A. BESPALOV, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet J. Numer. Anal. Math. Model., 3 (1988), pp. 163-178.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....these are not the desired answer. We are left with the task of steering the iterations away from the kernels. One option is regularization, i.e. adding a term corresponding to a weak version of graddiv. for (1.1) and curlcurl. for (1.2) to the differential operator (cf. 1, Sect. 4. 1] and [8]) This will make the kernel visible and convert the problem into a standard positive definite one. Thus it becomes amenable to shift and invert techniques combined with, e.g. an implicitly restarred Lanczos method. The resulting indefinite linear systems of equations can be solved by means ....

A. BESPALOV, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet J. Numer. Anal. Math. Model., 3 (1988), pp. 163-178.

....from the node elements are less sparse than those originating from the edge elements [14] 34] An advantage of the node elements is the availability of software for pre and post processing as e.g. generation of the finite element mesh and visualization of the computed results. Remark. Bespalov [7] proposed an approach similar to the penalty method to solve (5.14) If the positive definite matrix H in (A CHC T )x = Mx; 5.17) is chosen properly, the zero eigenvalues are shifted inside the spectrum of A relative to M and do not disturb the computations. The eigenvalues of (5.17) are the ....

A. N. Bespalov, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet Journal of Numerical Analysis and Mathematical Modelling 3 (1988), 163--178. EIGENVALUE SOLVERS FOR ELECTROMAGNETIC FIELDS IN CAVITIES 31

....degree. Then, the columns of M Gamma1 C form a basis for the nullspace of A and, in principle, it suffices to compute the eigenvalues of an eigenproblem formally equal to (3) but with A and M from (5) To get rid of the high dimensional eigenspace associated with the eigenvalue zero Bespalov [4] proposed to replace (5) by Ax = Mx; A = A CHC T ; 6) where H is a positive definite matrix chosen such that the zero eigenvalues are shifted to the right of the desired eigenvalues and do not disturb the computations. In this note we consider solvers for Ax = Mx with A and M from the ....

A. N. Bespalov, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet Journal of Numerical Analysis and Mathematical Modelling 3 (1988), 163--178.

....the condition Cx = 0 is equivalent with requiring x to be M orthogonal to the nullspace of A. So, the eigenpairs of (3) are precisely the eigenpairs of Ax = Mx (4) corresponding to positive eigenvalues. To get rid of the high dimensional eigenspace associated with the eigenvalue zero Bespalov [4] proposed to replace (4) by (A CHC T )x = Mx; 5) where H is a positive definite matrix chosen such that the zero eigenvalues are shifted to the right of the desired eigenvalues and do not disturb the computations. As suggested by Bespalov we set H = ffI, a multiple of the identity. 2 ....

....that have numerous degrees of freedom on the element interfaces. These are shared by only two finite elements. 3.2 Bespalov s penalty term With the results plotted in Fig. 3 we checked if introducing Bespalov s penalty term CHC T in (5) is advantageous. We chose H = ffI as suggested by Bespalov [4] and set ff = 50=h. The comparison of execution times again obtained with IRL and accuracy reveals that convergence is improved considerably. 3.3 IRL vs. JDQR In Fig. 4 the implicitly restarted Lanczos algorithm is compared with the Jacobi Davidson algorithm. The times for the linear elements ....

A. N. Bespalov, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet Journal of Numerical Analysis and Mathematical Modelling 3 (1988), 163--178.

....a whole) The second drawback: condition (5) is not valid for the mesh solution so it is unclear how to derive correctly a mesh analogue of (23) Thus, it can be said this kind of approximation is unnatural for the problem. To obtain a good finite element approximation the approach described in [7, 14, 32] can be used. It is based on the use of special piecewise polynomial vector basic functions whose normal component on boundaries of finite elements is (generally speaking) discontinuous but tangential one is always continuous. It is easy to check that r Theta Psi h is defined in the ....

....Their coefficients (i.e. mesh unknowns) correspond to the orthogonal projections of electric field onto the edges. It is easy to see that the boundary condition on Omega being approximated in the obvious straightforward way does not lead to parasite condition (28) It has been also shown in [7, 14, 32] that a mesh solution satisfies a mesh weak analogue of (5) basing on it, the mesh problem corresponding to (4) can be exactly transformed to a mesh analogue of (23) 7] Suppose that we are going to apply this way of approximation using a rectangular locally fitted mesh (mentioned in Subsection ....

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A. Bespalov, "Finite element method for the eigenmode problem of a RF cavity resonator", Sov. J. Numer. Anal. and Math. Modelling (1988) 3, pp. 163-178

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