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Optimizing the evaluation of finite element matrices

Journal article
Authors R.C. Kirby
M.G. Knepley
Anders Logg
L.R. Scott
Published in SIAM Journal on Scientific Computing
Volume 27
Issue 3
Pages 741-758
ISSN 1064-8275
Publication year 2006
Published at Department of Mathematical Sciences, Mathematics
Pages 741-758
Language en
Links dx.doi.org/10.1137/040607824
Keywords Compiler , Finite element , Variational form
Subject categories Mathematics, Computational Mathematics

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 operators. 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 pairs. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising. © 2005 Society for Industrial and Applied Mathematics.

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