Multigrid On A Sphere

Sean Buckeridge

Oct. 16, 2007, 4:10 p.m. T 1010

The solution of 2D elliptic equations in spherical coordinates is of interest for many applications, for example in Meteorology. The particular issues which have to be dealt with when solving these equations on the surface of a sphere are the singularities at the poles and the strong anisotropy of the Laplacian towards the poles.
The treatment of the singularities begins with the discretisation. Through the use of the Finite Volume discretisation, a discrete equation is naturally derived at these points.
Multigrid methods are known to be robust to anisotropies in the equation, so are a prime candidate to use as a solver for these problems. The multigrid techniques must be adapted such that they deal with the varying strengths of anisotropy across the mesh. In order to achieve this, a particular coarsening strategy must be applied at each grid.

Similar issues are present in the 3D elliptic equations on the sphere, where we observe anisotropy in the direction perpendicular to the surface. Here the coarsening strategies must be combined with appropriate relaxation techniques for the treatment of the anisotropy in 3D.