The Origin of Spurious Solutions in Computational Electromagnetics

Jiang, Bo‐Nan; Wu, Jie; Povinelli, Louis A.

doi:10.1006/jcph.1996.0082

Cited by 159 publications

(147 citation statements)

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“…This paper is about the application of the second-order formulation of Maxwell's equations proposed by Jiang et al [32] to the solution of the full Vlasov-Maxwell system.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, it is incorrect to consider the divergence equations redundant, on the basis that the divergence conditions will be satisfied if they are initially, as is often done [24]. It has been shown that ignoring the divergence conditions leads to incorrect solutions of Maxwell's equations [32,50].…”

Section: Introductionmentioning

confidence: 99%

“…The secondorder Maxwell's equations are derived from the first-order equations by applying the curl operator: The equations so obtained admit more solutions than do their progenitors. The spurious solutions, in particular, do not satisfy the divergence laws [32]. In this context, it is shown that boundary conditions play a crucial role in satisfying the constraints on the solution of Maxwell's equations [1,32].…”

Section: Introductionmentioning

confidence: 99%

“…In this context, it is shown that boundary conditions play a crucial role in satisfying the constraints on the solution of Maxwell's equations [1,32]. Jiang et al show that if one satisfies Gauss's law on a part of the boundary at all times, the solution of Faraday's and Ampère's laws will satisfy Gauss's law throughout the volume [32].…”

Section: Introductionmentioning

confidence: 99%

“…Within this context, in their paper Jiang et al describe a node-based least-squares finite element method able to achieve the satisfaction of the divergence conditions [32].…”

Section: Introductionmentioning

confidence: 99%

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A Simplified Implicit Maxwell Solver

Ricci

Lapenta

Brackbill

2002

Journal of Computational Physics

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We apply the second-order formulation of Maxwell's equations proposed by Jiang et al. (1996, J. Comput. Phys. 125, 104) to the solution of the implicit formulation of the three-dimensional, time-dependent Vlasov-Maxwell's system. An implicit finite difference algorithm is developed to solve the Maxwell's equations in a bounded domain with physical boundary conditions comprising electrically conducting walls (perfect conductors) and constant magnetic flux walls. We formulate the boundary conditions for Maxwell's equations to satisfy Poisson's equation throughout the domain by solving it only on the boundary. This eliminates the need for a separate projection step. We compare numerical results with analytical solutions for electromagnetic waves in vacuo, and using the implicit particle-in-cell code CELESTE3D, we test the new solver on the geospace environment modeling magnetic reconnection challenge problem. c 2002 Elsevier Science (USA)

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“…This paper is about the application of the second-order formulation of Maxwell's equations proposed by Jiang et al [32] to the solution of the full Vlasov-Maxwell system.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Within this context, in their paper Jiang et al describe a node-based least-squares finite element method able to achieve the satisfaction of the divergence conditions [32].…”

Section: Introductionmentioning

confidence: 99%

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A Simplified Implicit Maxwell Solver

Ricci

Lapenta

Brackbill

2002

Journal of Computational Physics

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Improved vector FEM solutions of Maxwell's equations using grid pre-conditioning

White

Rodrigue²

1997

Int. J. Numer. Meth. Engng.

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The Time Domain Vector Finite Element Method is a promising new approach for solving Maxwell's equations on unstructured triangular grids. This method is sensitive to the quality, or condition, of the grid. In this study grid pre-conditioning techniques, such as edge swapping, Laplacian smoothing, and energy minimization, are shown to improve the accuracy of the solution and also reduce the overall computational effort. approximation can give poor results. Nevertheless the FDTD is extremely efficient and it is often used as a benchmark to which new methods are compared.Whereas FDTD methods are defined on Cartesian grids, Finite Element Methods (FEM) are designed to solve partial differential equations on unstructured grids. Typically curved boundaries are approximated as piecewise linear, and an unstructured mesh is used within each region. The classic FEM using nodal elements has been quite successful in solving static electromagnetic problems where the continuous electrostatic potential can be employed. -Historically the use of nodal finite elements has been less successful for solving for the electric and/or magnetic fields directly. The use of nodal elements for solving frequency domain Maxwell's equations can lead to spurious modes, or numerical solutions that do not satisfy the divergence properties of the fields. Inclusion of divergence conditions into the variational problem can reduce these spurious modes, this is an area of current research. Time domain finite element methods may have similar difficulties with spurious modes. If the divergence conditions are neglected, then the divergence of the fields may grow with time, even if the source terms are divergence free. In this case the method does not conserve charge, and is not 'divergence preserving'. In addition nodal finite element methods are not appropriate for inhomogeneous volumes because the electric and magnetic fields are not continuous across a material interface, and it is difficult to correctly model this discontinuity using nodal elements.Recently developed vector elements, also known as edge elements, Whitney 1-forms, or H(curl) elements,have been used to solve Maxwell's equations for the electric and/or magnetic fields directly. These elements have degrees of freedom along the edges of the grid. Since there are in general more edges than nodes, the use of vector finite elements is slightly more expensive than nodal elements for the same grid. However the use of these elements eliminates spurious modes. These elements enforce tangential continuity of the fields but allow for jump discontinuity in the normal component of the fields, which is a requirement for accurate modelling of fields in inhomogeneous volumes. Vector finite element methods have been successfully used in the frequency domain to analyse resonant cavities, compute waveguide modes, and perform scattering calculations. -The Time Domain Vector Finite Element Method (TDVFEM), which is derived in Section 2, uses vector finite elements as basis functions in a Galerkin approximati...

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