  We derive and analyze a fully computable discrete scheme for fractional
partial differential equations posed on the full space $\mathbb{R}^d$ . Based
on a reformulation using the well-known Caffarelli-Silvestre extension, we
study a modified variational formulation to obtain well-posedness of the
discrete problem. Our scheme is obtained by combining a diagonalization
procedure with a reformulation using boundary integral equations and a coupling
of finite elements and boundary elements. For our discrete method we present
a-priori estimates as well as numerical examples.
