Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

Abstract

Abstract.By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements can be systematically constructed. The tools here developed are used to define a new family of spuriousfree edge elements which, in some sense, are complementary to those defined in 1986 by Nedelec.Mathematics Subject Classification. 65N25, 65N30, 65N12.

A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems

“…The compactness of the solution operator T and the existence of the uniform estimates (5.6) and (5.7) greatly simplify the analysis of the Maxwell eigenproblem. These ingredients are absent from Maxwell spectral approximations based on the full curl-curl problem and hence their justifications are much more involved [10,11,8,20,21,9,19,12].…”

The discrete compactness property for anisotropic edge elements on polyhedral domains

“…Precisely, they proved that the edge elements give spuriousfree approximations of the eigenvalue problem, in the sense of [5]. We point out that in [6] a quite general setting is considered, including anisotropic and discontinuous materials and mixed boundary conditions. Also in [2], for the eigenvalue problem (with a different boundary condition), the good approximation properties of edge elements of any order are shown on regular tetrahedral meshes.…”

“…In [6] the authors proved, using and inductive approach, the validity of the discrete compactness property for tetrahedral edge elements (actually, for Nédélec elements of first and second class) of any order on regular triangulations of a polyhedron. Using this result, they have obtained the convergence of the corresponding approximations of the Maxwell eigenvalue problem.…”

“…Also in [2], for the eigenvalue problem (with a different boundary condition), the good approximation properties of edge elements of any order are shown on regular tetrahedral meshes. There, the author consider a mixed method, which is equivalent, in the present situation, to the primal method of [6] (see [3], Sect. 5).…”

Reliability and Performances of Finite Element CAD Tools for the Solution of Microwave Problems

“…Moreover, it is now known that all elements of both Nedelec's families defined on triangles or tetrahedral guarantee spurious-free approximations [6]. This latter result implies that nowadays all problems of interest can be reliably solved by using the finite element method when it is based on Nedelec's elements.…”