Inconsistent Initial Conditions and Instantaneous Switching

If you start off a time-dependent calculation with initial conditions that are inconsistent, or turn on boundary values instantaneously at the start time (or some later time), you induce strong transient signals in the system.  This will cause the time step, and probably the mesh size as well, to be cut to tiny values to track the transients.  

Unless it is specifically the details of these transients that you want to know, you should start with initial conditions that are a consistent solution to a steady problem, and then turn on the boundary values, sources or driving fluxes over a time interval that is meaningful in your problem.  

It is a common mistake to think that simply turning on a source is a smooth operation.  It is not.  Mathematically, the turn-on time is significantly less that a femtosecond (zero, in fact), with attendant terahertz transients.  If that's the problem you pose, then that's the problem FlexPDE will try to solve.  More realistically, you should turn on your sources over a finite time.  Electrical switches take milliseconds, solid state switches take microseconds.  But if you only want to see what happens after a second or two, then fuzz the turn-on.

Turning on a driving flux or a volume source is somewhat more gentle than a boundary value, because it implies a finite time to raise the boundary value to a given level.  But there is still a meaningful time interval over which to turn it on.