Firedrake: automating the finite element method by composing abstractions

TL;DR

Firedrake automates finite element PDE solutions through a Python-based abstraction framework, enabling flexible, optimized, and extensible scientific computing with a clear separation of concerns.

Contribution

It introduces a Python-centric implementation of the FEniCS domain-specific language, enhancing modularity, optimization, and user extensibility in finite element computations.

Findings

Supports automatic optimizations like loop vectorization and block matrix operations

Provides a simple API for custom operations outside variational formulations

Facilitates collaboration among computer scientists, numerical analysts, and application specialists

Abstract

Firedrake is a new tool for automating the numerical solution of partial differential equations. Firedrake adopts the domain-specific language for the finite element method of the FEniCS project, but with a pure Python runtime-only implementation centred on the composition of several existing and new abstractions for particular aspects of scientific computing. The result is a more complete separation of concerns which eases the incorporation of separate contributions from computer scientists, numerical analysts and application specialists. These contributions may add functionality, or improve performance. Firedrake benefits from automatically applying new optimisations. This includes factorising mixed function spaces, transforming and vectorising inner loops, and intrinsically supporting block matrix operations. Importantly, Firedrake presents a simple public API for escaping the UFL abstraction. This allows users to implement common operations that fall outside pure variational formulations, such as flux-limiters.