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Exercise 10: Estimates of accuracy.

In considering the solution you calculated using the finite element method in excercise 9, ask yourself the following difficult questions: ``How accurate is my solution?'' and ``How do I figure out how to find out how accurate my solution is?'' These embarrassingly simple, but tremendously important questions can be difficult to answer. The best way to answer the first question is to know what the answer should be, by comparing it with either an experimental result or an analytical solution. However, for the majority of cases there isn't an analytical solution and/or there are no experimental results. This is one reason why the field of computational science has become so important to so many research areas. In many instances, either experiments cannot be performed or it would be too expensive or time consuming to perform an experiment to confirm computational results. However, we still need to estimate the accuracy of our solutions - and while we're at it, figure out how we can make our solution more accurate.