zbMATH — the first resource for mathematics

Efficient matrix-free high-order finite element evaluation for simplicial elements. (English) Zbl 1440.65223
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
68W10 Parallel algorithms in computer science
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] M. Ainsworth, G. Andriamaro, and O. Davydov, Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures, SIAM J. Sci. Comput., 33 (2011), pp. 3087-3109. · Zbl 1237.65120
[2] W. Bangerth, C. Burstedde, T. Heister, and M. Kronbichler, Algorithms and data structures for massively parallel generic adaptive finite element codes, ACM Trans. Math. Softw., 38 (2011), 14. · Zbl 1365.65247
[3] W. Bangerth, R. Hartmann, and G. Kanschat, Deal. II– a general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 24. · Zbl 1365.65248
[4] P. Bastian, C. Engwer, J. Fahlke, M. Geveler, D. Göddeke, O. Iliev, O. Ippisch, R. Milk, J. Mohring, and S. Müthing, Hardware-based Efficiency Advances in the EXA-DUNE Project, in Software for Exascale Computing-SPPEXA 2013-2015, Springer, New York, 2016, pp. 3-23.
[5] P. Bastian, C. Engwer, D. Göddeke, O. Iliev, O. Ippisch, M. Ohlberger, S. Turek, J. Fahlke, S. Kaulmann, and S. Müthing, EXA-DUNE: Flexible PDE Solvers, Numerical Methods, and Applications, in European Conference on Parallel Processing, Springer, New York, 2014, pp. 530-541.
[6] P. Bastian, F. Heimann, and S. Marnach, Generic implementation of finite element methods in the distributed and unified numerics environment (DUNE), Kybernetika, 46 (2010), pp. 294-315. · Zbl 1195.65130
[7] C. Burstedde, L. C. Wilcox, and O. Ghattas, P4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM J. Sci. Comput., 33 (2011), pp. 1103-1133. · Zbl 1230.65106
[8] C. D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. de Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R. M. Kirby, and S. J. Sherwin, Nektar++: An open-source spectral/hp element framework, Comput. Phys. Commun., 192 (2015), pp. 205-219. · Zbl 1380.65465
[9] C. D. Cantwell, S. J. Sherwin, R. M. Kirby, and P. H. J. Kelly, From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements, Comput. Fluids, 43 (2011), pp. 23-28. · Zbl 1452.76168
[10] C. Chevalier and F. Pellegrini, PT-Scotch: A tool for efficient parallel graph ordering, Parallel Comput., 34 (2008), pp. 318-331.
[11] M. Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput., 6 (1991), pp. 345-390. · Zbl 0742.76059
[12] M. G. Duffy, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal., 19 (1982), pp. 1260-1262. · Zbl 0493.65011
[13] N. Fehn, W. A. Wall, and M. Kronbichler, Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows, Int. J. Numer. Methods Fluids, 88 (2018), pp. 32-54. · Zbl 1415.76451
[14] N. Fehn, W. A. Wall, and M. Kronbichler, Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows, J. Comput. Phys., 372 (2018), pp. 667-693. · Zbl 1415.76451
[15] P. F. Fischer, An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations, J. Comput. Phys., 133 (1997), pp. 84-101. · Zbl 0904.76057
[16] C. Geuzaine and J.-F. Remacle, GMSH: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Engrg., 79 (2009), pp. 1309-1331. · Zbl 1176.74181
[17] L. Grinberg, D. Pekurovsky, S. J. Sherwin, and G. E. Karniadakis, Parallel performance of the coarse space linear vertex solver and low energy basis preconditioner for spectral/hp elements, Parallel Comput., 35 (2009), pp. 284-304.
[18] J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer Science & Business Media, New York, 2007. · Zbl 1134.65068
[19] I. Huismann, J. Stiller, and J. Fröhlich, Factorizing the factorization-a spectral-element solver for elliptic equations with linear operation count, J. Comput. Phys., 346 (2017), pp. 437-448. · Zbl 1380.65373
[20] G. Karniadakis and S. Sherwin, Spectral/Hp Element Methods for Computational Fluid Dynamics, Oxford University Press, Oxford, 2013. · Zbl 1256.76003
[21] G. E. Karniadakis, M. Israeli, and S. A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), pp. 414-443. · Zbl 0738.76050
[22] R. C. Kirby, Fast simplicial finite element algorithms using Bernstein polynomials, Numer. Math., 117 (2011), pp. 631-652. · Zbl 1211.65156
[23] R. M. Kirby and G. E. Karniadakis, De-aliasing on non-uniform grids: Algorithms and applications, J. Comput. Phys., 191 (2003), pp. 249-264. · Zbl 1161.76534
[24] B. Krank, N. Fehn, W. A. Wall, and M. Kronbichler, A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow, J. Comput. Phys., 348 (2017), pp. 634-659. · Zbl 1380.76040
[25] M. Kronbichler and W. A. Wall, A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers, preprint, arXiv:1611.03029, 2016. · Zbl 1402.65163
[26] M. Kronbichler and W. A. Wall, A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers, SIAM J. Sci. Comput., 40 (2018), pp. A3423-A3448. · Zbl 1402.65163
[27] J.-E. W. Lombard, D. Moxey, S. J. Sherwin, J. F. A. Hoessler, S. Dhandapani, and M. J. Taylor, Implicit large-eddy simulation of a wingtip vortex, AIAA J., 54 (2016), pp. 506-518.
[28] G. R. Markall, A. Slemmer, D. A. Ham, P. H. J. Kelly, C. D. Cantwell, and S. J. Sherwin, Finite element assembly strategies on multi-core and many-core architectures, Int. J. Numer. Methods Fluids, 71 (2013), pp. 80-97. · Zbl 1431.65217
[29] G. Mengaldo, D. de Grazia, D. Moxey, P. E. Vincent, and S. J. Sherwin, Dealiasing techniques for high-order spectral element methods on regular and irregular grids, J. Comput. Phys., 299 (2015), pp. 56-81. · Zbl 1352.65396
[30] K. Mittal, S. Dutta, and P. Fischer, Nonconforming Schwarz-spectral element methods for incompressible flow, Comput. Fluids, 191 (2019), 104237. · Zbl 07124550
[31] D. Moxey, C. D. Cantwell, Y. Bao, A. Cassinelli, G. Castiglioni, S. Chun, E. Juda, E. Kazemi, K. Lackhove, J. Marcon, G. Mengaldo, D. Serson, M. Turner, H. Xu, J. Peiró, R. M. Kirby, and S. J. Sherwin, Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods, Comput. Phys. Commun., (2019), 107110.
[32] D. Moxey, C. D. Cantwell, R. M. Kirby, and S. J. Sherwin, Optimizing the performance of the spectral/hp element method with collective linear algebra operations, Comput. Methods Appl. Mech. Engrg., 310 (2016), pp. 628-645. · Zbl 1439.65206
[33] D. Moxey, D. Ekelschot, Ü. Keskin, S. J. Sherwin, and J. Peiró, High-order curvilinear meshing using a thermo-elastic analogy, Computer-Aided Design, 72 (2016), pp. 130-139.
[34] D. Moxey, M. D. Green, S. J. Sherwin, and J. Peiró, An isoparametric approach to high-order curvilinear boundary-layer meshing, Comput. Methods Appl. Mech. Engrg., 283 (2015), pp. 636-650. · Zbl 1423.74908
[35] A. Peplinski, P. F. Fischer, and P. Schlatter, Parallel performance of h-type Adaptive Mesh Refinement for Nek5000, in Proceedings of the Exascale Applications and Software Conference 2016, ACM, 2016, 4.
[36] M. D. Samson, H. Li, and L.-L. Wang, A new triangular spectral element method I: Implementation and analysis on a triangle, Numer. Algorithms, 64 (2013), pp. 519-547. · Zbl 1280.65131
[37] S. J. Sherwin and G. E. Karniadakis, A new triangular and tetrahedral basis for high-order (hp) finite element methods, Int. J. Numer. Methods Engrg., 38 (1995), pp. 3775-3802. · Zbl 0837.73075
[38] J. Treibig, G. Hager, and G. Wellein, Likwid: A lightweight performance-oriented tool suite for x86 multicore environments, in Parallel Processing Workshops (ICPPW), 2010 39th International Conference On, IEEE, 2010, pp. 207-216.
[39] M. Turner, J. Peiró, and D. Moxey, Curvilinear mesh generation using a variational framework, Computer-Aided Design, 103 (2018), pp. 73-91.
[40] P. E. Vos, S. J. Sherwin, and R. M. Kirby, From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low-and high-order discretisations, J. Comput. Phys., 229 (2010), pp. 5161-5181. · Zbl 1194.65138
[41] S. Williams, A. Waterman, and D. Patterson, Roofline: An insightful visual performance model for multicore architectures, Commun. ACM, 52 (2009), pp. 65-76.
[42] F. D. Witherden, A. M. Farrington, and P. E. Vincent, PyFR: An open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach, Comput. Phys. Commun., 185 (2014), pp. 3028-3040. · Zbl 1348.65005
[43] S. Yakovlev, D. Moxey, S. J. Sherwin, and R. M. Kirby, To CG or to HDG: A comparative study in 3D, J. Sci. Comput., 67 (2016), pp. 192-220. · Zbl 1339.65225
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.