By Gabriel N. Gatica
The major objective of this e-book is to supply an easy and obtainable creation to the combined finite aspect approach as a basic device to numerically remedy a large category of boundary price difficulties coming up in physics and engineering sciences. The ebook is predicated on fabric that was once taught in corresponding undergraduate and graduate classes on the Universidad de Concepcion, Concepcion, Chile, over the last 7 years. in comparison with a number of different classical books within the topic, the most gains of the current one need to do, on one hand, with an test of proposing and explaining lots of the info within the proofs and within the assorted functions. specifically a number of effects and facets of the corresponding research which are often on hand in simple terms in papers or lawsuits are integrated here.
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Extra info for A Simple Introduction to the Mixed Finite Element Method: Theory and Applications
9). For the uniqueness, let (σ , u) ∈ H × Q be a solution of the homogeneous problem A(σ ) + B∗ (u) = 0 , B(σ ) = 0. It is clear from the second equation that σ ∈ V , and then, applying the projector Π to the first one, and recalling that B∗ (u) ∈ V ⊥ , we obtain Π A(σ ) = 0. Thus, since Π A : V → V is a bijection, it follows that σ = 0, and then from the first equation we obtain that B∗ (u) = 0. Finally, since B∗ : Q → V ⊥ is also a bijection, we conclude that u = 0. 1 are also necessary. Indeed, we have the following result.
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 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. 3. In fact, we first observe that a and b are clearly bounded with A ≤ 1 B ≤ 1, and where A : H → H and B : H → Q are the operators induced by a and b, respectively. Next, it is clear that B(τ ) := div τ ∀ τ ∈ H, and hence V := N(B) = τ ∈ H : B(τ ) = 0 = τ ∈ H(div; Ω ) : div τ = 0 in Ω .
A Simple Introduction to the Mixed Finite Element Method: Theory and Applications by Gabriel N. Gatica