Abstract
This text introduces to the main ingredients of the discontinuous Galerkin method, combining the framework of high-order finite element methods with Riemann solvers for the information exchange between the elements. The concepts are explained by the example of linear transport and convergence is evaluated in one, two, and three space dimensions. Finally, the construction of schemes for second derivatives is explained and detailed for the symmetric interior penalty method.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Commit 385c588, retrieved on August 21, 2020.
References
Ainsworth, M. (2004). Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. Journal of Computational Physics, 198(1), 106–130. https://doi.org/10.1016/j.jcp.2004.01.004.
Ainsworth, M., & Wajid, H. A. (2009). Dispersive and dissipative behavior of the spectral element method. SIAM Journal on Numerical Analysis, 47(5), 3910–3937. https://doi.org/10.1137/080724976.
Alzetta, G., Arndt, D., Bangerth, W., Boddu, V., Brands, B., Davydov, D., Gassmoeller, R., Heister, T., Heltai, L., Kormann, K., Kronbichler, M., Maier, M., Pelteret, J.-P., Turcksin, B., & Wells, D. (2018). The deal.II library, version 9.0. Journal of Numerical Mathematics, 26(4), 173–184. https://doi.org/10.1515/jnma-2018-0054.
Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.-P., Turcksin, B., & Wells, D. (2020). The deal.II finite element library: design, features, and insights. Computers & Mathematics with Applications. In press. https://doi.org/10.1016/j.camwa.2020.02.022.
Arnold, D. N. (1982). An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, 19(4), 742–760. https://doi.org/10.1137/0719052.
Arnold, D. N., Brezzi, F., Cockburn, B., & Marini, L. D. (2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39(5), 1749–1779. https://doi.org/10.1137/s0036142901384162.
Bassi, F., & Rebay, S. (1997a). High-order accurate discontinuous finite element solution of the 2D Euler equations. Journal of Computational Physics, 138(2), 251–285. https://doi.org/10.1006/jcph.1997.5454.
Bassi, F., & Rebay, S. (1997b). A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics, 131(2), 267–279. https://doi.org/10.1006/jcph.1996.5572.
Bastian, P. (2014). A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure. Computational Geosciences, 18, 779–796. https://doi.org/10.1007/s10596-014-9426-y.
Berrut, J.-P., & Trefethen, L. N. (2004). Barycentric Lagrange interpolation. SIAM Review, 46(3), 501–517. https://doi.org/10.1137/s0036144502417715.
Brenner, S. C., & Scott, R. L. (2002). The mathematical theory of finite elements (2nd ed.). Berlin: Springer.
Brown, J. (2010). Efficient nonlinear solvers for nodal high-order finite elements in 3D. Journal of Scientific Computing, 45(1–3), 48–63.
Chan, J. (2018). On discretely entropy conservative and entropy stable discontinuous Galerkin methods. Journal of Computational Physics, 362, 346–374. https://doi.org/10.1016/j.jcp.2018.02.033.
Chang, J., Fabien, M. S., Knepley, M. G., & Mills, R. T. (2018). Comparative study of finite element methods using the time-accuracy-size (TAS) spectrum analysis. SIAM Journal on Scientific Computing, 40(6), C779–C802. https://doi.org/10.1137/18m1172260.
Cockburn, B., & Shu, C.-W. (1991). The Runge-Kutta local projection \(p^1\)-discontinuous-Galerkin finite element method for scalar conservation laws. Mathematical Modelling and Numerical Analysis, 25(3), 337–361.
Cockburn, B., & Shu, C.-W. (1998a). The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35(6), 2440–2463. https://doi.org/10.1137/s0036142997316712.
Cockburn, B., & Shu, C.-W. (1998b). The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Journal of Computational Physics, 141(2), 199–224. https://doi.org/10.1006/jcph.1998.5892.
Cockburn, B., Karniadakis, G. E., & Shu., C.-W. (2000). The development of discontinuous Galerkin methods. Lecture notes in computational science and engineering (pp. 3–50). Berlin: Springer. https://doi.org/10.1007/978-3-642-59721-3_1.
Coons, S. A. (1967). Surfaces for computer-aided design of space forms. Technical report MAC-TR-41, MIT.
Deville, M. O., Fischer, P. F., & Mund, E. H. (2002). High-order methods for incompressible fluid flow (Vol. 9). Cambridge: Cambridge University Press.
Dumbser, M., & Käser, M. (2006). An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - II. The three-dimensional isotropic case. Geophysical Journal International, 167(1), 319–336. https://doi.org/10.1111/j.1365-246x.2006.03120.x.
Dumbser, M., Käser, M., & Toro, E. F. (2007). An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time step** and p-adaptivity. Geophysical Journal International, 171(2), 695–717. https://doi.org/10.1111/j.1365-246x.2007.03427.x.
Dumbser, M., Fambri, F., Tavelli, M., Bader, M., & Weinzierl, T. (2018). Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Axioms, 7(3), 63. https://doi.org/10.3390/axioms7030063.
Durufle, M., Grob, P., & Joly, P. (2009). Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numerical Methods for Partial Differential Equations, 25(3), 526–551. https://doi.org/10.1002/num.20353.
Epshteyn, Y., & Rivière, B. (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206, 843–872. https://doi.org/10.1016/j.cam.2006.08.029.
Fehn, N., Wall, W. A., & Kronbichler, M. (2018). Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 88(1), 32–54. https://doi.org/10.1002/fld.4511.
Fehn, N., Kronbichler, M., Lehrenfeld, C., Lube, G., & Schroeder, P. W. (2019a). High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of \( {L}^{2}\) and \({H }\)(div) methods. International Journal for Numerical Methods in Fluids, 91(11), 533–556. https://doi.org/10.1002/fld.4763.
Fehn, N., Wall, W. A., & Kronbichler, M. (2019b). A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 89(3), 71–102. https://doi.org/10.1002/fld.4683.
Fischer, P., Min, M., Rathnayake, T., Dutta, S., Kolev, T., Dobrev, V., Camier, J.-S., Kronbichler, M., Warburton, T., Świrydowicz, K., & Brown, J. (2020). Scalability of high-performance PDE solvers. International Journal of High Performance Computing Applications, 34(5), 562–586. https://doi.org/10.1177/1094342020915762.
Gassner, G., & Kopriva, D. A. (2011). A comparison of the dispersion and dissipation errors of Gauss and Gauss-Lobatto discontinuous Galerkin spectral element methods. SIAM Journal on Scientific Computing, 33(5), 2560–2579. https://doi.org/10.1137/100807211.
Gassner, G. J. (2013). A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM Journal on Scientific Computing, 35(3), A1233–A1253. https://doi.org/10.1137/120890144.
Gassner, G. J. (2014). A kinetic energy preserving nodal discontinuous Galerkin spectral element method. International Journal for Numerical Methods in Fluids, 76(1), 28–50. https://doi.org/10.1002/fld.3923.
Gassner, G. J., & Beck, A. D. (2012). On the accuracy of high-order discretizations for underresolved turbulence simulations. Theoretical and Computational Fluid Dynamics, 27(3–4), 221–237. https://doi.org/10.1007/s00162-011-0253-7.
Gassner, G. J., Winters, A. R., & Kopriva, D. A. (2016). Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327, 39–66. https://doi.org/10.1016/j.jcp.2016.09.013.
Gordon, W. J., & Thiel, L. C. (1982). Transfinite map**s and their application to grid generation. Applied Mathematics and Computation, 10, 171–233. https://doi.org/10.1016/0096-3003(82)90191-6.
Gustafsson, B. (2008). High order difference methods for time dependent PDE. Berlin: Springer. https://doi.org/10.1007/978-3-540-74993-6.
Gustafsson, B., Kreiss, H.-O., & Oliger, J. (2013). Time dependent problems and difference methods (2nd ed.). New York: Wiley.
Hairer, E., & Wanner, G. (1991). Solving ordinary differential equations II. Stiff and differential-algebraic problems. Berlin: Springer.
Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving ordinary differential equations I. Nonstiff problems (2nd ed.). Berlin: Springer.
Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: Structure-preserving algorithms for ordinary differential equations (2nd ed.). Berlin: Springer.
Heltai, L., Bangerth, W., Kronbichler, M., & Mola, A. (2019). Using exact geometry information in finite element computations. Technical report. ar**v:1910.09824.
Hesthaven, J. S., & Warburton, T. (2008). Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Berlin: Springer. https://doi.org/10.1007/978-0-387-72067-8.
Hesthaven, J. S., Gottlieb, S., & Gottlieb, D. (2006). Spectral methods for time-dependent problems. Cambridge: Cambridge University Press.
Hu, F. Q., Hussaini, M. Y., & Rasetarinera, P. (1999). An analysis of the discontinuous Galerkin method for wave propagation problems. Journal of Computational Physics, 151(2), 921–946. https://doi.org/10.1006/jcph.1999.6227.
Hughes, T. J. R., Cottrell, J. A., & Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39–41), 4135–4195. https://doi.org/10.1016/j.cma.2004.10.008.
Huismann, I., Stiller, J., & Fröhlich, J. (2019). Scaling to the stars - a linearly scaling elliptic solver for p-multigrid. Journal of Computational Physics, 398, 108868. https://doi.org/10.1016/j.jcp.2019.108868.
Huynh, H. T., Wang, Z. J., & Vincent, P. E. (2014). High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids. Computers & Fluids, 98, 209–220. https://doi.org/10.1016/j.compfluid.2013.12.007.
Johnson, C., & Pitkäranta, J. (1986). An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Mathematics of Computation, 46(173), 1. https://doi.org/10.1090/s0025-5718-1986-0815828-4.
Karniadakis, G., & Sherwin, S. J. (2005). Spectral/hp element methods for computational fluid dynamics (2nd ed.). Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198528692.001.0001.
Kennedy, C. A., Carpenter, M. H., & Lewis, R. M. (2000). Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Applied Numerical Mathematics, 35(3), 177–219. https://doi.org/10.1016/s0168-9274(99)00141-5.
Ketcheson, D. I., LeVeque, R. J., & del Razo, M. J. (2020). Riemann problems and Jupyter solutions. Philadelphia: Society for Industrial and Applied Mathematics. 10(1137/1), 9781611976212.
Klose, B. F., Jacobs, G. B., & Kopriva, D. A. (2020). Assessing standard and kinetic energy conserving volume fluxes in discontinuous Galerkin formulations for marginally resolved Navier-Stokes flows. Computers & Fluids, 205, 104557. https://doi.org/10.1016/j.compfluid.2020.104557.
Kopriva, D. A. (2006). Metric identities and the discontinuous spectral element method on curvilinear meshes. Journal of Scientific Computing, 26(3), 301–327. https://doi.org/10.1007/s10915-005-9070-8.
Kopriva, D. A. (2009). Implementing spectral methods for partial differential equations. Berlin: Springer.
Kopriva, D. A., & Gassner, G. J. (2016). Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable. Applied Mathematics and Computation, 272, 274–290. https://doi.org/10.1016/j.amc.2015.08.047.
Kormann, K. (2016). A time-space adaptive method for the Schrödinger equation. Communications in Computational Physics, 20(1), 60–85.
Kronbichler, M., & Kormann, K. (2012). A generic interface for parallel cell-based finite element operator application. Computers & Fluids, 63, 135–147.
Kronbichler, M., & Kormann, K. (2019). Fast matrix-free evaluation of discontinuous Galerkin finite element operators. ACM Transactions on Mathematical Software, 45(3), 29:1–29:40. https://doi.org/10.1145/3325864.
Kronbichler, M., & Wall, W. A. (2018). A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers. SIAM Journal on Scientific Computing, 40(5), A3423–A3448. https://doi.org/10.1137/16M110455X.
Kubatko, E. J., Yeager, B. A., & Ketcheson, D. I. (2014). Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods. Journal of Scientific Computing, 60(2), 313–344. https://doi.org/10.1007/s10915-013-9796-7.
Lesaint, P., & Raviart, P. A. (1974). On a finite element method for solving the neutron transport equation. In Mathematical aspects of finite elements in partial differential equations (pp. 89–123). Amsterdam: Elsevier. https://doi.org/10.1016/b978-0-12-208350-1.50008-x.
LeVeque, R. J. (2002). Finite volume methods for hyberbolic problems. Cambridge Texts in Applied Mathematics. Cambridge.
Löhner, R. (2011). Error and work estimates for high-order elements. International Journal for Numerical Methods in Fluids, 67(12), 2184–2188. https://doi.org/10.1002/fld.2488.
Löhner, R. (2013). Improved error and work estimates for high-order elements. International Journal for Numerical Methods in Fluids, 72(11), 1207–1218. https://doi.org/10.1002/fld.3783.
Mengaldo, G., De Grazia, D., Moxey, D., Vincent, P. E., & Sherwin, S. J. (2015). Dealiasing techniques for high-order spectral element methods on regular and irregular grids. Journal of Computational Physics, 299, 56–81. https://doi.org/10.1016/j.jcp.2015.06.032.
Moura, R. C., Sherwin, S. J., & Peiró, J. (2015). Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/\(hp\) methods. Journal of Computational Physics, 298, 695–710. https://doi.org/10.1016/j.jcp.2015.06.020.
Nitsche, J. (1971). Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36(1), 9–15.
Noventa, G., Massa, F., Bassi, F., Colombo, A., Franchina, N., & Ghidoni, A. (2016). A high-order discontinuous Galerkin solver for unsteady incompressible turbulent flows. Computers & Fluids, 139, 248–260. https://doi.org/10.1016/j.compfluid.2016.03.007.
Persson, P.-O., & Peraire, J. (2006). Sub-cell shock capturing for discontinuous Galerkin methods. In 44th AIAA Aerospace Sciences Meeting and Exhibit. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2006-112.
Peterson, T. E. (1991). A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM Journal on Numerical Analysis, 28(1), 133–140. https://doi.org/10.1137/0728006.
Reed, W.H., & Hill, T. R. (1973). Triangluar mesh methods for the neutron transport equation. Technical report, Los Alamos Scientific Lab, New Mexico (USA). http://www.osti.gov/scitech/servlets/purl/4491151.
Reinarz, A., Charrier, D. E., Bader, M., Bovard, L., Dumbser, M., Duru, K., Fambri, F., Gabriel, A.-A., Gallard, J.-M., Köppel, S., Krenz, L., Rannabauer, L., Rezzolla, L., Samfass, P., Tavelli, M., & Weinzierl, T. (2020). ExaHyPE: An engine for parallel dynamically adaptive simulations of wave problems. Computer Physics Communications, 254, 107251. https://doi.org/10.1016/j.cpc.2020.107251.
Rivière, B., Wheeler, M. F., & Girault, V. (1999). Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Computational Geosciences, 3(3/4), 337–360. https://doi.org/10.1023/a:1011591328604.
Schoeder, S., Kormann, K., Wall, W. A., & Kronbichler, M. (2018a). Efficient explicit time step** of high order discontinuous Galerkin schemes for waves. SIAM Journal on Scientific Computing, 40(6), C803–C826. https://doi.org/10.1137/18M1185399.
Schoeder, S., Kronbichler, M., & Wall, W. A. (2018b). Arbitrary high-order explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. Journal of Scientific Computing, 76, 969–1006. https://doi.org/10.1007/s10915-018-0649-2.
Sevilla, R., Fernández-Méndez, S., & Huerta, A. (2008). NURBS-enhanced finite element method (NEFEM). International Journal for Numerical Methods in Engineering, 76(1), 56–83. https://doi.org/10.1002/nme.2311.
Shahbazi, K. (2005). An explicit expression for the penalty parameter of the interior penalty method. Journal of Computational Physics, 205, 401–407. https://doi.org/10.1016/j.jcp.2004.11.017.
Shu, C.-W. (1988). Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing, 9(6), 1073–1084. https://doi.org/10.1137/0909073.
Shu, C.-W. (2018). Bound-preserving high-order schemes for hyperbolic equations: Survey and recent developments. In Theory, numerics and applications of hyperbolic problems II (pp. 591–603). Berlin: Springer International Publishing. https://doi.org/10.1007/978-3-319-91548-7_44.
Å olÃn, P., Segeth, K., & Doležel, I. (2004). High-order finite element methods. Boca Raton, FL, USA: Chaptman & Hall/CRC.
Strang, G., & Fix, G. F. (1988). An analysis of the finite element method. Wellesley, MA, USA: Wellesley-Cambridge Press.
Sudirham, J. J., van der Vegt, J. J. W., & van Damme, R. M. J. (2006). Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains. Applied Numerical Mathematics, 56(12), 1491–1518. https://doi.org/10.1016/j.apnum.2005.11.003.
Taylor, M. A., Wingate, B. A., & Vincent, R. E. (2000). An algorithm for computing Fekete points in the triangle. SIAM Journal on Numerical Analysis, 38(5), 1707–1720. https://doi.org/10.1137/s0036142998337247.
Toulorge, T., & Desmet, W. (2012). Optimal Runge-Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems. Journal of Computational Physics, 231(4), 2067–2091. https://doi.org/10.1016/j.jcp.2011.11.024.
Tselios, K., & Simos, T. E. (2007). Optimized Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. Physics Letters A, 363(1–2), 38–47. https://doi.org/10.1016/j.physleta.2006.10.072.
Wang, Z. J. (2007). High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Progress in Aerospace Sciences, 43(1–3), 1–41. https://doi.org/10.1016/j.paerosci.2007.05.001.
Wang, Z. J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H. T., Kroll, N., May, G., Persson, P.-O., van Leer, B., & Visbal, M. (2013). High-order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 72(8), 811–845. https://doi.org/10.1002/fld.3767.
Warburton, T., & Hesthaven, J. S. (2003). On the constants in \(hp\)-finite element trace inverse inequalities. Computer Methods in Applied Mechanics and Engineering, 192, 2765–2773. https://doi.org/10.1016/S0045-7825(03)00294.
Winters, A. R., Moura, R. C., Mengaldo, G., Gassner, G. J., Walch, S., Peiró, J., & Sherwin, S. J. (2018). A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. Journal of Computational Physics, 372, 1–21. https://doi.org/10.1016/j.jcp.2018.06.016.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 CISM International Centre for Mechanical Sciences, Udine
About this chapter
Cite this chapter
Kronbichler, M. (2021). The Discontinuous Galerkin Method: Derivation and Properties. In: Kronbichler, M., Persson, PO. (eds) Efficient High-Order Discretizations for Computational Fluid Dynamics. CISM International Centre for Mechanical Sciences, vol 602. Springer, Cham. https://doi.org/10.1007/978-3-030-60610-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-60610-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-60609-1
Online ISBN: 978-3-030-60610-7
eBook Packages: EngineeringEngineering (R0)