Multigrid solution of the semigeostrophic invertibility relation

In semigeostrophic theory, one can recover the balanced wind and mass fields from the potential vorticity by solving a nonlinear elliptic boundary-value problem. This paper describes the efficient solution of this invertibility relation in two dimensions by a multigrid algorithm. Numerical results show this method is competitive with a generalized Buneman algorithm for the linearized cases to which the latter applies. The fully nonlinear problem is solved to well below the level of truncation error 50-80 times faster than by simple relaxation on a single grid.