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3 consider the linear boundary value problem pu 00 qu

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Unformatted text preview: roduce N equally spaced elements on 0 x 1 with nodes xj = jh, j = 0 1 : : : N (h = 1=N ). Approximate u by U having the form U (x) = N X j =1 ck k (x) where j (x), j = 1 2 : : : N , is the piecewise linear basis (1.3.4), and use Galerkin's method to obtain the global sti ness and mass matrices and the load vector for this problem. (Again, the approximation U (x) does not satisfy the natural boundary condition u (1) = 0 nor does it have to. We will discuss this issue in Chapter 2.) 0 22 Introduction 3.3. Write a program to solve this problem using the nite element method developed in Part 3.2b and the tridiagonal algorithm of Problem 2. Execute your program with p = 1, q = 1, and f (x) = x and f (x) = x2 . In each case, use N = 4, 8, 16, and 32. Let e(x) = u(x) ; U (x) and, for each value of N , compute jej , je (xN )j, and kekA according to (1.3.22) and (1.3.23a). You may (optionally) also compute kek0 as de ned by (1.3.23c). In each case, estimate the rate of convergence of the nite element solution...
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