Differential Equations Solutions 132

# Differential Equations Solutions 132 - 142 Chapter 23...

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Unformatted text preview: 142 Chapter 23. Solutions: Case Study: Finite Diﬀerences and Finite Elements good approximations, although all of them return a reasonable answer (See Figure 23.1) that could be mistaken for what we are looking for. The ﬁnite diﬀerence approximations lose accuracy because their error term depends on u . The ﬁnite element equations were derived from the integrated (weak) formulation of our problem, and when we used integration by parts, we left oﬀ the boundary term that we would have gotten at x = 2/3, so our equations are wrong. This is a case of, “Be careful what you ask for.” • The entries in the ﬁnite element matrices are only approximations to the true values, due to inaccuracy in estimation of the integrals. This means that as the grid size is decreased, we need to reduce the tolerance that we send to quad in order to keep the matrix accurate enough. • The theoretical convergence rate only holds down to the rounding level of the machine, so if we took even ﬁner grids (much larger M ), we would fail to see the expected rate. On these simple 1-dimensional examples, we uncovered many pitfalls in naive use of ﬁnite diﬀerences and ﬁnite elements. Nevertheless, both methods are quite useful when used with care. ...
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