  Fractional diffusion has become a fundamental tool for the modeling of
multiscale and heterogeneous phenomena. However, due to its nonlocal nature,
its accurate numerical approximation is delicate. We survey our research
program on the design and analysis of efficient solution techniques for
problems involving fractional powers of elliptic operators. Starting from a
localization PDE result for these operators, we develop local techniques for
their solution: a priori and a posteriori error analyses, adaptivity and
multilevel methods. We show the flexibility of our approach by proposing and
analyzing local solution techniques for a space-time fractional parabolic
equation.
