Tensor Product Meshes and Elements for Discretizing Models Across the Geophysical Sciences using the Finite Element Modeling Package Firedrake

Abstract:

The motion of the atmosphere, oceans, and glaciers can all be described as thin-film flows, i.e. the horizontal length scale is orders of magnitude greater than the vertical length scale. These systems also typically have an easily describable, quasi-stable vertical structure, for example hydrostatic balance. This structure can be leveraged both to discretize these 3D models directly, or to define and discretize simplified 2.5D models. We will describe how the finite element modeling package Firedrake supports these use cases by implementing (1) extrusion of 2D domains into the vertical dimension, (2) definition of tensor product elements on these extruded domains, (3) specialized language extensions for problems on extruded domains to the Unified Form Language for describing PDEs, and (4) sum-factorization, a low-level optimization to the generated C code that reduces the cost of matrix assembly on extruded domains. Finally, we will describe how these features are used in the atmosphere model Gusto, the ocean model Thetis, and the glacier flow model Icepack.