  The hot spots conjecture is only known to be true for special geometries. It
can be shown numerically that the hot spots conjecture can fail to be true for
easy to construct bounded domains with one hole. The underlying eigenvalue
problem for the Laplace equation with Neumann boundary condition is solved with
boundary integral equations yielding a non-linear eigenvalue problem. Its
discretization via the boundary element collocation method in combination with
the algorithm by Beyn yields highly accurate results both for the first
non-zero eigenvalue and its corresponding eigenfunction which is due to
superconvergence. Additionally, it can be shown numerically that the ratio
between the maximal/minimal value inside the domain and its maximal/minimal
value on the boundary can be larger than $1+10^{-3}$. Finally, numerical
examples for easy to construct domains with up to five holes are provided which
fail the hot spots conjecture as well.
