I have always been fascinated by the so called taxicab geometry first considered by Hermann Minkowski. The metric that has to be used here is a L1 distance which e.g. means that the lenght of the diagonal in a unit square is 2 and not √2 - and this holds true no matter how fine the mesh is! Basically the solution doesn't converge when approximated dicretely.
If you see the mesh as an analogy of solving a differential equation numerically the difference between their true solution and the numerical outcome is obviously huge and won't be acceptable in most applications.
My questions: Do you know of certain classes of differential equations where no closed form solutions exist AND that misbehave in an corresponding way? How are they called and where do I find more information on those? How do you identify them (what characteristics do they have) and how do you tackle such misbehaved equations?
Addition: When I think about it: What could be even stranger is the case when a closed form solution exists but the differential equation can't be solved numerically in the abovementioned sense.