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

By Gabriel N. Gatica

The major function of this ebook is to supply an easy and obtainable advent to the combined finite point technique as a primary software to numerically clear up a large category of boundary price difficulties coming up in physics and engineering sciences. The booklet relies on fabric that used to be taught in corresponding undergraduate and graduate classes on the Universidad de Concepcion, Concepcion, Chile, over the past 7 years. in comparison with a number of different classical books within the topic, the most good points of the current one need to do, on one hand, with an try out of featuring and explaining many of the information within the proofs and within the diverse purposes. particularly a number of effects and facets of the corresponding research which are frequently to be had in simple terms in papers or complaints are incorporated here.

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27), we deduce that |d| n −1/2,00,ΓN = γ n (τ 0 )|ΓN −1/2,00,ΓN ≤ γ n (τ 0 ) −1/2,Γ ≤ τ0 div,Ω , that is, |d| ≤ 1 n −1/2,00,ΓN τ0 div,Ω . The preceding inequality, together with the fact that τ 2 div,Ω = τ0 2 div,Ω + n d 2 |Ω | , implies that τ 2 div,Ω ≤ 1+ n|Ω | n 2 −1/2,00,ΓN τ0 2 div,Ω , which completes the proof. We now observe that B : H → Q, the operator induced by the present bilinear form b : H × Q → R [cf. 62)], is given by 1 B(τ ) := (div τ , (τ − τ t ), R00 γ n (τ )|ΓN ) ∀ τ ∈ H , 2 −1/2 1/2 where R00 : H00 (ΓN ) → H00 (ΓN ) is the corresponding Riesz mapping.

33) has a unique solution z ∈ HΓ1D (Ω ). 34) where Cˆ > 0 is a constant arising from that inequality and the trace inequality (cf. 4). Thus, defining τˆ := ∇z in Ω , we have that div τˆ = v in Ω , −1 which yields τˆ ∈ H(div; Ω ). In addition, there holds γn (ˆτ )|ΓN = R00 (η ) on ΓN , from which it is clear that R00 γn (ˆτ )|ΓN = η on ΓN . 32), that B(ˆτ ) = (v, η ), and therefore B is surjective. Finally, it is easy to see that G is bounded with G Q ≤ f 0,Ω + g −1/2,00,ΓN , −1/2 where · −1/2,00,ΓN stands for the norm of H00 (ΓN ).

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 ≤ .

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