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.

翻译：兼容有限元离散化方法在大气运动方程中的应用近期引起了广泛关注。半隐式时间步进方法需要重复求解大型鞍点线性方程组。由于速度质量矩阵是非对角矩阵，导致舒尔补稠密，因此该系统的预处理具有挑战性。可杂交离散化方法克服了这一问题：通过拉格朗日乘子弱约束速度场的连续性，可得到稀疏方程组，其结构与传统方法中的压力舒尔补类似。本文描述了如何利用非嵌套两层预处理器对杂交稀疏系统进行预处理。对于粗网格系统，我们采用了部分作者先前提出的近似舒尔补方法中的多重网格压力求解器。该方法显著减少了求解器迭代次数，在英国气象局下一代气候与天气预报模型LFRic中展现出卓越性能，并可扩展至大规模核数。