A Simple Introduction to the Mixed Finite Element Method: by Gabriel N. Gatica

By Gabriel N. Gatica

The most function of this e-book is to supply an easy and obtainable advent to the mixed finite aspect procedure as a basic instrument to numerically resolve a large classification of boundary worth difficulties coming up in physics and engineering sciences. The e-book is based on fabric that used to be taught in corresponding undergraduate and graduate courses on the Universidad de Concepcion, Concepcion, Chile, over the past 7 years. compared with a number of different classical books within the topic, the most gains of the present one need to do, on one hand, with an try of proposing and explaining most of the main points within the proofs and within the diversified functions. particularly numerous results and features of the corresponding research which are frequently on hand in basic terms in papers or proceedings are incorporated right here.

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Example text

We use the trace inequality (cf. 4) and the fact that D(Ω¯ ) is dense in H 1 (Ω ) with respect to the norm · 1,Ω . In fact, given v ∈ H 1 (Ω ), we consider a sequence {ϕ j } j∈N ⊆ D(Ω¯ ) such that ϕj − v 1,Ω →0 j → +∞ . 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 .

43) after replacing H by H0 , that is: find (σ , (u,ρ )) ∈ H0 × Q such that a(σ , τ ) + b(τ , (u, ρ )) = F(τ ) ∀ τ ∈ H0 , b(σ , (v, η )) = G(v, η ) ∀ (v, η ) ∈ Q . 50). 2. 50). Proof. 43). 50). 50). 43). 43). 50). 51) from which it is clear that B is bounded with B ≤ 1. Then, from the definition of the bilinear form a [cf. 44)], applying the Cauchy–Schwarz inequality, utilizing √ nλ ≤ 1, we that tr(τ ) 0,Ω ≤ n τ 0,Ω ∀ τ ∈ L2 (Ω ), and noting that (n λ + 2 μ ) deduce that λ tr(ζ ) tr(τ ) 2μ (n λ + 2 μ ) Ω Ω 1 λ tr(ζ ) 0,Ω tr(τ ) 0,Ω ζ 0,Ω τ 0,Ω + 2 μ (n λ + 2 μ ) 1 ζ 0,Ω τ 0,Ω ≤ ζ div,Ω τ div,Ω ∀ ζ , τ ∈ H0 , μ |a(ζ , τ )| = 1 2μ 1 ≤ μ ≤ 1 2μ ζ :τ− which proves that A : H0 → H0 , the operator induced by a, is also bounded with 1 A ≤ .

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 ) . 41) Finally, the equilibrium equation is rewritten as Ω v · div σ = − Ω ∀ v ∈ L2 (Ω ). 45) for all (τ , (v, η )) ∈ H × Q, and the functionals F ∈ H and G ∈ Q are given by F(τ ) := 0 ∀ τ ∈ H, G(v, η ) := − Ω f· v ∀ (v, η ) ∈ Q .

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