Gabriel N. Gatica's 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 publication is to supply an easy and obtainable advent to the combined finite point strategy as a primary device to numerically remedy a large type of boundary worth difficulties bobbing up in physics and engineering sciences. The booklet relies on fabric that was once 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 gains of the current one need to do, on one hand, with an try of offering and explaining lots of the information within the proofs and within the various purposes. particularly a number of effects and facets of the corresponding research which are frequently to be had in simple terms in papers or court cases are incorporated here.

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

Example text

3) where B∗ : Q → H is the adjoint operator of B. 4) = RQ (G) , which, denoting the null operator by 0, reduces to the following matrix operator equation: find (σ , u) ∈ H × Q such that A B∗ σ B 0 u RH (F) = RQ (G) . 5)] to be well-posed. 2 The inf-sup Condition We recall first that this condition was already introduced in Sect. 1 [cf. 19)]. Indeed, we say that the bounded bilinear form b : H × Q → R satisfies the continuous inf-sup condition if there exists a constant β > 0 such that sup τ ∈H τ =0 b(τ , v) ≥ β v τ H Q ∀v ∈ Q.

Since [C0∞ (Ω¯ )]n is dense in H(div; Ω ), there exists a sequence {zk }k∈N ⊆ [C0∞ (Ω¯ )]n such that zk − τ lim k→ +∞ div,Ω = 0. 49) to w and zk we obtain γn (zk ), γ0 (w) := γ0 (zk ) · n, γ0 (w) 0,Γ = Ω zk · ∇w + Ω w div (zk ) ∀k ∈ N, whence, taking limit when k → +∞, and employing the continuity of γn [cf. 50). We find it important to remark here that when τ ∈ H(div; Ω ), the evaluation γn (τ ), γ0 (w) cannot be replaced by the expression γ0 (τ ) · n, γ0 (w) 0,Γ since the latter only makes sense if τ ∈ [H 1 (Ω )]n .

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


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