Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field

and Applied Analysis 3 In scientific and engineering computations, many eigenvalue problems have the following first type of mixed variational formulation: find λ, u, σ ∈ R × V × W , u, σ / 0, 0 , such that a ( u, ψ ) b ( ψ, σ ) λ ( u, ψ ) D, ∀ψ ∈ V, 2.3 b u, v 0, ∀v ∈ W. 2.4 In order to solve problem 2.3 2.4 , one should construct finite element spaces Vh ⊂ V and Wh ⊂ W . Restricting 2.3 2.4 on Vh ×Wh, we get the conforming mixed finite element approximation as follows: find λh, uh, σh ∈ R × Vh ×Wh, uh, σh / 0, 0 , such that a ( uh, ψ ) b ( ψ, σh ) λh ( uh, ψ ) D, ∀ψ ∈ Vh, 2.5 b uh, v 0, ∀v ∈ Wh. 2.6 Consider the associated source and approximate source problems. Given f ∈ D, find w, p ∈ V ×W satisfying a ( w,ψ ) b ( ψ, p ) ( f, ψ ) D, ∀ψ ∈ V, b w,v 0, ∀v ∈ W. 2.7 Given f ∈ D, find wh, ph ∈ Vh ×Wh satisfying a ( wh, ψ ) b ( ψ, ph ) ( f, ψ ) D, ∀ψ ∈ Vh, b wh, v 0, ∀v ∈ Wh. 2.8 Note that the source term f is independent of the solution. As for themixed finite elementmethod for boundary value problems, Brezzi and Fortinand so forth have established a systematic theory see 14 . By Brezzi-Babuska Theorem, we have the following. Lemma 2.1 Brezzi-Babuska . Suppose that C1 2.1 2.2 hold; C2 inf-sup condition holds, namely, there exists a constant ν1 > 0, such that sup ψ∈V,ψ / 0 b ( ψ, v ) ∥ ∥ψ ∥ ∥ V ≥ ν1‖v‖W, ∀v ∈ W, 2.9 then there exists a unique solution w, p to the problem 2.7 and ‖w‖a ∥ ∥p ∥ ∥ W ≤ Cr ∥ ∥f ∥ ∥ D, 2.10 where Cr just depends on ν, ν1, and M1. Moreover, suppose; 4 Abstract and Applied Analysis C3 discrete inf-sup condition holds, namely, there exists a constant ν2 > 0 independent of h, such that sup ψ∈Vh,ψ / 0 b ( ψ, v ) ∥ ∥ψ ∥ ∥ V ≥ ν2‖v‖W, ∀v ∈ Wh, 2.11 then there exists a unique solution wh, ph to the problem 2.8 and the following error estimate is valid: ‖w −wh‖a ∥ ∥p − ph ∥ ∥ W ≤ Ce { inf q∈Vh ∥ ∥w − q∥∥a inf v∈Wh ∥ ∥p − v∥∥W }