Hybridised multigrid preconditioners for a compatible finite element dynamical core

TL;DR

This paper introduces a hybridised multigrid preconditioning approach for compatible finite element discretisations in atmospheric models, significantly improving solver efficiency and scalability in climate and weather prediction simulations.

Contribution

It develops a novel non-nested two-level preconditioner using multigrid for the coarse system, enhancing solver performance for saddle-point problems in atmospheric modeling.

Findings

Reduces solver iterations substantially.

Achieves excellent scalability on large core counts.

Demonstrates effectiveness in the LFRic climate model.

Abstract

Compatible finite element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging since the velocity mass matrix is non-diagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate- and weather prediction model LFRic.