  Only a few numerical methods can treat boundary value problems on polygonal
and polyhedral meshes. The BEM-based Finite Element Method is one of the new
discretization strategies, which make use of and benefits from the flexibility
of these general meshes that incorporate hanging nodes naturally. The article
in hand addresses quasi-interpolation operators for the approximation space
over polygonal meshes. To prove interpolation estimates the Poincar\'e constant
is bounded uniformly for patches of star-shaped elements. These results give
rise to the residual based error estimate for high order BEM-based FEM and its
reliability as well as its efficiency are proven. Such a posteriori error
estimates can be used to gauge the approximation quality and to implement
adaptive FEM strategies. Numerical experiments show optimal rates of
convergence for meshes with non-convex elements on uniformly as well as on
adaptively refined meshes.
