Exploiting batch processing on streaming architectures to solve 2D elliptic finite element problems: a hybridized discontinuous Galerkin (HDG) case study. 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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  1. Frontin, Cory V.; Walters, Gage S.; Witherden, Freddie D.; Lee, Carl W.; Williams, David M.; Darmofal, David L.: Foundations of space-time finite element methods: polytopes, interpolation, and integration (2021)
  2. Jiang, Hongyi: Three-dimensional wake transition of a diamond-shaped cylinder (2021)
  3. Jiang, Hongyi: Formation mechanism of a secondary vortex street in a cylinder wake (2021)
  4. Krais, Nico; Beck, Andrea; Bolemann, Thomas; Frank, Hannes; Flad, David; Gassner, Gregor; Hindenlang, Florian; Hoffmann, Malte; Kuhn, Thomas; Sonntag, Matthias; Munz, Claus-Dieter: FLEXI: a high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws (2021)
  5. Pintore, Moreno; Pichi, Federico; Hess, Martin; Rozza, Gianluigi; Canuto, Claudio: Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method (2021)
  6. Tonicello, Niccolò; Lodato, Guido; Vervisch, Luc: A comparative study from spectral analyses of high-order methods with non-constant advection velocities (2021)
  7. Yan, Zhen-Guo; Pan, Yu; Castiglioni, Giacomo; Hillewaert, Koen; Peiró, Joaquim; Moxey, David; Sherwin, Spencer J.: Nektar++: design and implementation of an implicit, spectral/(hp) element, compressible flow solver using a Jacobian-free Newton Krylov approach (2021)
  8. Cheng, Liang; Ju, Xiaoying; Tong, Feifei; An, Hongwei: Transition to chaos through period doublings of a forced oscillating cylinder in steady current (2020)
  9. Gupta, Vikrant; He, Wei; Wan, Minping; Chen, Shiyi; Li, Larry K. B.: A Ginzburg-Landau model for linear global modes in open shear flows (2020)
  10. Ju, Xiaoying; An, Hongwei; Cheng, Liang; Tong, Feifei: Modes of synchronisation around a near-wall oscillating cylinder in streamwise directions (2020)
  11. Kumar, Abhishek; Pothérat, Alban: Mixed baroclinic convection in a cavity (2020)
  12. Matteo Giacomini, Ruben Sevilla, Antonio Huerta: HDGlab: An open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB (2020) arXiv
  13. Moratilla-Vega, M. A.; Lackhove, K.; Janicka, J.; Xia, H.; Page, G. J.: Jet noise analysis using an efficient LES/high-order acoustic coupling method (2020)
  14. Moura, Rodrigo C.; Aman, Mansoor; Peiró, Joaquim; Sherwin, Spencer J.: Spatial eigenanalysis of spectral/\textithpcontinuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows (2020)
  15. Moxey, David; Amici, Roman; Kirby, Mike: Efficient matrix-free high-order finite element evaluation for simplicial elements (2020)
  16. Nordström, Jan; Hagstrom, Thomas M.: The number of boundary conditions for initial boundary value problems (2020)
  17. Önder, Asim; Liu, Philip L.-F.: Stability of the solitary wave boundary layer subject to finite-amplitude disturbances (2020)
  18. Puligilla, Shivakanth Chary; Jayaraman, Balaji: Assessment of end-to-end and sequential data-driven learning for non-intrusive modeling of fluid flows (2020)
  19. Xiong, Chengwang; Qi, Xiang; Gao, Ankang; Xu, Hui; Ren, Chengjiao; Cheng, Liang: The bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime (2020)
  20. Zhang, Kai; Hayostek, Shelby; Amitay, Michael; He, Wei; Theofilis, Vassilios; Taira, Kunihiko: On the formation of three-dimensional separated flows over wings under tip effects (2020)

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