On the Convergence of Galerkin Finite Element Approximations of Electromagnetic Eigenproblems

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

“…We will actually achieve this target in an indirect way, by exploiting again some results proved in [13]. As a matter of fact, it is proved in [13] that if the sequence of spaces {V h } satisfies (CAS), (CDK), and (DCP) for = 1 = µ (involved in Problems 2 and 5), then it satisfies the same conditions for and µ involved in both Problems 1 and 4.…”

“…, r, such that r k=1Ω k =Ω and all the coefficients of and µ (which may be globally discontinuous) are Lipschitz continuous in each of them. Notice that these assumptions will allow us to use results in [19] and [13].…”

“…We define the following conditions on the sequence {X h } [13] (CAS) "Completeness of the approximating subspace"…”

“…As a matter of fact, it is proved in [13] that it is necessary and sufficient that {V h } satisfies the above three characteristic conditions in order that Problem 4 can be regarded as a "spurious-free approximation" [13] of the following restriction of Problem 1, which provides all the significant (i.e., non static, namely having ω = 0) solutions of the Maxwell system without introducing any extraneous one:…”

The discrete compactness property for anisotropic edge elements on polyhedral domains

“…Using this result, they have obtained the convergence of the corresponding approximations of the Maxwell eigenvalue problem. 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.…”