This paper provides an overview over our results on the construction and analysis of a non-standard finite element method that is based on the use of boundary integral operators for constructing the element stiffness matrices. This approach permits polyhedral element shapes as well as meshes with hanging nodes. We consider the diffusion equation and convection-diffusion-reaction problems as our model problems, but the method can also be generalized to more general problems like systems of partial differential equations. We provide a rigorous
-error analysis of the method for smooth and non-smooth solutions. This a priori discretization error analysis is only done for the diffusion equation. However, our numerical results also show good performance of our method for convection-dominated diffusion problems.