The Closest Point Method and multigrid solvers for elliptic equations on surfaces
This work provides a provably convergent numerical method and efficient solver for elliptic PDEs on surfaces, benefiting computational geometry and applied mathematics.
The paper applies the Closest Point Method to solve elliptic equations on curved surfaces, proving convergence for Poisson's equation on smooth closed curves and proposing a geometric multigrid solver that demonstrates effectiveness in accuracy and speed.
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.