  We investigate an approach for the numerical solution of differential
equations which is based on the perfect discretization of actions. Such perfect
discretizations show up at the fixed points of renormalization group
transformations. This technique of integrating out the high momentum degrees of
freedom with a path integral has been mainly used in lattice field theory,
therefore our study of its application to PDE's explores new possibilities. We
calculate the perfect discretized Laplace operator for several non-trivial
boundary conditions analytically and numerically. Then we construct a
parametrization of the perfect Laplace operator and we show that with this
parametrization discretization errors -- or computation time -- can be reduced
dramatically compared to the standard discretization.
