Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria

doi:10.1090/s0025-5718-2012-02627-5

Cited by 70 publications

(70 citation statements)

References 44 publications

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“…Bloch waves propagating in various directions are a particular example of Trefftz functions. While mathematical issues related to convergence of Trefftz approximations are quite technical [20][21][22], there is ample evidence in the literature that the approximation errors are reasonably low even for bases of small size [20,21,23,24], as long as the illumination conditions are within a certain range. For example, consider any Maxwellian field that results from illumination of a composite slab by an arbitrary superposition of plane waves with any incidence angles in the range [−θ max , θ max ].…”

Section: Non-asymptotic (Trefftz) Homogenizationmentioning

confidence: 99%

A non-asymptotic homogenization theory for periodic electromagnetic structures

Tsukerman

Markel

2014

Proc. R. Soc. A.

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Homogenization of electromagnetic periodic composites is treated as a two-scale problem and solved by approximating the fields on both scales with eigenmodes that satisfy Maxwell's equations and boundary conditions as accurately as possible. Built into this homogenization methodology is an error indicator whose value characterizes the accuracy of homogenization. The proposed theory allows one to define not only bulk, but also position-dependent material parameters (e.g. in proximity to a physical boundary) and to quantify the trade-off between the accuracy of homogenization and its range of applicability to various illumination conditions.

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Section: Non-asymptotic (Trefftz) Homogenizationmentioning

confidence: 99%

A non-asymptotic homogenization theory for periodic electromagnetic structures

Tsukerman

Markel

2014

Proc. R. Soc. A.

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“…The benefit of TD-DGMs over time-domain finite-element methods (FEM) is that DGM implementations result in a global mass matrix that is locally invertible, thereby facilitating the implementation of explicit time-stepping schemes [1,4,5]. More recently, DGM formulations have been applied to elliptic partial differential equations (PDEs) [6][7][8][9][10][11][12][13][14], including the time-harmonic Maxwell's equations. In the frequency domain, both DGM and FEM formulations result in large sparse systems of equations that must be solved to determine the electromagnetic fields of interest.…”

Section: Introductionmentioning

confidence: 99%

“…Existing work focuses on mathematical issues such as convergence [12][13][14]. To the best of our knowledge, no existing work presents DGM formulations for the time-harmonic electromagnetic vector wave equations with both electric and magnetic sources and locally varying constitutive parameters.…”

Section: Introductionmentioning

confidence: 99%

The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications

Jeffrey¹,

Geddert²,

Brown³

et al. 2015

PIER

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Abstract-Novel microwave imaging systems require flexible forward solvers capable of incorporating arbitrary boundary conditions and inhomogeneous background constitutive parameters. In this work we focus on the implementation of a time-harmonic Discontinuous Galerkin Method (DGM) forward solver with a number of features that aim to benefit tomographic microwave imaging algorithms: locally varying high-order polynomial field expansions, locally varying high-order representations of the complex constitutive parameters, and exact radiating boundary conditions. The DGM formulated directly from Maxwell's curl equations facilitates including both electric and magnetic contrast functions, the latter being important when considering quantitative imaging with magnetic contrast agents. To improve forward solver performance we formulate the DGM for time-harmonic electric and magnetic vector wave equations driven by both electric and magnetic sources. Sufficient implementation details are provided to permit existing DGM codes based on nodal expansions of Maxwell's curl equations to be converted to the wave equation formulations. Results are shown to validate the DGM forward solver framework for transverse magnetic problems that might typically be found in tomographic imaging systems, illustrating how high-order expansions of the constitutive parameters can be used to improve forward solver performance.

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“…Furthermore, Hiptmair et al [13,14,26] have proved error estimates for the more general plane wave based discontinuous Galerkin method (DGM) of the Helmholtz equation [13,26] and Maxwell's equations [14], results that are also applicable to the UWVF. The error estimates in [13,14,26] are derived using the approximation properties of plane waves. Recently, Keywords and phrases.…”

Section: Introductionmentioning

confidence: 99%

“…Moiola [24,25] has proved best approximation estimates for plane wave approximation in linear elasticity using the approximation properties of the plane wave basis functions in acoustics [13] and electromagnetism [14]. Approximation properties of plane waves are also investigated for the partition unity finite element method (PUFEM) [23], and the least-square and collocation formulations [29].…”

Section: Introductionmentioning

confidence: 99%

Error estimates for the ultra weak variational formulation in linear elasticity

2012

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Abstract. We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier's equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L 2 (Ω) norm in terms of the best approximation error. Our final result is an L 2 (Ω) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.Mathematics Subject Classification. 65N15, 65N30, 74J05, 74S30.

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Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

Cited by 70 publications

References 44 publications

A non-asymptotic homogenization theory for periodic electromagnetic structures

A non-asymptotic homogenization theory for periodic electromagnetic structures

The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications

Error estimates for the ultra weak variational formulation in linear elasticity

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