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31 which is 12 k d cl where 2 1 61 d qh 6 6 4 1 3 7 7

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Unformatted text preview: on's rule weights. The diagonal matrix D is sometimes called a \lumped" approximation of the consistent mass matrix M. Both nite di erence and nite element solutions behave similarly for the present problem and have the same order of accuracy at the nodes of a uniform mesh. Example 1.3.1. Consider the nite element solution of ;u 00 +u=x which has the exact solution 0<x<1 u(0) = u(1) = 0 u(x) = x ; sinh x : sinh 1 1.3. A Simple Finite Element Problem 17 Relative to the more general problem (1.3.1), this example has p = q = 1 and f (x) = x. We solve it using the piecewise-linear nite element method developed in this section on uniform meshes with spacing h = 1=N for N = 4 8 : : : 128. Before presenting results, it is worthwhile mentioning that the load vector (1.3.15) is exact for this example. Even though we replaced f (x) by its piecewise linear interpolant according to (1.3.14), this introduced no error since f (x) is a linear function of x. Letting e(x) = u(x) ; U (x) (1.3.21) denote the discretization error, in Table 1.3.1 we display the maximum error of the nite element solution and of its r...
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