  Elliptic partial differential equations (PDEs) arise in many areas of
computational sciences such as computational fluid dynamics, biophysics,
engineering, geophysics and more. They are difficult to solve due to their
global nature and sometimes ill-conditioned operators. We review common
discretization methods for elliptic PDEs such as the finite difference, finite
volume, finite element, and spectral methods and the linear systems they form.
We also provide an overview of classic to modern solution methods for the
linear systems formed by these discretization methods. These methods include
splitting and Krylov methods, direct methods, and hierarchical methods.
Finally, we show applications that would benefit from fast and efficient
solvers for elliptic PDEs, including projection methods for the incompressible
Navier-Stokes equations and the shallow water wave equations with dispersive
corrections.
