Efficiency analysis of continuous and discontinuous Galerkin finite element methods

Hanifa Hanif; Jan Nordström; Arnaud. G. Malan; Hanifa Hanif; Jan Nordström; Arnaud. G. Malan

doi:10.3934/math.20251005

AIMS Mathematics

2025, Volume 10, Issue 9: 22579-22597. doi: 10.3934/math.20251005

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Efficiency analysis of continuous and discontinuous Galerkin finite element methods

1.
InCFD Research Group, Department of Mechanical Engineering, University of Cape Town, Cape Town 7700, South Africa
2.
Department of Mathematics, Sardar Bahadur Khan Women's University, Quetta, Pakistan
3.
Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden
4.
Department of Mathematics, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa

Received: 09 July 2025 Revised: 14 September 2025 Accepted: 24 September 2025 Published: 29 September 2025
MSC : 65M12, 65M22, 65M60

We present a comparative analysis of the continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element (FE) methods, focusing on their efficiency in solving advection–diffusion problems in the Summation By Parts–Simultaneous Approximation Term (SBP–SAT) framework. Our findings provide insights into the practical advantages and limitations of each approach, guiding future applications in high–order numerical simulations of real world applications.
- stability, convergence,
- efficiency,
- summation by parts,
- continuous Galerkin method,
- discontinuous Galerkin method
Citation: Hanifa Hanif, Jan Nordström, Arnaud. G. Malan. Efficiency analysis of continuous and discontinuous Galerkin finite element methods[J]. AIMS Mathematics, 2025, 10(9): 22579-22597. doi: 10.3934/math.20251005

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Abstract

We present a comparative analysis of the continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element (FE) methods, focusing on their efficiency in solving advection–diffusion problems in the Summation By Parts–Simultaneous Approximation Term (SBP–SAT) framework. Our findings provide insights into the practical advantages and limitations of each approach, guiding future applications in high–order numerical simulations of real world applications.

References

[1]	J. Nordström, J. Gong, E. Van der Weide, M. Svärd, A stable and conservative high order multi–block method for the compressible Navier–Stokes equations, J. Comput. Phys., 228 (2009), 9020–9035. https://doi.org/10.1016/j.jcp.2009.09.005 doi: 10.1016/j.jcp.2009.09.005
[2]	M. Svärd, M. H. Carpenter, J. Nordström, A stable high–order finite difference scheme for the compressible Navier–Stokes equations, far–field boundary conditions, J. Comput. Phys., 225 (2007), 1020–1038. https://doi.org/10.1016/j.jcp.2007.01.023 doi: 10.1016/j.jcp.2007.01.023
[3]	J. Nordström, K. Forsberg, C. Adamsson, P. Eliasson, Finite volume methods, unstructured meshes and strict stability for hyperbolic problems, Appl. Numer. Math., 45 (2003), 453–473. https://doi.org/10.1016/S0168-9274(02)00239-8 doi: 10.1016/S0168-9274(02)00239-8
[4]	J. Nordström, S. Eriksson, P. Eliasson, Weak and strong wall boundary procedures and convergence to steady–state of the Navier–Stokes equations, J. Comput. Phys., 231 (2012), 4867–4884. https://doi.org/10.1016/j.jcp.2012.04.007 doi: 10.1016/j.jcp.2012.04.007
[5]	P. Castonguay, D. M. Williams, P. E. Vincent, A. Jameson, Energy stable flux reconstruction schemes for advection–diffusion problems, Comput. Methods Appl. Mech. Eng., 267 (2013), 400–417. https://doi.org/10.1016/j.cma.2013.08.012 doi: 10.1016/j.cma.2013.08.012
[6]	H. T. Huynh, A flux reconstruction approach to high–order schemes including discontinuous Galerkin methods, In: 18th AIAA Computational Fluid Dynamics Conference, 2007, 4079. https://doi.org/10.2514/6.2007-4079
[7]	M. H. Carpenter, D. Gottlieb, Spectral methods on arbitrary grids, J. Comput. Phys., 129 (1996), 74–86. https://doi.org/10.1006/jcph.1996.0234 doi: 10.1006/jcph.1996.0234
[8]	M. H. Carpenter, T. C. Fisher, E. J. Nielsen, S. H. Frankel, Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous interfaces, SIAM J. Sci. Comput., 36 (2014), B835–B867. https://doi.org/10.1137/130932193 doi: 10.1137/130932193
[9]	O. Karakashian, C. Makridakis, A space–time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), 1779–1807. https://doi.org/10.1137/S0036142997330111 doi: 10.1137/S0036142997330111
[10]	A. V. Idesman, A new high–order accurate continuous Galerkin method for linear elastodynamics problems, Comput. Mech., 40 (2007), 261–279. https://doi.org/10.1007/s00466-006-0096-z doi: 10.1007/s00466-006-0096-z
[11]	G. J. Gassner, A skew–symmetric discontinuous Galerkin spectral element discretization and its relation to SBP–SAT finite difference methods, SIAM J. Sci. Comput., 35 (2013), A1233–A1253. https://doi.org/10.1137/120890144 doi: 10.1137/120890144
[12]	D. A. Kopriva, G. J. Gassner, J. Nordström, Stability of discontinuous Galerkin spectral element schemes for wave propagation when the coefficient matrices have jumps, J. Sci. Comput., 88 (2021), 3. https://doi.org/10.1007/s10915-021-01516-w doi: 10.1007/s10915-021-01516-w
[13]	M. Zakerzadeh, G. May, On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws, SIAM J. Numer. Anal., 54 (2016), 874–898. https://doi.org/10.1137/14096503X doi: 10.1137/14096503X
[14]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva, Analysis of the SBP–SAT stabilization for finite element methods part I: Linear problems, J. Sci. Comput., 85 (2020), 43. https://doi.org/10.1007/s10915-020-01349-z doi: 10.1007/s10915-020-01349-z
[15]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva, Analysis of the SBP–SAT stabilization for finite element methods part II: Entropy stability, Commun. Appl. Math. Comput., 5 (2023), 573–595. https://doi.org/10.1007/s42967-020-00086-2 doi: 10.1007/s42967-020-00086-2
[16]	H. O. Kreiss, G. Scherer, Finite element and finite difference methods for hyperbolic partial differential equations, In: Mathematical aspects of finite elements in partial differential equations, 1974,195–212. https://doi.org/10.1016/B978-0-12-208350-1.50012-1
[17]	K. Mattsson, J. Nordström, High order finite difference methods for wave propagation in discontinuous media, J. Comput. Phys., 220 (2006), 249–269. https://doi.org/10.1016/j.jcp.2006.05.007 doi: 10.1016/j.jcp.2006.05.007
[18]	M. H. Carpenter, J. Nordström, D. Gottlieb, A stable and conservative interface treatment of arbitrary spatial accuracy, J. Comput. Phys., 148 (1999), 341–365. https://doi.org/10.1006/jcph.1998.6114 doi: 10.1006/jcph.1998.6114
[19]	J. Nordström, T. Lundquist, Summation–by–parts in time, J. Comput. Phys., 251 (2013), 487–499. https://doi.org/10.1016/j.jcp.2013.05.042 doi: 10.1016/j.jcp.2013.05.042
[20]	D. C. Del Rey Fernández, J. E. Hicken, D. W. Zingg, Review of summation–by–parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, 95 (2014), 171–196. http://doi.org/10.1016/j.compfluid.2014.02.016 doi: 10.1016/j.compfluid.2014.02.016
[21]	M. Svärd, J. Nordström, Review of summation–by–parts schemes for initial–boundary-value problems, J. Comput. Phys., 268 (2014), 17–38. https://doi.org/10.1016/j.jcp.2014.02.031 doi: 10.1016/j.jcp.2014.02.031
[22]	J. Gong, J. Nordström, Interface procedures for finite difference approximations of the advection–diffusion equation, J. Comput. Appl. Math., 236 (2011), 602–620. https://doi.org/10.1016/j.cam.2011.08.009 doi: 10.1016/j.cam.2011.08.009
[23]	R. L. Taylor, O. C. Zienkiewicz, The finite element method, Oxford: Butterworth-Heinemann, 2013.
[24]	B. Cockburn, C. W. Shu, The local discontinuous Galerkin method for time–dependent convection–diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440–2463. https://doi.org/10.1137/S0036142997316712 doi: 10.1137/S0036142997316712
[25]	C. E. Baumann, J. T. Oden, A discontinuous hp finite element method for convection–diffusion problems, Comput. Methods Appl. Mech. Eng., 175 (1999), 311–341. https://doi.org/10.1016/S0045-7825(98)00359-4 doi: 10.1016/S0045-7825(98)00359-4
[26]	B. Cockburn, C. W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), 199–224. https://doi.org/10.1006/jcph.1998.5892 doi: 10.1006/jcph.1998.5892
[27]	S. B. Hazara, V. Schulz, Simultaneous pseudo-timestepping for PDE-model based optimization problems, Bit. Numer. Math., 44 (2004), 457–472. https://doi.org/10.1023/B:BITN.0000046815.96929.b8 doi: 10.1023/B:BITN.0000046815.96929.b8
[28]	J. Nordström, A. A. Ruggiu, Dual time-stepping using second derivatives, J. Sci. Comput., 81 (2019), 1050–1071. https://doi.org/10.1007/s10915-019-01047-5 doi: 10.1007/s10915-019-01047-5

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