Finite and boundary element methods belong to the most used numerical discretization methods for the approximate solution of elliptic boundary value problems. Finite element methods (FEM) are based on a variational formulation of the partial differential equation to be solved. The definition of a conforming finite dimensional trial space requires an appropriate decomposition of the computational domain into finite elements. The advantage of using finite element methods is their almost universal applicability, e.g. when considering nonlinear partial differential equations. Contrary, the use of boundary element methods (BEM) requires the explicit knowledge of a fundamental solution, which allows the transformation of the partial differential equation to a boundary integral equation to be solved. The approximate solution then only requires a decomposition of the boundary into boundary elements. Boundary element methods are often used to solve partial differential equations with (piecewise) constant coefficients, and to find solutions of boundary value problems in exterior unbounded domains. In addition, direct boundary element methods provide a direct computation of the complete Cauchy data which are the real target functions in many applications. In finite element methods, the Cauchy data can be computed by using Lagrange multipliers and by solving related saddle point problems. By combining both discretization methods it is possible to profit from the advantages of both methods.