Unified Form Language: A domain-specific language for weak formulations of partial differential equations

TL;DR

The paper introduces the Unified Form Language (UFL), a domain-specific language designed to express weak formulations of PDEs for numerical methods, enabling compact and readable finite element implementations.

Contribution

UFL provides a novel, expressive language for representing variational forms of PDEs, supporting automatic differentiation and complex function spaces, facilitating easier finite element method development.

Findings

UFL enables concise expression of complex PDE systems.

The language supports automatic differentiation of forms.

Open-source implementation available for use and extension.

Abstract

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multi-field problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.