By Gabriel N. Gatica
The major goal of this e-book is to supply an easy and available creation to the combined finite point procedure as a primary instrument to numerically resolve a large classification of boundary worth difficulties coming up in physics and engineering sciences. The e-book relies 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 a number of different classical books within the topic, the most positive aspects of the current one need to do, on one hand, with an test of proposing and explaining lots of the information within the proofs and within the assorted functions. particularly numerous effects and points of the corresponding research which are often on hand merely in papers or complaints are incorporated here.
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Additional info for A Simple Introduction to the Mixed Finite Element Method: Theory and Applications
Certainly, when a is a symmetric bilinear form on V × V , the operator Π A becomes self-adjoint, and in this case (i-2) and (i-2) are redundant and therefore unnecessary. Furthermore, it is important to remark that a sufficient (but not necessary) condition for (i), which appears very often in applications, is the V -ellipticity of the bilinear form a, which means (cf. 3) that there exists α > 0 such that a(τ , τ ) ≥ α τ ∀τ ∈ V . 1, is the following. 3. Let V := N(B) and assume that: (i) The bilinear form a is V -elliptic [cf.
Moreover, there exists a constant C > 0, which depends on A , α , and β , such that (σ , u) H×Q ≤C F H + G Q . Proof. 1. , [13, 26, 31, 45]). In addition, interesting characterizations of the inf-sup condition for bilinear forms defined on product spaces can be found in  and . 4 Application Examples In this section we illustrate the applicability of the Babuˇska–Brezzi theory with the classical examples given by the Poisson and elasticity problems. 1 Poisson Problem Let Ω be a bounded domain of Rn , n ≥ 2, with Lipschitz-continuous boundary Γ .
5)]. Moreover, there exists a constant C > 0, which depends on A , (Π A)−1 , and β , such that (σ , u) H×Q ≤C F H + G . 9) Proof. 8), there holds σg H ≤ 1 B(σg ) β Q = 1 G β Q . 11) Next, since Π A : V → V is a bijection and Π (RH (F) − A(σg )) belongs to V , there exists a unique σ0 ∈ V such that Π A(σ0 ) = Π (RH (F) − A(σg )). 3 Main Result 31 from which, using the bound for σg σ0 H ≤ C˜ H, F we obtain that H + 1 A β G Q . 12) Now, thanks to the orthogonality condition of the projector Π , it is easy to see that the identity Π A(σ0 ) = Π (RH (F) − A(σg )) is equivalent to saying that the vector A(σ0 + σg ) − RH (F) belongs to V ⊥ .
A Simple Introduction to the Mixed Finite Element Method: Theory and Applications by Gabriel N. Gatica