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Elliptic partial differential equations (PDEs) arise in many areas ofcomputational sciences such as computational fluid dynamics, biophysics,engineering, geophysics and more. They are difficult to solve due to theirglobal nature and sometimes ill-conditioned operators. We review commondiscretization methods for elliptic PDEs such as the finite difference, finitevolume, finite element, and spectral methods and the linear systems they form.We also provide an overview of classic to modern solution methods for thelinear systems formed by these discretization methods. These methods includesplitting and Krylov methods, direct methods, and hierarchical methods.Finally, we show applications that would benefit from fast and efficientsolvers for elliptic PDEs, including projection methods for the incompressibleNavier-Stokes equations and the shallow water wave equations with dispersivecorrections.

Overview of solution methods for elliptic partial differential equations on cartesian and hierarchical grids