How to categorize different numerical methods for PDE?

by Yi Zhang

All the numerical methods for PDE can be boiled down two things: the projection from infinite dimension to finite dimension where the numerical solution is seeked, and the discretization of the equaiton itself. Quote from page 1 of Hesthaven and Warburton’s book “Nodal Discontinuous Galerkin Methods“:

The construction of any numerical method for solving a partial differential equation requires one to consider the two choices:

• How does one represent the solution by an approximate solution?
• In which sense will the approximate solution satisfy the partial differential equation?