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Exploiting Batch Processing on Streaming Architectures to Solve 2D Elliptic Finite Element Problems: A Hybridized Discontinuous Galerkin (HDG) Case Study

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Abstract

Numerical methods for elliptic partial differential equations (PDEs) within both continuous and hybridized discontinuous Galerkin (HDG) frameworks share the same general structure: local (elemental) matrix generation followed by a global linear system assembly and solve. The lack of inter-element communication and easily parallelizable nature of the local matrix generation stage coupled with the parallelization techniques developed for the linear system solvers make a numerical scheme for elliptic PDEs a good candidate for implementation on streaming architectures such as modern graphical processing units (GPUs). We propose an algorithmic pipeline for mapping an elliptic finite element method to the GPU and perform a case study for a particular method within the HDG framework. This study provides comparison between CPU and GPU implementations of the method as well as highlights certain performance-crucial implementation details. The choice of the HDG method for the case study was dictated by the computationally-heavy local matrix generation stage as well as the reduced trace-based communication pattern, which together make the method amenable to the fine-grained parallelism of GPUs. We demonstrate that the HDG method is well-suited for GPU implementation, obtaining total speedups on the order of 30–35 times over a serial CPU implementation for moderately sized problems.

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Acknowledgments

We would like to thank Professor B. Cockburn (U. Minnesota) for the helpful discussions on this topic. This work was supposed by the Department of Energy (DOE NETL DE-EE0004449) and under NSF OCI-1148291.

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Authors and Affiliations

  1. School of Computing and Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, USA

    James King, Zhisong Fu & Robert M. Kirby

  2. Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, USA

    Sergey Yakovlev

  3. Department of Aeronautics, Imperial College London, London, UK

    Spencer J. Sherwin

Authors
  1. James King
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  2. Sergey Yakovlev
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  3. Zhisong Fu
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  4. Robert M. Kirby
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  5. Spencer J. Sherwin
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Correspondence to Robert M. Kirby.

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King, J., Yakovlev, S., Fu, Z. et al. Exploiting Batch Processing on Streaming Architectures to Solve 2D Elliptic Finite Element Problems: A Hybridized Discontinuous Galerkin (HDG) Case Study. J Sci Comput 60, 457–482 (2014). https://doi.org/10.1007/s10915-013-9805-x

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  • DOI: https://doi.org/10.1007/s10915-013-9805-x

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