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Get A Simple Introduction to the Mixed Finite Element Method: PDF

By Gabriel N. Gatica

ISBN-10: 3319036947

ISBN-13: 9783319036946

ISBN-10: 3319036955

ISBN-13: 9783319036953

The major objective of this e-book is to supply an easy and obtainable advent to the combined finite point approach as a primary software to numerically resolve a large type of boundary price difficulties coming up in physics and engineering sciences. The booklet is predicated on fabric that used to be taught in corresponding undergraduate and graduate classes on the Universidad de Concepcion, Concepcion, Chile, over the last 7 years. compared with numerous different classical books within the topic, the most good points of the current one need to do, on one hand, with an test of featuring and explaining lots of the info within the proofs and within the diversified functions. particularly numerous effects and features of the corresponding research which are frequently on hand simply in papers or lawsuits are integrated here.

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Additional resources for A Simple Introduction to the Mixed Finite Element Method: Theory and Applications

Example text

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 ≤ .

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 ≤ .

4 and that ·, · denotes the duality between H −1/2 (Γ ) and H 1/2 (Γ ) with respect to the inner product of L2 (Γ ). 31). On the other hand, the equilibrium equation div σ = − f in Ω , is rewritten as Ω v div σ = − Ω fv ∀ v ∈ L2 (Ω ). 4 Application Examples 35 where H := H(div; Ω ), Q := L2 (Ω ), a and b are the bilinear forms defined by a(σ , τ ) := b(τ , v) := Ω Ω σ ·τ ∀ (σ , τ ) ∈ H × H, v div τ ∀ (τ , v) ∈ H × Q, and the functionals F ∈ H and G ∈ Q are given by F(τ ) := γn (τ ), g ∀ τ ∈ H, G(v) := − Ω fv ∀ v ∈ Q.

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


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