Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.