A note on using the mass matrix as a preconditioner for the Poisson equation

TL;DR

This paper demonstrates that the finite element mass matrix can serve as an effective preconditioner for iterative solutions of the Poisson equation, reducing iteration counts and improving efficiency in multiple dimensions.

Contribution

It analytically derives the condition number of the preconditioned operator and validates the scheme's effectiveness through numerical experiments.

Findings

Condition number ratio is 8/3 in 1D, 9/2 in 2D, and approximately 6.3 in 3D.

Expected iteration count is less than half of unpreconditioned CG in 2D and 3D.

Numerical experiments confirm the scheme's simplicity and efficiency.

Abstract

We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is $8/3$ in one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three dimensions. From this it follows that the expected iteration count for achieving a fixed reduction of the norm of the residual is smaller than a half of the number of the iterations of unpreconditioned CG in 2D and 3D. The scheme is easy to implement, and numerical experiments show its efficiency.