Solving the Poisson equation on small aspect ratio domains using unstructured meshes

TL;DR

This paper introduces a novel multigrid preconditioner for solving the Poisson equation on unstructured meshes in small aspect ratio domains, significantly improving solver efficiency for nonhydrostatic ocean modeling.

Contribution

A new multigrid preconditioner tailored for unstructured meshes that maintains condition number independence from aspect ratio, enabling efficient nonhydrostatic ocean simulations.

Findings

Preconditioner dramatically outperforms standard multigrid methods.

Additive smoother yields better convergence than SOR smoothing.

Method facilitates feasible nonhydrostatic ocean modeling on unstructured meshes.

Abstract

We discuss the ill conditioning of the matrix for the discretised Poisson equation in the small aspect ratio limit, and motivate this problem in the context of nonhydrostatic ocean modelling. Efficient iterative solvers for the Poisson equation in small aspect ratio domains are crucial for the successful development of nonhydrostatic ocean models on unstructured meshes. We introduce a new multigrid preconditioner for the Poisson problem which can be used with finite element discretisations on general unstructured meshes; this preconditioner is motivated by the fact that the Poisson problem has a condition number which is independent of aspect ratio when Dirichlet boundary conditions are imposed on the top surface of the domain. This leads to the first level in an algebraic multigrid solver (which can be extended by further conventional algebraic multigrid stages), and an additive smoother. We illustrate the method with numerical tests on unstructured meshes, which show that the preconditioner makes a dramatic improvement on a more standard multigrid preconditioner approach, and also show that the additive smoother produces better results than standard SOR smoothing. This new solver method makes it feasible to run nonhydrostatic unstructured mesh ocean models in small aspect ratio domains.