  Elliptic partial differential equations are important both from application
and analysis points of views. In this paper we apply the Closest Point Method
to solving elliptic equations on general curved surfaces. Based on the closest
point representation of the underlying surface, we formulate an embedding
equation for the surface elliptic problem, then discretize it using standard
finite differences and interpolation schemes on banded, but uniform Cartesian
grids. We prove the convergence of the difference scheme for the Poisson's
equation on a smooth closed curve. In order to solve the resulting large sparse
linear systems, we propose a specific geometric multigrid method in the setting
of the Closest Point Method. Convergence studies both in the accuracy of the
difference scheme and the speed of the multigrid algorithm show that our
approaches are effective.
