  In this work, we present two numerical schemes for a free boundary problem
called one phase quadrature domain. In the first method by applying the
proprieties of given free boundary problem, we derive a method that leads to a
fast iterative solver. The iteration procedure is adapted in order to work in
the case when topology changes. The second method is based on shape
reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various
numerical experiments confirm the efficiency of the derived numerical methods.
