The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions

• Autorzy: Jaśkowiec J.
• Języki publikacji: EN

Abstrakty

EN

This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.

Słowa kluczowe

EN

discontinuous Galerkin method; finite differences; compatibility condition; approximation basis

PL

nieciągła metoda Galerkina; różnice skończone; stan zgodności

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Methods in Engineering and Science
• Rocznik: 2016
• Tom: Vol. 23, no. 2/3
• Strony: 109--132
• Opis fizyczny: Bibliogr. 56 poz., rys., wykr.

Twórcy

autor

Jaśkowiec J.

• Cracow University of Technology, Faculty of Civil Engineering Institute for Computational Civil Engineering Warszawska 24, 31-155 Kraków, Poland

Bibliografia

• [1] D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4): 175–186, 1982.
• [2] D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5): 1749–1779, May 2001.
• [3] I. Babuška, J.M. Melenk. The partition of unity method. International Journal for Numerical Methods in Engineering, 40(4): 727–758, 1997.
• [4] G. Billet, J. Ryan. A Runge-Kutta discontinuous Galerkin approach to solve reactive flows: The hyperbolic operator. Journal of Computational Physics, 230(4): 1064–1083, 2011.
• [5] S. Brogniez, C. Farhat, E. Hachem. A high-order discontinuous Galerkin method with Lagrange multipliers for advection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 264: 49–66, 2013.
• [6] T. Bui-Thanh, O. Ghattas. Analysis of an hp-nonconforming discontinuous Galerkin spectral element method for wave propagation. SIAM Journal on Numerical Analysis, 50(3): 1801–1826, 2012.
• [7] N.K. Burgess, D.J. Mavriplis. hp-adaptive discontinuous Galerkin solver for the Navier-Stokes equations. American Institute of Aeronautics and Astronautics, 50(12): 2682–2692, 2012.
• [8] Y. Chen, J. Huang, Y. Huang, Y. Xu. On the local discontinuous Galerkin method for linear elasticity. Mathematical Problems in Engineering, p. 20, 2010.
• [9] B. Cockburn, S. Hou, C-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case. Mathematics of Computation, 54: 545–581, 1990.
• [10] B. Cockburn, S. Lin, C-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One-dimensional systems. Journal of Computational Physics, 84(1): 90–113, 1989.
• [11] B. Cockburn, C-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of Computation, 52(186): 411–435, 1989.
• [12] B. Cockburn, C-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35(6): 2440–2463, 1998.
• [13] C. Dawson, Sh. Sun, M.F. Wheeler. Compatible algorithms for coupled flow and transport. Computer Methods in Applied Mechanics and Engineering, 193(23–26): 2565–2580, 2004.
• [14] L. Demkowicz. Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1 One and Two Dimensional Elliptic and Maxwell Problems. Chapman & Hall/CRC Applied Mathematics & Nonlinear Science. CRC Press, 2006.
• [15] P.N. Dorival, B.P.S. Persival. Generalized finite element method in linear and nonlinear structural dynamic analyses. Engineering Computations, 33(3): 806–830, 2016.
• [16] M. Elliotis, G. Georgiou, Ch. Xenophontos. Solving Laplacian problems with boundary singularities: a comparison of a singular function boundary integral method with the p/hp version of the finite element method. Applied Mathematics and Computation, 169(1): 485–499, 2005.
• [17] G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, R.L. Taylor. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Computer Methods in Applied Mechanics and Engineering, 191(34): 3669–3750, 2002.
• [18] Y. Epshteyn, B. Rivire. Analysis of discontinuous Galerkin methods for incompressible two-phase flow. Journal of Computational and Applied Mathematics, 225(2): 487–509, 2009.
• [19] Ch. Farhat, I. Harari, U. Hetmaniuk. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Computer Methods in Applied Mechanics and Engineering, 192(11–12): 1389–1419, 2003.
• [20] D. Fournier, R. Herbin, R. Tellier. Discontinuous Galerkin discretization and hp-refinement for the resolution of the neutron transport equation. SIAM Journal on Scientific Computing, 35(2): A936–A956, 2013.
• [21] E. Georgoulis, E. Hall, P. Houston. Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM Journal on Scientific Computing, 30(1): 246–271, 2008.
• [22] C. Gürkan, E. Sala-Lardies, M. Kronbichler, S. Fernández-Méndez. eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems, pp. 103–122. Springer International Publishing, Cham, 2016.
• [23] P. Hansbo, M.G. Larson. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Computer Methods in Applied Mechanics and Engineering, 191(17–18): 1895–1908, 2002.
• [24] J. Jaśkowiec. The hp nonconforming mesh refinement based on Zienkiewicz-Zhu error estimation in discontinuous Galerkin finite element method. Computer Assisted Methods in Engineering and Science, 23(1): 43–67, 2016.
• [25] J. Jaśkowiec, F.P. van der Meer. A consistent iterative scheme for 2D and 3D cohesive crack analysis in XFEM. Computers & Structures, 136: 98–107, 2014.
• [26] L. Ji, Y. Xu. Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on cartesian meshes. J. Sci. Comput., 51(1): 1–27, April 2012.
• [27] H. Kaneko, K.S. Bey, G.J.W. Hou. A discontinuous Galerkin method for parabolic problems with modified hp-finite element approximation technique. Applied Mathematics and Computation, 182(2): 1405–1417, 2006.
• [28] H. Kaneko, K.S. Bey, Y. Lenbury, P. Toghaw. Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates. Applied Mathematics and Computation, 201(1–2): 414–430, 2008.
• [29] M-Y. Kim. A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions. Computers & Mathematics with Applications, 70(4): 488–506, 2015.
• [30] F. Kummer. Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization. International Journal for Numerical Methods in Engineering, 109(2): 259–289, 2017.
• [31] D. Kuzmin. A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. Journal of Computational and Applied Mathematics, 233(12): 3077–3085, 2010.
• [32] H. Luo, J.D. Baum, R. Löhner. A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. Journal of Computational Physics, 227(20): 8875–8893, 2008.
• [33] H. Luo, L. Luo, R. Nourgaliev, V.A. Mousseau, N. Dinh. A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids. Journal of Computational Physics, 229(19): 6961–6978, 2010.
• [34] J.M. Melenk, I. Babuška. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1–4): 289–314, 1996.
• [35] N. Moës, T. Belytschko. Extended finite element method for cohesive crack growth. Engineering Fracture Mechanics, 69(7): 813–833, 2002.
• [36] N.C. Nguyen, J. Peraire, B. Cockburn. Hybridizable Discontinuous Galerkin Methods, pp. 63–84. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.
• [37] J. Nitsche. 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, 1971.
• [38] D.A. Di Pietro, S. Nicaise. A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media. Applied Numerical Mathematics, 63: 105–116, 2013.
• [39] W.H. Reed, T.R. Hill. Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
• [40] B. Riviére. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics, 2008.
• [41] B. Riviére, M.F. Wheeler, K. Banaś. Part II. Discontinuous Galerkin method applied to a single phase flow in porous media. Computational Geosciences, 4(4): 337–349, 2000.
• [42] B. Riviére, M. Wheeler, V. Girault. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM Journal on Numerical Analysis, 39(3): 902–931, 2001.
• [43] Y. Shen, A. Lew. Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics. Computer Methods in Applied Mechanics and Engineering, 199(37–40): 2360– 2382, 2010.
• [44] Y. Shen, A.J. Lew. A locking-free and optimally convergent discontinuous-Galerkin-based extended finite element method for cracked nearly incompressible solids. Computer Methods in Applied Mechanics and Engineering, 273: 119–142, 2014.
• [45] F. Stan. Discontinuous Galerkin method for interface crack propagation. International Journal of Material Forming, 1(1): 1127–1130, 2008.
• [46] T. Strouboulis, I. Babuška, K. Copps. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 181(1–3): 43–69, 2000.
• [47] T. Strouboulis, K. Copps, I. Babuška. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 190(32–33): 4081–4193, 2001.
• [48] I. Toulopoulos. An interior penalty discontinuous Galerkin finite element method for quasilinear parabolic problems. Finite Elements in Analysis and Design, 95: 42–50, 2015.
• [49] F. Vilar, P-H. Maire, R. Abgrall. A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids. Journal of Computational Physics, 276: 188–234, 2014.
• [50] M. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis, 15(1): 152–161, 1978.
• [51] Y. Xia, H. Luo, M. Frisbey, R. Nourgaliev. A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. Computers & Fluids, 98: 134–151, 2014.
• [52] L. Yuan, C-W. Shu. Discontinuous Galerkin method based on non-polynomial approximation spaces. Journal of Computational Physics, 218(1): 295–323, 2006.
• [53] G. Zboiński, L. Demkowicz. Application of the 3D hpq Adaptive Finite Element for Plate and Shell Analysis. Technical Report TICAM Report 94-13, Texas Institute for Computational and Applied Mathematics. The University of Texas at Austin, Austin (Texas), 1994.
• [54] G. Zboinski. Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximations. Computer Methods in Applied Mechanics and Engineering, 199(45–48): 2913–2940, 2010.
• [55] Q. Zhang, C-W. Shu. Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal., 48(3): 1038–1063, July 2010.
• [56] T. Zhang, Sh. Yu. The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method. Applied Numerical Mathematics, 85: 128–141, 2014.