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

  • Authors:

  • Kristian B. Ølgaard;Garth N. Wells

  • Affiliations:

  • Delft University of Technology, Delft, The Netherlands;University of Cambridge, United Kingdom

  • Venue:
  • ACM Transactions on Mathematical Software (TOMS)
  • Year:
  • 2010

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Abstract

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.