Weighted-norm preconditioners for a multi-layer tide model

TL;DR

This paper develops a robust preconditioning method for a multi-layer tide model using weighted norms, effectively handling inter-layer coupling to improve computational efficiency in finite element discretizations.

Contribution

It introduces a new preconditioning approach that accounts for inter-layer coupling, enhancing the robustness and efficiency of solving multi-layer tide models.

Findings

Preconditioner remains effective as the number of layers increases.

Numerical results confirm the theoretical robustness of the method.

Reformulated system simplifies inter-layer coupling handling.

Abstract

We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case~\cite{kirby2021preconditioning} a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake confirm the theory.