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# Before presenting results it is worthwhile mentioning

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Unformatted text preview: st derivative at the nodes of a mesh, i.e., jej 1 := 0max je(xj )j <j<N je j 0 1 := 1max je (xj )j: <j<N 0 (1.3.22) ; We have seen that U (x) is a piecewise constant function with jumps at nodes. Data in Table 1.3.1 were obtained by using derivatives from the left, i.e., xj = lim 0 xj ; . With this interpretation, the results of second and fourth columns of Table 1.3.1 indicate that jej =h2 and je j =h are (essentially) constants hence, we may conclude that jej = O(h2 ) and je j = O(h). 0 ; ! 0 1 1 1 0 1 N jej 1 jej 1 =h2 je j 0 je j 0 1 1 =h 4 0.269(-3) 0.430(-2) 0.111( 0) 0.444 8 0.688(-4) 0.441(-2) 0.589(-1) 0.471 16 0.172(-4) 0.441(-2) 0.303(-1) 0.485 32 0.432(-5) 0.442(-2) 0.154(-1) 0.492 64 0.108(-5) 0.442(-2) 0.775(-2) 0.496 128 0.270(-6) 0.442(-2) 0.389(-2) 0.498 Table 1.3.1: Maximum nodal errors of the piecewise-linear nite element solution and its derivative for Example 1.3.1. (Numbers in parenthesis indicate a power of 10.) The nite element and exact solutions of this...
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