Optimizing the Evaluation of Finite Element Matrices

TL;DR

This paper introduces a heuristic graph algorithm that optimizes the evaluation of finite element matrices, significantly reducing computational costs for Laplace and Navier-Stokes operators in various dimensions.

Contribution

It presents a novel graph-based method to identify redundancies in matrix assembly, enabling faster computation of local stiffness matrices in finite element methods.

Findings

Reduces assembly cost for 2D Laplace matrices to less than one multiply-add per entry.

Achieves assembly in under two operations for cubic degree matrices.

Preliminary results show promising efficiency gains in 3D Poisson and Navier-Stokes operators.

Abstract

Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for Navier-Stokes. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiply-add pair per entry. Up to sixth degree, we can do it in less than about two. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising.