformulated in a novel way, with the aim of implementation on Graphical Processing Units (GPUs). Numerical tests show that the GPU implementation of the resulting scheme yields correct results, while also offering an order of magnitude in speedup and still preserving all the main properties of the original Finite Difference Time Domain (FDTD) algorithm.

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Conservation Properties of the Discontinuous Galerkin Method for Maxwell Equations

Cited by 8 publications

References 6 publications

A projection penalization approach for the high‐order DG‐FEM in the time domain

A projection penalization approach for the high‐order DG‐FEM in the time domain

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

GPU Accelerated Time-Domain Discrete Geometric Approach Method for Maxwell’s Equations on Tetrahedral Grids

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