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This paper develops a fully discrete modified characteristic finite elementmethod for a coupled system consisting of the fully nonlinear Monge-Amp\'ereequation and a transport equation. The system is the Eulerian formulation inthe dual space for the B. J. Hoskins' semigeostrophic flow equations, which arewidely used in meteorology to model slowly varying flows constrained byrotation and stratification. To overcome the difficulty caused by the strongnonlinearity, we first formulate (at the differential level) a vanishing momentapproximation of the semigeostrophic flow equations, a methodology recentlyproposed by the authors \cite{Feng1,Feng2}, which involves approximating thefully nonlinear Monge-Amp\'ere equation by a family of fourth order quasilinearequations. We then construct a fully discrete modified characteristic finiteelement method for the regularized problem. It is shown that under certain meshand time stepping constraints, the proposed numerical method converges with anoptimal order rate of convergence. In particular, the obtained error boundsshow explicit dependence on the regularization parameter $\vepsi$. Numericaltests are also presented to validate the theoretical results and to gauge theefficiency of the proposed fully discrete modified characteristic finiteelement method.

A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations