17c 1317d 16 introduction the matrix m and the vector

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Unformatted text preview: (1.3.19) Thus, the nodal values ck , k = 1 2 : : : N ; 1, of the nite element solution are determined by solving a linear algebraic system. With c known, the piecewise linear nite element U can be evaluated for any x using (1.2.3a). The matrix K + M is symmetric, positive de nite, and tridiagonal. Such systems may be solved by the tridiagonal algorithm (cf. Problem 2 at the end of this section) in O(N ) operations, where an operation is a scalar multiply followed by an addition. The discrete system (1.3.19) is similar to the one that would be obtained from a centered nite di erence approximation of (1.3.1), which is 12] (K + D)^ = ^ cl where 2 1 61 D = qh 6 6 ... 4 1 3 7 7 7 5 2 6 ^= h6 l6 4 (1.3.20a) 3 f1 f2 7 7 ... 7 5 fN 2 6 ^6 c=6 4 1 ; 3 c1 ^ c2 7 ^7 ... 7 : 5 cN ^ (1.3.20b) 1 ; Thus, the qu and f terms in (1.3.1) are approximated by diagonal matrices with the nite di erence method. In the nite element method, they are \smoothed" by coupling diagonal terms with their nearest neighbors using Simps...
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