## To CG or to HDG: a comparative study in 3D.(English)Zbl 1339.65225

Summary: Since the inception of discontinuous Galerkin (DG) methods for elliptic problems, there has existed a question of whether DG methods can be made more computationally efficient than continuous Galerkin (CG) methods. Fewer degrees of freedom, approximation properties for elliptic problems together with the number of optimization techniques, such as static condensation, available within CG framework made it challenging for DG methods to be competitive until recently. However, with the introduction of a static-condensation-amenable DG method – the hybridizable discontinuous Galerkin (HDG) method – it has become possible to perform a realistic comparison of CG and HDG methods when applied to elliptic problems. In this work, we extend upon an earlier 2D comparative study, providing numerical results and discussion of the CG and HDG method performance in three dimensions. The comparison categories covered include steady-state elliptic and time-dependent parabolic problems, various element types and serial and parallel performance. The postprocessing technique, which allows for superconvergence in the HDG case, is also discussed. Depending on the direct linear system solver used and the type of the problem (steady-state vs. time-dependent) in question the HDG method either outperforms or demonstrates a comparable performance when compared with the CG method. The HDG method however falls behind performance-wise when the iterative solver is used, which indicates the need for an effective preconditioning strategy for the method.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65Y05 Parallel numerical computation 35K05 Heat equation 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

### Software:

Nektar++; RODAS; PT-Scotch; MUMPS
Full Text:

### References:

 [1] Amestoy, PR; Duff, IS; Koster, J; L’Excellent, J-Y, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 15-41, (2001) · Zbl 0992.65018 [2] Arnold, DN; Brezzi, F; Cockburn, B; Marini, D, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779, (2002) · Zbl 1008.65080 [3] Cantwell, CD; Moxey, D; Comerford, A; Bolis, A; Rocco, G; Mengaldo, G; Grazia, D; Yakovlev, S; Lombard, J-E; Ekelschot, D; Xu, H; Mohamied, Y; Eskilsson, C; Nelson, B; Vos, P; Biotto, C; Kirby, RM; Sherwin, SJ, Nektar++: an open-source spectral/hp element framework, Comput. Phys. Commun., 192, 205-219, (2015) · Zbl 1380.65465 [4] Celiker, F; Cockburn, B; Shi, K, Hybridizable discontinuous Galerkin methods for Timoshenko beams, J. Sci. Comput., 44, 1-37, (2010) · Zbl 1203.74078 [5] Celiker, F; Cockburn, B; Shi, K, A projection-based error analysis of HDG methods for Timoshenko beams, Math. Comput., 81, 277, (2012) · Zbl 1427.74163 [6] Cesmelioglu, A; Cockburn, B; Nguyen, NC; Peraire, J, Analysis of HDG methods for Oseen equations, J. Sci. Comput., 55, 392-431, (2013) · Zbl 1366.76048 [7] Chen, Y; Cockburn, B, Analysis of variable-degree HDG methods for convection-diffusion equations. part I: general nonconforming meshes, IMA J. Numer. Anal., 32, 1267-1293, (2012) · Zbl 1277.65094 [8] Chevalier, C; Pellegrini, F, PT-scotch: a tool for efficient parallel graph ordering, Parallel Comput., 34, 318-331, (2008) [9] Cockburn, B; Dong, B; Guzmán, J, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comput., 77, 1887-1916, (2007) · Zbl 1198.65193 [10] Cockburn, B; Guzmán, J; Wang, H, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comput., 78, 1-24, (2009) · Zbl 1198.65194 [11] Cockburn, B; Shu, C-W, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261, (2001) · Zbl 1065.76135 [12] Cockburn, B; Cui, J, Divergence-free HDG methods for the vorticity-velocity formulation of the Stokes problem, J. Sci. Comput., 52, 256-270, (2012) · Zbl 1311.76054 [13] Cockburn, B; Dong, B; Guzmán, J; Restelli, M; Sacco, Riccardo, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput., 31, 3827-3846, (2009) · Zbl 1200.65093 [14] Cockburn, B; Dubois, O; Gopalakrishnan, J; Tan, S, Multigrid for an HDG method, IMA J. Numer. Anal., 34, 1386-1425, (2014) · Zbl 1304.65260 [15] Cockburn, B; Gopalakrishnan, J; Lazarov, R, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 1319-1365, (2009) · Zbl 1205.65312 [16] Cockburn, B; Gopalakrishnan, J; Sayas, F-J, A projection-based error analysis of HDG methods, Math. Comput., 79, 1351-1367, (2010) · Zbl 1197.65173 [17] Cockburn, B; Nguyen, NC; Peraire, J, A comparison of HDG methods for Stokes flow, J. Sci. Comput., 45, 215-237, (2010) · Zbl 1203.76079 [18] Cockburn, B; Qiu, W; Shi, K, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comput., 81, 279, (2012) · Zbl 1251.65158 [19] Cockburn, B; Shi, K, Conditions for superconvergence of HDG methods for Stokes flow, Math. Comput., 82, 282, (2013) · Zbl 1322.65104 [20] Demmel, JW; Heath, MT; Vorst, HA, Parallel numerical linear algebra, Acta Numer., 2, 111-197, (1993) · Zbl 0793.65011 [21] Dubiner, M, Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390, (1991) · Zbl 0742.76059 [22] Grinberg, L; Pekurovsky, D; Sherwin, SJ; Karniadakis, GE, Parallel performance of the coarse space linear vertex solver and low energy basis preconditioner for spectral/hp elements, Parallel Comput., 35, 284-304, (2009) [23] Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, New York (1991) · Zbl 0729.65051 [24] Huerta, A; Angeloski, A; Roca, X; Peraire, J, Efficiency of high-order elements for continuous and discontinuous Galerkin methods, Int. J. Numer. Meth. Eng., 96, 529-560, (2013) · Zbl 1352.65512 [25] Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987) · Zbl 0634.73056 [26] Jaust, A., Schuetz, J., Woopen, M.: A hybridyzed discontinuous galerkin method for unsteady flows with Shock-Capturing. In: 44th AIAA Fluid Dynamics Conference, AIAA Aviation. American Institute of Aeronautics and Astronautics (2014) [27] Karniadakis, GE; Israeli, M; Orszag, SA, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 414-443, (1991) · Zbl 0738.76050 [28] Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD, 2nd edn. OXFORD University Press, Oxford (2005) · Zbl 1116.76002 [29] Karypis, G; Kumar, V, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20, 359-392, (1999) · Zbl 0915.68129 [30] Kirby, RM; Sherwin, SJ; Cockburn, B, To CG or to HDG: a comparative study, J. Sci. Comput., 51, 183-212, (2012) · Zbl 1244.65174 [31] Lange, M., Gorman, G., Weiland, M., Mitchell, L., Southern, J.: Achieving efficient strong scaling with PETSc using hybrid MPI/OpenMP optimisation. In: Supercomputing, pp. 97-108. Springer, New York (2013) · Zbl 0742.76059 [32] Lanteri, S., Perrussel, R.: An implicit hybridized discontinuous Galerkin method for time-domain Maxwell’s equations. Rapport de recherche RR-7578, INRIA, (March 2011) · Zbl 1304.65260 [33] Li, L; Lanteri, S; Perrussel, R; Cangiani, A (ed.); Davidchack, RL (ed.); Georgoulis, E (ed.); Gorban, AN (ed.); Levesley, Jeremy (ed.); Tretyakov, Michael V (ed.), A hybridizable discontinuous Galerkin method for solving 3D time-harmonic maxwells equations, 119-128, (2013), Berlin · Zbl 1269.78018 [34] Nguyen, NC; Peraire, J; Cockburn, B, An implicit high-order hybridizable discontinuous Galerkin method for linear convection diffusion equations, J. Comput. Phys., 228, 3232-3254, (2009) · Zbl 1187.65110 [35] Nguyen, NC; Peraire, J; Cockburn, B, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection diffusion equations, J. Comput. Phys., 228, 8841-8855, (2009) · Zbl 1177.65150 [36] Nguyen, NC; Peraire, J; Cockburn, B, Hybridizable discontinuous Galerkin methods for the time-harmonic maxwell’s equations, J. Comput. Phys., 230, 7151-7175, (2011) · Zbl 1230.78031 [37] Nguyen, NC; Peraire, J; Cockburn, B, An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 1147-1170, (2011) · Zbl 1391.76353 [38] Persson, P-O; Peraire, J, Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput., 30, 2709-2733, (2008) · Zbl 1362.76052 [39] Rhebergen, S; Cockburn, B, A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys., 231, 4185-4204, (2012) · Zbl 1426.76298 [40] Rhebergen, S; Cockburn, B; Vegt, JJW, A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 233, 339-358, (2013) · Zbl 1286.76033 [41] Roca, X., Nguyen, N.C., Peraire, J.: Scalable parallelization of the hybridized discontinuous galerkin method for compressible flow. In: 21st AIAA Computational Fluid Dynamics Conference (2013) · Zbl 1311.76054 [42] Sherwin, SJ, Hierarchical hp finite elements in hybrid domains, Finite Elment Anal. Design, 27, 109, (1997) · Zbl 0896.65074 [43] Sherwin, SJ; Karniadakis, GE, A new triangular and tetrahedral basis for high-order (hp) finite element methods, Int. J. Numer. Meth. Eng., 38, 3775-3802, (1995) · Zbl 0837.73075 [44] Sherwin, SJ; Casarin, M, Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretization, J. Comput. Phys., 171, 394-417, (2001) · Zbl 0985.65143 [45] Soon, S-C; Cockburn, B; Stolarski, HK, A hybridizable discontinuous Galerkin method for linear elasticity, Int. J. Numer. Meth. Eng., 80, 1058-1092, (2009) · Zbl 1176.74196 [46] Tufo, HM; Fischer, PF, Fast parallel direct solvers for coarse grid problems, J. Parallel Distrib. Comput., 61, 151-177, (2001) · Zbl 0972.68191 [47] Vos, P.: From h to p efficiently: optimising the implementation of spectral/$$hp$$ element methods. PhD thesis, Imperial College London (2011) [48] Vos, PEJ; Sherwin, SJ; Kirby, RM, 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, 5161-5181, (2010) · Zbl 1194.65138 [49] Woopen, M., Ludescher, T., May, G.: A hybridyzed discontinuous Galerkin method for turbulent compressible flow. In: 44th AIAA Fluid Dynamics Conference, AIAA Aviation. American Institute of Aeronautics and Astronautics (2014) [50] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Fluid Mechanics, vol. 3, 5th edn. Butterworth-Heinemann, Oxford (2000) · Zbl 0991.74003 [51] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Solid Mechanics, vol. 2, 5th edn. Butterworth-Heinemann, Oxford (2000) · Zbl 0991.74003 [52] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: The Basis, vol. 1, 5th edn. Butterworth-Heinemann, Oxford (2000) · Zbl 0991.74002
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