Hodge decomposition to solve singular static Maxwell's equations in a non-convex polygon

Assous, Franck; Michaeli, Michael

doi:10.1016/j.apnum.2009.09.004

Cited by 8 publications

(4 citation statements)

References 18 publications

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“…Then one can deduce the solution to the static Maxwell equation. This method, that is well adapted for static problems, has been tested in [7] on the same numerical example. The result is reproduced on Fig.…”

Section: Electric Regular Casementioning

confidence: 98%

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Solving Maxwell’s equations in singular domains with a Nitsche type method

Assous

Michaeli

2011

Journal of Computational Physics

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Section: Electric Regular Casementioning

confidence: 98%

“…As for the case of the Hodge decomposition method (see [7]), it is better to solve (58) and (59) with a P 2 finite element method, to obtain a P 1 discrete approximation of u.…”

Section: Magnetic Singular Casementioning

confidence: 99%

Solving Maxwell’s equations in singular domains with a Nitsche type method

Assous

Michaeli

2011

Journal of Computational Physics

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“…References [30]- [35] examined triangulation algorithms for polygons. References [36]- [41] provided useful algorithms for the rectangular and trapezoidal division of polygons. References [42]- [44] introduced algorithms for dividing polygons according to a given area size.…”

mentioning

confidence: 99%

Area Decomposition Algorithm for Large Region Maritime Search

2020

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In this paper, an area decomposition algorithm is presented that is suitable for multiple vessels and multiple aircraft participating in maritime searches over a large region. The algorithm can decompose the entire sea region to be searched (deemed as polygonal area) into nonoverlapping subpolygons (search subareas) according to the sizes of the areas covered by various search facilities while considering their search capabilities as well as their corresponding commence search points. The algorithm draws on the concept of a polygon division algorithm in computational geometry. The main novelty in this study is the optimization of the classic polygon division algorithm by introducing a ''maximizing-minimum-angle'' strategy, which can effectively compensate for the deficiency of the traditional algorithm, as reflected in the area decomposition result. This improved algorithm can produce rectangle-like subareas, especially for a rectangular search region, which is commonly used in maritime search operations. For nonrectangular search regions, a rightangle division can be achieved so that the shapes of the search subareas are more conducive to planning specific search routes for search facilities. Fast maritime search coverage over a large region can be achieved. The effectiveness of the algorithm is validated by comparing decomposition results before and after the improvement.

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“…As a matter of fact, non-H 1 -space solutions are commonplace in electromagnetism. A main cause is due to the reentrant corners and edges along the domain boundary Γ and across the interfaces of different materials occupying the domain Ω (see [30,34,29,33,4,14] for details). For a non-H 1 -space solution, it has been widely recognized that the H 1 -conforming finite element solution of the eigenproblem (1.3) cannot correctly converge (e.g., see [17,45,49]).…”

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confidence: 99%

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

Duan¹,

Lin²,

Tan³

2016

SIAM J. Numer. Anal.

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Abstract. In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-H 1 . We would like to approximate such a solution in an H 1 -conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard H 1 -conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds O(h 2r 0 ) for eigenvalues which correspond to eigenfunctions in J j=1 (H r (Ω j )) 3 → (H r 0 (Ω)) 3 space, where the piecewise regularity r and the global regularity r 0 may belong to the most interesting interval [0,1].

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Hodge decomposition to solve singular static Maxwell's equations in a non-convex polygon

Cited by 8 publications

References 18 publications

Solving Maxwell’s equations in singular domains with a Nitsche type method

Solving Maxwell’s equations in singular domains with a Nitsche type method

Area Decomposition Algorithm for Large Region Maritime Search

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

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