Finite Element Approximation for the eigenvalue problem associated to Maxwell’s equations.

Maxwell’s system of vector-valued equations For an open bounded lipschitz domain Ω⊆R^3 with outward normal unit vector n ⃗ , the eigenvalue problem for Maxwell’s equations is given by: Finding (λ,▁u) such that: {█(▁curl (▁curl ▁u)=λ▁u in Ω@div(▁u)=0 in Ω@▁u × ▁n=0 on ∂Ω@@ )┤ ……… (1) Weak formulation for system (1): Finding λ∈R s.t ∃ ▁u≠0,▁u∈Σ∶ (▁curl ▁u,▁curl ▁v)=λ (▁u,▁v) ∀ ▁v∈Σ …….. (2) Where Σ= H_0 (▁curl; Ω)∩H(div^0; Ω) In system (1) , if λ≠0 then one can drop the second equation and getting the followed system : {█(▁curl (▁curl ▁u)=λ▁u in Ω@▁u × ▁n=0 on ∂Ω)┤ ……..(3) With weak formulation: Finding λ≠0,λ∈R s.t ∃ ▁u≠0,▁u∈H_0 (▁curl; Ω) ∶ (▁curl ▁u,▁curl ▁v)=λ (▁u,▁v) ∀ ▁v∈H_0 (▁curl; Ω) …… (4) Remark: λ=0 is considered eigenvalue of the weak form (4). Def. Space of Harmonic fields: H(Ω)= {▁u∈〖L^2 (Ω)〗^3: ▁curl ▁u=0,div(▁u)=0,▁u.▁n on the boudary=0} ……. (5) Remark: The space of Harmonic field H(Ω) is related to the egienvalue problem (1) when λ=0 . In other words: for λ=0 , there exists corresponding harmonic field ▁( u)∈H(Ω). Remark: If Ω is simply connected domain then H(Ω)={0} in other words: the zero eigenvalue corresponds only trivial eigen function ▁u=0 GOAL OF STUDY: When Ω is ▁NOT simply connected domain, the zero eigenvalue λ=0 corresponds a nontrivial harmonic field ▁( u)∈H(Ω). OUR GOAL : Employing the Edge Finite Element to construct an approximation of the double (0,▁u), and comparing our numerical results with the work in the paper: “ Construction of a finite element basis of the first De Rham cohomology group and numerical solution of 3 D magnetostatic problems” for Alonso, Bertalozzi, and Valli. Two possible Mixed formulation for problem (1): 1- KIKUCHI 2- B-F-G-P: Boffi-Fernandes-Gastaldi-Perugia Paper: Computation models of Electromagnetic resonators: Analysis of Edge Element Approximation.

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