E 1 if xk xj 134b j xk j k 0 otherwise using

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Unformatted text preview: , we consider approximations of the form U (x) = Let's examine this result more closely. N1 X ; j =1 cj j (x): (1.3.5) 10 Introduction U(x) cj cj-1 cj+1 φj-1 (x) 1 φj (x) x x0 xj-1 xj xj+1 xN Figure 1.3.2: Piecewise linear nite element solution U (x). 1. Since each j (x) is a continuous piecewise linear function of x, their summation U is also continuous and piecewise linear. Evaluating U at a node xk of the mesh using (1.3.4b) yields U (xk ) = N1 X ; j =1 cj j (xk ) = ck : Thus, the coe cients ck , k = 1 2 : : : N ; 1, are the values of U at the interior nodes of the mesh (Figure 1.3.2). 2. By selecting the lower and upper summation indices as 1 and N ; 1 we have ensured that (1.3.5) satis es the prescribed boundary conditions U (0) = U (1) = 0: As an alternative, we could have added basis elements approximation and written the nite element solution as U (x) = N X j =0 0 (x) and cj j (x): N (x) to the (1.3.6) Since, using (1.3.4b), U (x0 ) = c0 and U (xN ) = cN , the boundary conditions are satis ed by requiring c0 =...
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