We describe a fully discrete mixed finite element method for the linearized rotating shallow water model, possibly with damping. While Crank–Nicolson time-stepping conserves energy in the absence of drag or forcing terms and is not subject to a CFL-like stability condition, it requires the inversion of a linear system at each step. We develop weighted-norm preconditioners for this algebraic system that are nearly robust with respect to the physical and discretization parameters in the system. Numerical experiments using Firedrake support the theoretical results.

中文翻译：

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潮汐模型的混合有限元预处理

我们为线性旋转浅水模型描述了一种完全离散的混合有限元方法，可能带有阻尼。尽管Crank–Nicolson时间步长在没有阻力或强迫项的情况下可以节省能量，并且不受CFL一样的稳定性条件的影响，但它需要在每一步中反转线性系统。我们为该代数系统开发了加权范数前置条件，这些条件对于系统中的物理参数和离散参数几乎是可靠的。使用Firedrake进行的数值实验支持了理论结果。