Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

Bonito, Andrea; Guermond, Jean‐Luc

doi:10.1090/s0025-5718-2011-02464-6

Cited by 56 publications

(68 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…REMARK 3.3. Let us note that a similar method has recently been proposed in [6] for electromagnetic eigenvalue problems. The method in [6] depends on a coefficient α and corresponds to the method proposed herein for α = 1 with the only difference that no restriction over the FE spaces or meshes is assumed.…”

Section: Allows Us To Obtainmentioning

confidence: 99%

“…The method in [6] depends on a coefficient α and corresponds to the method proposed herein for α = 1 with the only difference that no restriction over the FE spaces or meshes is assumed. Unfortunately, the convergence of the proposed algorithm is deteriorating in the limit α → 1 and the corresponding numerical analysis in [6] does not apply for the limit case considered in this work.…”

Section: Allows Us To Obtainmentioning

confidence: 99%

“…We also refer to [6] for a similar approach, regarded to the eigenvalue problem. The compatibility condition associated to the inf-sup condition can be avoided by the introduction of the stabilization and exact penalty terms.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

Badia¹,

Codina²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract.A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement.1. Introduction. The simulation of electromagnetic phenomena with increasing complexity demands accurate and efficient numerical methods suitable for large-scale computing. Finite element (FE) methods are commonly used in this context because they can easily handle complicated geometries by using unstructured grids, provide a rigorous mathematical framework and allow adaptation.In many applications of current interest, the electromagnetic problem is coupled to other physical processes. Salient examples of multiphysics phenomena that include electromagnetics are magnetohydrodynamics (MHD) and plasma physics. These two problems have experienced increasing attention due to the need to develop numerical laboratories in fusion technology design. The simulation of these problems (and many others) would benefit of an all-purpose FE method that would be suitable for the different sub-problems at hand, simplifying the implementation issues and the enforcement of the coupling conditions. In particular, an all-purpose continuous nodal-based formulation would be a favored candidate. E.g. the Navier-Stokes equations are commonly solved with stabilized FE approximations that can deal with the singularly perturbed nature of the system for high Reynolds numbers and circumvent the restrictions related to the corresponding inf-sup condition (see e.g. [13,14]). In plasma physics, fields computed by discontinuous FE Maxwell solvers create a considerable numerical noise when embedded in a plasma code, e.g. using the particle-in-cell method (see [2]). Furthermore, nodal approximations are particularly well-suited for time-dependent electromagnetic problems because the mass matrix can be consistently lumped without loss of accuracy, leading to inexpensive transient solvers.The Maxwell operator has a saddle-point structure, with the particularity that the Lagrange multiplier introduced to enforce the divergence-free constraint is identically zero. Existing FE methods that satisfy the discrete counterpart of the inherent inf-sup condition for this problem are based on Nedelec's or edge elements (see e.g. [27,33]); edge elements lead to fields with discontinuous normal component on element edges or faces. We also refer to alternative formulations based on discontinuous Galerkin approximations [28,24,23,34]. With the aim to solve the Maxwell problem with Lagrangian finite elements (FEs), the differential operator of the problem can be transformed into an elliptic on...

show abstract

Section: Allows Us To Obtainmentioning

confidence: 99%

Section: Allows Us To Obtainmentioning

confidence: 99%

See 1 more Smart Citation

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

Badia¹,

Codina²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Let us note that a method that introduces similar stabilization terms has recently been proposed in [9] for electromagnetic eigenvalue problems. The method in [9] depends on a coefficient α and corresponds to the method proposed in [5] for α = 1 with the only difference that no restriction over the FE spaces or meshes is assumed.…”

Section: Remarkmentioning

confidence: 99%

On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

Badia

Codina

Planas

2013

Journal of Computational Physics

View full text Add to dashboard Cite

In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even for singular ones. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh. A detailed set of numerical experiments have been performed in order to validate our approach.

show abstract

“…Although for many years there have been many attempts on how to use the H 1 -conforming finite element method to correctly approximate a non-H 1 -space solution, it is only during the last decade that we have seen a few successful methods. For (1.2), there are the weighted method [31,21,48,25], the H −α -method for some 1/2 < α ≤ 1 [18,11,15], and the L 2 projection method [36,35]. We should also mention the earlier work on the singular complement method [5,6,7,8,9], which deals with Maxwell equations in a domain with reentrant corners as well.…”

mentioning

confidence: 99%

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

Duan¹,

Lin²,

Tan³

2016

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

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].

show abstract

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

Cited by 56 publications

References 35 publications

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

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

Contact Info

Product

Resources

About