A simple embedding method for solving partial differential equations on surfaces

It is increasingly common to encounter partial differential equations (PDEs) posed on surfaces, and standard numerical methods are not available for such novel situations. Herein, we develop a simple method for the numerical solution of such equations which embeds the problem within a Cartesian analog of the original equation, posed on the entire space containing the surface. This allows the immediate use of familiar finite difference methods for the discretization and numerical solution. The particular simplicity of our approach results from using the closest point operator to extend the problem from the surface to the surrounding space. The resulting method is quite general in scope, and in particular allows for boundary conditions at surface boundaries, and immediately generalizes beyond surfaces embedded in R 3 , to objects of any dimension embedded in any R n . The procedure is also computationally efficient, since the computation is naturally only carried out on a grid near the surface of interest. We present the motivation and the details of the method, illustrate its numerical convergence properties for model problems and also illustrate its application to several complex model equations.