Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond

TL;DR

This paper introduces a domain-specific abstraction in the Firedrake library that enables efficient hybridization techniques for solving complex geophysical flow problems, improving solver performance and flexibility.

Contribution

It extends Firedrake with a new abstraction for hybridization, allowing rapid, flexible implementation of advanced solvers as runtime-configurable preconditioners.

Findings

Hybridization improves solver efficiency for geophysical flow models.

The framework integrates seamlessly with Firedrake and PETSc.

Examples demonstrate performance gains in elliptic and fluid dynamics problems.

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 on-going 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 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 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.