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

By Gabriel N. Gatica

The major objective of this booklet is to supply an easy and available advent to the combined finite aspect technique as a primary instrument to numerically resolve a large category of boundary worth difficulties bobbing up in physics and engineering sciences. The e-book relies 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 numerous different classical books within the topic, the most gains of the current one need to do, on one hand, with an try out of featuring and explaining lots of the info within the proofs and within the various purposes. particularly a number of effects and points of the corresponding research which are often on hand purely in papers or complaints are incorporated here.

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

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5 Galerkin Scheme Let {Hh }h>0 and {Qh }h>0 be sequences of finite-dimensional subspaces of H and Q, respectively. 1) reads as follows: find (σh , uh ) ∈ Hh × Qh such that a(σh , τh ) + b(τh , uh ) = F(τh ) ∀ τh ∈ Hh , b(σh , vh ) ∀ v h ∈ Qh . 71) we basically follow the same approach of Sect. 1.

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 . However, we must make sure that this is well defined in the sense that the resulting ξ is independent of the chosen sequence. Indeed, let us consider another sequence {ϕ˜ j } j∈N ⊆ D(Ω¯ ) such that ϕ˜ j − v 1,Ω →0 when j → +∞ .

50) through the rows of each tensor, we obtain: C −1 σ : τ = Ω =− Ω Ω ∇u : τ − Ω u · div τ + γ n (τ ), γ 0 (u) − ρ :τ Ω ρ : τ, where γ n : H(div; Ω ) → H−1/2(Γ ) and γ 0 : H1 (Ω ) → H1/2 (Γ ) are the natural tensor and vector extensions of the respective trace operators defined in Sect. 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 ) .

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