Background

Partial differential equations (PDEs) on surfaces occur in many applications in applied science, and it is important to have accurate and efficient numerical methods to solve them. The Closest Point Method (CPM) is a simple and reliable technique to solve PDEs on general smooth surfaces.

Techniques and Challenges

CPM embeds the original surface PDE into a higher-dimensional space, so in practice it is crucial to solve the embedding PDE quickly and accurately. For time-dependent problems involving diffusion on the surface, for example the Laplace-Beltrami operator or higher-order differential operators, implicit schemes are often used to relax the numerical time-step restriction which leads to systems of algebraic equations to solve at each time-step. Such systems also arise from elliptic problems.

Results

We have implemented the Geometric Multigrid Method to solve the linear systems arising from the discretisation of certain surface elliptic or time-dependent PDEs by the Implicit Closest Point Method.  Our approach consists of building a hierarchical series of banded grids around the surface, and on each level of grids establishing the necessary relaxation, restriction and prolongation operators using the closest point extension.

The Future

We are working on testing the accuracy and convergence of the multigrid solver on the Poisson, heat and reaction-diffusion equations on a variety of closed and open curves and surfaces. We plan to analyse the theoretical properties of the multigrid approach.

References

Macdonald C.B., Ruuth S.J.: The implicit Closest Point Method for the numerical solution of partial differential equations: SIAM J. Sci. Comput. 31(6):4330-4350, 2009

Briggs W.L., Henson V.E., McCormick S.F.: A Multigrid Tutorial, Second Edition