Abstract
Abstract. Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle point and elliptic systems arising from these discretizations continue to be an ongoing challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models whilst maintaining a separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to the Portable, Extensible Toolkit for Scientific Computation (PETSc), petsc4py. We provide examples derived from second-order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows.
Highlights
The development of simulation software is an increasingly important aspect of modern scientific computing, in the geosciences in particular
The finite element method (FEM) is a mathematically robust framework for computing numerical solutions of partial differential equations (PDEs) that has become increasingly popular in fluids and solids models across the geosciences, with a formulation that is highly amenable to code generation techniques
Slate is heavily influenced by the Unified Form Language (UFL) (Alnæs et al, 2014; Logg et al, 2012a), a domain-specific language (DSL) embedded in Python which provides symbolic representations of finite element forms
Summary
The development of simulation software is an increasingly important aspect of modern scientific computing, in the geosciences in particular. Software projects developing automatic code generation systems have become quite popular in recent years, as such systems help create a separation of concerns which focuses on a particular complexity independent from the rest This allows for agile collaboration between computer scientists with hardware and software expertise, computational scientists with numerical algorithm expertise, and domain scientists such as meteorologists, oceanographers and climate scientists. This automated translation facilitates the separation of concerns between applications scientists and computational/computer scientists and facilitates the automated optimization of compiled code This framework provides an environment for the development and testing of numerics relevant to the Gung-Ho Project, an initiative by the UK Met Office in designing the next-generation atmospheric dynamical core using mixed finite element methods (Melvin et al, 2019).
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