21 is satis ed exactly at n distinct points on 0 1 2

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Unformatted text preview: iecewise continuous test functions having the basis 1 if x 2 (xj 1=2 xj+1=2) : j (x) := 0 otherwise where xj 1=2 = (xj + xj 1)=2. Using (1.2.6), show that the approximate solution U (x) satis es the di erential equation (1.2.1a) on the average on each subinterval (xj 1=2 xj+1=2), j = 1 2 : : : N . ; ; ; ; 8 Introduction 3. Consider the two-point boundary value problem ;u 00 +u=x which has the exact solution 0<x<1 u(0) = u(1) = 0 u(x) = x ; sinh x : sinh 1 Solve this problem using Galerkin's method (1.2.12c) using the trial function U (x) = c1 sin x: Thus, N = 1, 1(x) = 1 (x) = sin x in (1.2.3). Calculate the error in strain energy as A(u u) ; A(U U ), where A(u v) is given by (1.2.9b). 1.3 A Simple Finite Element Problem Finite element methods are weighted residuals methods that use bases of piecewise polynomials having small support. Thus, the functions (x) and (x) of (1.2.3, 1.2.4) are nonzero only on a small portion of problem domain. Since continuity may be di cult to impose, bases will typically use the minimum continuity ne...
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