Optimizations for quadrature representations of finite element tensors through automated code generation

• Published in: ACM transactions on mathematical software 2010-01, Vol.37 (1), p.1-23
• Main Authors: Ølgaard, Kristian B., Wells, Garth N.
• Format: Article
• Language: English
• ISSN: 0098-3500; 1557-7295
• DOI: 10.1145/1644001.1644009

We examine aspects of the computation of finite element matrices and vectors that are made possible by automated code generation. Given a variational form in a syntax that resembles standard mathematical notation, the low-level computer code for building finite element tensors, typically matrices, vectors and scalars, can be generated automatically via a form compiler. In particular, the generation of code for computing finite element matrices using a quadrature approach is addressed. For quadrature representations, a number of optimization strategies which are made possible by automated code generation are presented. The relative performance of two different automatically generated representations of finite element matrices is examined, with a particular emphasis on complicated variational forms. It is shown that approaches which perform best for simple forms are not tractable for more complicated problems in terms of run-time performance, the time required to generate the code or the size of the generated code. The approach and optimizations elaborated here are effective for a range of variational forms.