tailieunhanh - FEM for Elliptic problems
FEM for Elliptic problems presented Introduction; variational formulation; existence of solutions: lax-milgram theorem; FEM problem,. Invite you to consult the documentation | FEM for Elliptic Problems Sebastian Gonzalez Pintor November, 2016 Proof. Multiply equation (D) by v ∈ V and integrate over the whole domain Z 1 Z 1 00 f v dx, u v dx = − We follow the results from [Joh12] and [And15]. 1 Introduction then we integrate by parts and use the boundary conditions on the left side to obtain Z 1 Z 1 − u00 v dx = − [u0 v]10 + u0 v 0 dx Outline: Variational form. and Minimization prob. • • • • 0 0 Definition of (D), (V) and (M) Equivalence (D) ⇒ (V ) ⇔ (M ) If u ∈ C 2 then (D) ⇐ (V ) Uniqueness of (V ). 0 0 = −u0 (1)v(1) + u0 (0)v(0) +(u0 , v 0 ) {z } | v(1)=v(0)=0 0 0 = (u , v ). We notice that u ∈ C 2 and u(0) = u(1) = 0 implies that u ∈ V , and because the choice of v ∈ V is arbitrary, the function u satisfies the equation Lets consider the following two-point boundary value problem − u00 (x) = f (x) for x ∈ (0, 1) (D) (u0 , v 0 ) = (f, v) ∀v ∈ V, u(0) = u(1) = 0 where u0 = ux and f is a given continuous function. We introduce the notation Z 1 (u, v) = v(x)w(x) dx what is the same as problem (V). Proposition 2 (V ⇒ M ). If u ∈ V is a solution of the variational problem (V), then u is a solution of the minimization problem (M) . 0 for real valued piecewise continuous bounded functions, and the linear space Z 1 V = {v : v ∈ C 0 ([0, 1]), |v 0 |2 dx 0, ∀v ∈ V, v 6= 0, Subtracting these two equations and v = u1 − u2 ∈ V gives Z 1 (u01 − u02 )2 dx = 0 and the norm associated with the scalar product is defined by ||v||a = (a(v, v))1/2 , ∀v ∈ V. 0 Moreover, if is a scalar product with corresponding norm || · ||, we have that the following CauchySchwarz’s inequality is satisfied which shows that u01 (x) − u02 (x) = 0 ∀x ∈ [0, 1], | | ≤ ||v|| ||w||, ∀v, w ∈ V. so u1 − u2 is constant in [0, 1], which together with the boundary conditions u1 (0) = u2 (0) = 0 gives u1 (x) = u2 (x), ∀x ∈ [0, 1], and the uniqueness follows. Definition 2. A linear space V with a scalar product and the corresponding norm || · || is said to
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