Two Valuable papers on current research areas in computational mechanics:

http://www.tam.northwestern.edu/tb/TBPublications.htm
http://www.nsf.gov/pubs/reports/sbes_final_report.pdf

Suppose if somebody asked you the following question, and more importantly, wanted the answer to an accuracy of 100-digits:

• Problem A: A particle at the center of a 10 x 1 rectangle undergoes Brownian motion (i.e., two-dimensional random walk with infinitesimal step lengths) until it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?

Or, this question (again, demanding the answer to an accuracy of 100-digits):

• Problem B: A square plate [-1,1]x[-1x1] is at a temperature u = 0. At time t=0 the temperature is increased to u=5 along one of the four sides while being held at u=0 along the other three sides, and heat then flows into the plate according to u_t = \nabla u. When does the temperature reach u=1 at the center of the plate?

You might be tempted to answer back "What's the point?", and, probably, you are justified. However, if you did that, you would have missed some wonderful oppurtunity to learn (and/or teach), among other things, the following lessons: