Automated generation and symbolic manipulation of tensor product finite elements

TL;DR

This paper introduces a symbolic algebra extension for finite element generation that supports tensor product structures on various cell types, enhancing the flexibility and automation in finite element software.

Contribution

It presents a novel symbolic algebra integrated with UFL and FIAT for automatic generation of tensor product finite elements, expanding capabilities beyond existing software.

Findings

Enabled automatic generation of tensor product finite elements.

Extended UFL and FIAT for new element types.

Demonstrated numerical examples within Firedrake.

Abstract

We describe and implement a symbolic algebra for scalar and vector-valued finite elements, enabling the computer generation of elements with tensor product structure on quadrilateral, hexahedral and triangular prismatic cells. The algebra is implemented as an extension to the domain-specific language UFL, the Unified Form Language. This allows users to construct many finite element spaces beyond those supported by existing software packages. We have made corresponding extensions to FIAT, the FInite element Automatic Tabulator, to enable numerical tabulation of such spaces. This tabulation is consequently used during the automatic generation of low-level code that carries out local assembly operations, within the wider context of solving finite element problems posed over such function spaces. We have done this work within the code-generation pipeline of the software package Firedrake; we make use of the full Firedrake package to present numerical examples.