D. BOFFI, P. FERNANDES, L. GASTALDI, I. PERUGIA. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1999) 1264--1290.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Unfortunately the choice of appropriate nite element functions is very delicate, mainly because of the continuous operator having in nitely many zero eigenvalues. The degenerate null space has to be re ected by the discrete operator as well, in order to obtain reasonable approximations, see [14,7,8,21] and [9] to name a few. If this is not done properly, so called spurious modes are likely to arise, i.e. eigenvalues with no physical meaning. A possible choice that avoids this spectral pollution consists in modelling the eld components by N ed elec edge and modi ed Lagrange nodal ....

D. Bo, P. Fernandes, L. Gastaldi, and I. Perugia. Computational models of electromagnetic resonators: Analysis of edge element approximations. SIAM J. Numer. Anal., 36(4):1264-1290, June 1999.

....good approximations for the positive eigenvalues of the continuous problem. However, the simple choice of Lagrange finite elements did not give good results. We now explain the good performance of the edge elements based on the middle square of the commutative diagram (3.3) Following Bo# et. al [4] we set P h = curl Q h and introduce the following mixed discrete eigenvalue problem: find # h (E h , p h ) P h such that F dx curl F p h dx = 0 for all F Q h , 5.13) q dx = p h q dx for all q P h . 5.14) It is then easy to verify that if # h , E h is a ....

....(E h , # 1 h curl E h ) is a solution to (5.13) and if # h , E h , p h ) is a solution to (5.13) then # h 0 and # h , E h is a solution to (5.12) In short, the two problems are equivalent except that the former admits a zero eigenspace which the mixed formulation suppresses. As explained in [4], the accuracy of the mixed eigenvalue problem (5.13) hinges on the stability of the corresponding mixed source problem. This is a saddle point problem of the sort studied by Brezzi, and so stability depends on conditions analogous to (S1) and (S2) The proof of these conditions in case Q h is the ....

D. Bo#, P. Fernandes, L. Gastaldi & I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal., 36 (1999), pp. 1264-1290.

.... of the vectorfields u, the discrete spectrum may feature eigenvalues that are not related to an eigenvalue of the continuous problem [12, 15, 30] On the con trary, in recent years rigorous arguments have been found, why discrete differential forms ensure a correct approximation of the spectrum [10, 13, 21, 22, 30, 48]. For qua siuniform and shape regular families of meshes convergence of the eigenvalues will be quadratic in the meshwidth [21] under mild assumptions on the smoothness of the eigenfunctions. A key role in the convergence theory is played by discrete potentials. They refer to an exceptional ....

D. BorrL P. FERNANDES, L. GASTALDL AND I. PERUGA, Computational models of electro- magnetic resonators: Analysis of edge element approximation, SIAM J. Numer. Anal., 36 (1999), pp. 1264-1290.

....and thus can be naturally posed in mixed form. Early work on the analysis of eigenvalues associated with discrete approximations to curl curl systems was done in [16, 17, 18] Recently, the behavior of the discrete eigenvalues associated with abstract mixed finite element formulations was studied [3, 4, 5, 6]. It was observed that even though the mixed discretization satisfies all of the usual conditions for convergence for the underlying problem, i.e. coercivity on the kernel and the discrete LBB condition, it can behave badly on the eigenvalue problem. For example, the numerical method can generate ....

....of (6.1) coincide with the positive eigenvalues associated with eigenvectors U # V # h satisfying a(U, #) # h (U, #) for all # # V # h . 6.2) The convergence of the approximate eigenvalue problem (6. 2) was studied in the case of # = 0, 0, 0) by Bo#, Fernandes, Gastaldi and Perugia [6]. There they showed that this problem is equivalent to the discrete eigenvalue problem: Find # h and (U h , P h ) # V # h # # h satisfying (U h , # h ) # 1 2 # # # h , P h ) 0, for all # h # V # h , # 1 2 # # U h , Q) # h (P h , Q) for all Q # # # h . ....

D. Bo#, P. Fernandes, L. Gastaldi and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal. 36 (1999) 1264--1290.

....k 2 0 . Therefore a method which minimizes the Rayleigh quotient will converge to this null space. As can be seen in equation (5) the null space consists in the continuous case of 3D curl free vector elds r ik z : This null space is closely tied to the divergence condition (4) see [3]. In fact, by solving the Poisson problem k 2 z = ik z H z r H ) one can split the magnetic eld into 3D div free and curl free parts (Helmholtz decomposition) H H z = r ik z z curl free H H z : z div free The curl free part is ....

Bo, D., Fernandes, P., Gastaldi, L. and Perugia, I.: Computational models of electromagnetic resonators: Analysis of edge element approximations. SIAM J. Numer. Anal. 36, (1999) 12641290

....of the system is too large for direct solvers was the focus of that work. Our eigenvalue problem involves a divergence free condition and thus can be naturally posed in mixed form. Recently, the behavior of the discrete eigenvalues associated with mixed finite element problems has been studied [3, 4, 5, 6]. It was observed that even though the mixed discretization satisfies all of the usual conditions for convergence for the underlying problem, i.e. coercivity on the kernel and the discrete LBB condition, it can behave badly on the eigenvalue problem. For example, the numerical method can generate ....

....of (6.1) coincide with the positive eigenvalues associated with eigenvectors U # V # h satisfying a(U, #) # h (U, #) for all # # V # h . 6.2) The convergence of the approximate eigenvalue problem (6. 2) was studied in the case of # = 0, 0, 0) by Bo#, Fernandes, Gastaldi and Perugia [6]. There they showed that this problem is equivalent to the discrete eigenvalue problem: Find # h and (U h , P h ) # V # h # # h satisfying (U h , # h ) # 1 2 # # # h , P h ) 0, for all # h # V # h , # 1 2 # # U h , Q) # h (P h , Q) for all Q # # # h . 6.3) ....

D. Bo#, P. Fernandes, L. Gastaldi and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal. 36 (1999) 1264--1290.

.... formulation coercive in H(curl) and discontinuous (edge) elements, see NEDELEC [13] It is well known, however, that this formulation presents problems for the computation of eigenvalues, due to the infinite dimensional null space of the curl operator, see BOFFI, FERNANDES, GASTALDI PERUGIA [6] for a discussion. There exist mixed formulations that overcome some of these problems, see BOFFI, BREZZI GASTALDI [5] Another standard way to avoid these problems is to use a variational formulation ( regularized formulation , see HAZARD LENOIR [12] containing the divergence explicitly and ....

D. BOFFI, P. FERNANDES, L. GASTALDI, I. PERUGIA. Computational models of electromagnetic resonators: analysis of edge element approximation. 1997.

....by the Brezzi Babuska condition for (4.5) sup v2H 0 (curl; Omega Gamma (v; grad p) kvk H(curl; Omega Gamma kqk H 1 0 ; 8q 2 H 1 0( Omega Gamma : 4.7) This inequality is obtained by setting v = grad q, q 2 H 1 0( Omega Gamma4 Another mixed formulation for (3. 4) can be found in [9]. 1 For a rectangular box with sides of length a , b and c , we have 1 = a Gamma2 b Gamma2 c Gamma2 . 8 STEFAN ADAM, PETER ARBENZ, AND ROMAN GEUS 5. Finite element formulations We are now going to discuss finite element approximations for the penalty method (4.2) and the mixed ....

D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia, Computational models of electromagnetic resonators: Analysis of edge element approximations, Preprint IAN-CNR No. 1039/1997, Department of Mathematics, University of Pavia, Pavia, Italy, September 1997.

....Introduction In the last few years the interest of the mathematical community in the Maxwell system has increased and important results have been obtained about its numerical approximation. In particular several papers consider the approximation of Maxwell interior eigenvalues (see, for instance, [14, 9, 17, 6, 3]) and the related problem of the approximation of the timeharmonic Maxwell equations (see [16, 11] In this framework the edge nite elements are the natural choice for getting performant schemes. In general the use of edge elements is the main way in order to achieve convergence and optimal ....

....regularity assumptions. Also [11, 17] deal with the time harmonic system (1) All these results, however, use an argument which relies on an inverse inequality, so that a quasiuniformity of the mesh is assumed. On the other hand the hypothesis on the mesh could be avoided by using the results of [6, 3, 4]. The arguments of our proofs hold for various edge element families, which share the same abstract properties as the N ed elec ones [18, 4] For instance, the hp adaptive nite element families presented in [10] t within our analysis as well. The outline of the paper is the following. In the ....

D. Bo, P. Fernandes, L. Gastaldi, and I. Perugia. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal., 36:1264-1290, 1998.

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D. BOFFI, P. FERNANDES, L. GASTALDI, I. PERUGIA. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1999) 1264--1290.

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D. Boffi, P. Fernandes, L. Gastaldi, I. Perugia. Computational models of electromagnetic resonators: analysis of edge element approximation. 1997.

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