Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM

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 Poincaré 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.