Topological optimization of the evaluation of finite element matrices
Journal article, 2006

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization. © 2006 Society for Industrial and Applied Mathematics.

Variational form

Finite element

Optimized algorithm

Minimum spanning tree


R.C. Kirby

Anders Logg

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

L. Ridgway Scott

A.R. Terrel

SIAM Journal of Scientific Computing

1064-8275 (ISSN) 1095-7197 (eISSN)

Vol. 28 1 224-240

Subject Categories


Computational Mathematics



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