TwoLevel.tex

Although the streamfunction formulation of the QGE contains only one flow
variable, $\psi$, it still suffers from having to solve a large nonlinear system
of equations. This is usually done by using a nonlinear solver, such as Newton's
method. These nonlinear solvers typically require solving large linear systems
multiple times to obtain the solution to the nonlinear system. Solving these
large linear systems multiple times can be time consuming. Thus, a two-level
algorithm can improve solve times greatly over the standard nonlinear solver,
since we need only solve the nonlinear system on a coarse mesh and then use that
solution to solve a linear system on a fine mesh. To this end, this section
discusses the two-level finite element discretization for the pure
streamfunction formulation of the SQGE and QGE.