Efficient Compilation of a Class of Variational Forms

TL;DR

This paper introduces an efficient method for compiling multilinear variational forms by representing element tensors as contractions of reference and geometry tensors, significantly speeding up compile times in finite element software.

Contribution

It presents a new representation theorem and an optimized algorithm for precomputing reference tensors, enhancing the efficiency of variational form compilation in FEniCS Form Compiler.

Findings

Implementation in FFC reduces compile times by several orders of magnitude.

The new algorithm outperforms previous loop-based methods.

Improves development cycles for finite element software users.

Abstract

We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loop-based implementation by several orders of magnitude, thus shortening compile-times and development cycles for users of FFC.