Problems with Singularities

Until now, we have encountered only problems whose exact solutions are piecewise smooth. These problems can admit discontinuities in the form of jumps in the field variable at interfaces where the properties of the constituent material can be discontinuous. By contrast, a few exercises we have posed in the Exercise sections have exact solutions whose first derivatives are unbounded even if the material does not experience discontinuities. We left it to the reader to make the pertinent observations. Since such problems do occur in practical applications and can lead to unacceptable physical behavior, it is important to give some attention explicitly to this issue of singular behavior. In this section we discuss two such specific examples, one in 1D and the second in 2D, in order to examine the salient points. Moreover, also from a numerical point of view, it is important to identify this type of problems, because in such cases the standard finite element method no longer converges according to the theoretically predicted rate but instead deteriorates and may converge with a significantly reduced rate.