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Additional resources for A Simple Introduction to the Mixed Finite Element Method: Theory and Applications
Sample text
5 Galerkin Scheme Let {Hh }h>0 and {Qh }h>0 be sequences of finite-dimensional subspaces of H and Q, respectively. 1) reads as follows: find (σh , uh ) ∈ Hh × Qh such that a(σh , τh ) + b(τh , uh ) = F(τh ) ∀ τh ∈ Hh , b(σh , vh ) ∀ v h ∈ Qh . 71) we basically follow the same approach of Sect. 1.
35) to (ϕ j − ϕk ), j, k ∈ N, we obtain γ0 (ϕ j ) − γ0 (ϕk ) 0,Γ = γ0 (ϕ j − ϕk ) 0,Γ ≤ C ϕ j − ϕk 1,Ω , which says that {γ0 (ϕ j )} j∈N is a Cauchy sequence in L2 (Γ ). Thus, since L2 (Γ ) is complete, there exists ξ ∈ L2 (Γ ) such that γ0 (ϕ j ) − ξ 0,Γ →0 when j → +∞ , which suggests setting γ0 (v) := ξ . However, we must make sure that this is well defined in the sense that the resulting ξ is independent of the chosen sequence. Indeed, let us consider another sequence {ϕ˜ j } j∈N ⊆ D(Ω¯ ) such that ϕ˜ j − v 1,Ω →0 when j → +∞ .
50) through the rows of each tensor, we obtain: C −1 σ : τ = Ω =− Ω Ω ∇u : τ − Ω u · div τ + γ n (τ ), γ 0 (u) − ρ :τ Ω ρ : τ, where γ n : H(div; Ω ) → H−1/2(Γ ) and γ 0 : H1 (Ω ) → H1/2 (Γ ) are the natural tensor and vector extensions of the respective trace operators defined in Sect. 3. Then, using that γ 0 (u) = 0 on Γ , we arrive at Ω C −1 σ : τ + Ω u · div τ + Ω ρ :τ =0 ∀ τ ∈ H(div; Ω ). Note here that ρ ∈ L2skew (Ω ), where L2skew (Ω ) := η ∈ L2 (Ω ) : η + ηt = 0 . 40) 2 BABU Sˇ KA –BREZZI THEORY 42 In addition, it is easy to see that the symmetry of σ can be imposed weakly through the equation Ω σ :η =0 ∀ η ∈ L2skew (Ω ) .