Why does smooth initial condition involving Piecewise cause mxsst warning?

Question

This is a very simple one-dimensional heat-conduct equation, the only special part of it is the piecewise initial condition:

    b = NDSolve[{D[tes[t, x], t] == D[tes[t, x], x, x] + Exp[-1/(tes[t, x])], 
                 tes[t, 0] == 1, tes[t, 1] == 1, 
                 tes[0, x] == Piecewise[{{-100 (x - 0.1)^2 + 2, 0 <= x <= 0.1}, 
                                        {2, 0.1 <= x <= 0.9}, 
                                        {-100 (x - 0.9)^2 + 2, 0.9 <= x <= 1}}]}, 
                {tes[t, x]}, {t, 0, 100}, {x, 0, 1}]

If you run the code, you will get this warning message:

NDSolve::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x

Why does this message come out? I read the help but I don't think my initial condition has that kind of fault: it's piecewise but smooth, right?

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I'd like to add another sample here since its behavior makes an interesting contrast to the sample above:

c = NDSolve[{D[tes[t, x], t] == D[tes[t, x], x, x] + Exp[-1/(tes[t, x])],
             tes[t, 0] == 1, tes[t, 1] == 1, 
             tes[0, x] == Piecewise[{{-Exp[-1000 x] + 2, 0 <= x <= 0.1},
                                     {2, 0.1 <= x <= 0.9}, 
                                     {-Exp[-1000 (1 - x)] + 2, 0.9 <= x <= 1}}]},
            {tes[t, x]}, {t, 0, 100}, {x, 0, 1}]

In this example, Exp is chosen to be the transition between initial condition and boundary conditions. This initial condition is not even continuous at x=0.1 and x=0.9 in the view of math, and the change of Exp is more drastic than the polynomial function in my view, but it causes no warning message. Well, I should say, after all this time (notice the time I posted this question) I've already treated the warning as some kind of bug, but I still expect an in-depth explanation.

Answer 1

I still get a warning but I think it's less worrying if I use an alternative form of your initial condition :

b=NDSolve[{D[tes[t,x],t]==D[tes[t,x],x,x]+Exp[-1/(tes[t,x])],
   tes[t,0]==1,tes[t,1]==1,
   tes[0,x]==1+UnitStep[x-0] UnitStep[1-x]},{tes[t,x]},
   {t,0,100},{x,0,1}]