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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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- Fall '11