Solution of Elliptic Equations

Abstract

Solution of the Poisson equation (2.8) is necessary whenever we employ the vorticity—stream function formulation of the incompressible fluid equations. A Poisson equation (2.10) is also encountered when we seek the solution for the pressure in the primitive equation formulation. Generalizations of the Poisson equation arise in other physical problems with similar features, e.g., motion of plasma in the ionosphere (Scannapieco et al., 1976; cf Figs. 3-8 through 3-13). All of these are elliptic equations. In most calculations the boundary conditions take the form of Neumann, Dirichlet, or “radiation” conditions; in a few cases of interest, absorbing boundaries which neither transmit nor reflect waves are employed.