An elliptic problem with integral constraints with application to large-scale geophysical flows

We present a weak formulation of a non-standard elliptic equation whose boundary values are determined in part by integral relations. Existence and uniqueness of its solution are proved, and a finite element discretization is described, analyzed, and implemented on a test problem. The equation is a generalization of one that is solved during integration of the three-dimensional Quasigeostrophic equations, which model large-scale rotating stratified flows, where the integral constraints represent conservation of physical properties.