A fast iterative solver for the variable coefficient diffusion equation on a disk

We present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk. The equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions. The resultant matrix is symmetric positive definite so the preconditioned conjugate gradient (PCG) method can be applied. Different preconditioners have been tested for comparison, in particular, a fast direct solver derived from the equation and the semi-coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly. The present elliptic solver has been applied to study the vortex dynamics of the Ginzburg-Landau equation with a variable diffusion coefficient.