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[Tapadi <>]

Here, you just found a failure case of the checking method we used so far
here. Let's remember:

1. We assume the density function to be constant over each triangle of the
polygon's discretization (that is an approximation)
2. To check the validity of this approximation, we compare the values at
each triangle's vertices. If they are equal, we deduce that the function has
good chances to be actually constant in the triangle.

But this is not always true, your case is a good example. The function may
have equal values at each vertice, but still have huge variations between
those vertices. How to overcome this case? There are two straight solutions:

a. Decide of an maximum area each triangle should have. If any triangle has
an area greater than this max value, then subdivide it no matter what are
the values at vertices. This max area should be chosen in function of the
maximum frequency you can find in your function. We can discuss this point
later if you want, for now just choose a max area value and decrease it
until you get a good integral result.

b. In addition to the comparison of function's values at each vertice, you
can check for the function's partial derivatives at each vertice. If they
are not near zero, subdivide the triangle. You can generalize this approach
by checking for zero values of function's second, third derivatives, etc.
Actually, this approach is considering the Taylor series of the function and
checking for zero values till a given rank.

Here you are!
Best regards,
Hugo Loi
PhD student at Inria - Maverick team



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