"Misbehaved" differential equations 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.
 A: I'm not a numerical analyst, and I didn't really understand your question, but one example that came to mind as I was reading it was oscillatory phenomena at discontinuities.  If you model a linear initial value problem with a jump discontinuity using a linear method with greater than first order accuracy, you will get oscillations near the jump, and this can be qualitatively at odds with predicted behavior.  It is a numerical analogue of the Gibbs phenomenon.  There are ways of damping this using nonlinear methods like flux limiters.
As far as I can tell, you are asking for a fairly generic phenomenon.  If you choose a random PDE with sufficient complexity, you should expect to encounter bad stability behavior and numerical intractability.  For well-behaved PDEs, there are ways of rigorously bounding error.
A: You'd probably be interested in reading about discrete differential geometry, as put forth by Bobenko et al.  Here's one paper about when certain geometric quantities defined in the discrete sense converge to the analogous quantities in the continuous sense.  Again, I don't know what connection you're trying to make to differential equations, (perhaps the results on convergence of finite element methods?) so perhaps you should spend some time trying to make that precise and then edit your question again.
