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. Basso, R. L. G.; Hwang, Y.; Assi, G. R. S.; Sherwin, S. J.: Instabilities and sensitivities in a flow over a rotationally flexible cylinder with a rigid splitter plate (2021)
  2. 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)
  3. Jiang, Hongyi: Formation mechanism of a secondary vortex street in a cylinder wake (2021)
  4. Jiang, Hongyi: Three-dimensional wake transition of a diamond-shaped cylinder (2021)
  5. Kopriva, David A.; Gassner, Gregor J.: A split-form, stable CG/DG-SEM for wave propagation modeled by linear hyperbolic systems (2021)
  6. 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)
  7. Laughton, Edward; Tabor, Gavin; Moxey, David: A comparison of interpolation techniques for non-conformal high-order discontinuous Galerkin methods (2021)
  8. Mengaldo, Gianmarco; Moxey, David; Turner, Michael; Moura, Rodrigo Costa; Jassim, Ayad; Taylor, Mark; Peiró, Joaquim; Sherwin, Spencer: Industry-relevant implicit large-eddy simulation of a high-performance road car via spectral/(hp) element methods (2021)
  9. Önder, Asim; Liu, Philip L.-F.: Receptivity and transition in a solitary wave boundary layer over random bottom topography (2021)
  10. Pech, Jan: Spectral/hp elements in fluid structure interaction. (2021)
  11. 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)
  12. Raposo, Henrique; Mughal, Shahid; Bensalah, Antoine; Ashworth, Richard: Acoustic-roughness receptivity in subsonic boundary-layer flows over aerofoils (2021)
  13. Ren, Chengjiao; Lu, Lin; Cheng, Liang; Chen, Tingguo: Hydrodynamic damping of an oscillating cylinder at small Keulegan-Carpenter numbers (2021)
  14. Tonicello, Niccolò; Lodato, Guido; Vervisch, Luc: A comparative study from spectral analyses of high-order methods with non-constant advection velocities (2021)
  15. 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)
  16. Cheng, Liang; Ju, Xiaoying; Tong, Feifei; An, Hongwei: Transition to chaos through period doublings of a forced oscillating cylinder in steady current (2020)
  17. 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)
  18. Ju, Xiaoying; An, Hongwei; Cheng, Liang; Tong, Feifei: Modes of synchronisation around a near-wall oscillating cylinder in streamwise directions (2020)
  19. Kumar, Abhishek; Pothérat, Alban: Mixed baroclinic convection in a cavity (2020)
  20. Matteo Giacomini, Ruben Sevilla, Antonio Huerta: HDGlab: An open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB (2020) arXiv

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