Compatible finite element methods for geophysical fluid dynamics

Abstract

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Rotational shallow water equations with viscous damping and boundary control: structure-preserving spatial discretization

“…Furthermore, the other part of the boundary observation given in (34) has to be discretized in the normal family:…”

Rotational Shallow Water Equations with viscous damping and boundary control: structure-preserving spatial discretization

“…As mentioned in [32], following e.g. [33], it can be interesting to adapt the chosen scalar product to the physical problem: indeed, since the shallow-water model performs an average in one dimension, i.e.…”

“…As mentioned in [34], following e.g., [35], it can be interesting to adapt the chosen scalar product to the physical problem: indeed, since the shallow-water model performs an average in one dimension, i.e., on the height of the water column, it is natural to introduce for the velocities the scalar product in L 2 h (Ω):…”

Spherical Gravity Forwarding of Global Discrete Grid Cells by Isoparametric Transformation

“…To solve optimization problems (including integrals) for dealing with complex structures, the finite element method (FEM), as a cornerstone of computational mechanics, has become an indispensable tool of computational in scientific and engineering calculations and numerical simulations [48][49][50][51][52], which plays a crucial role in geophysical research exploring the large-scale atmosphere, oceans, and crust [50]. The target object analyzed by FEM usually has a complex structure [49,53].…”