The hp nonconforming mesh refinement in discontinuous Galerkin finite element method based on Zienkiewicz-Zhu error estimation

• Autorzy: Jaśkowiec J.
• Języki publikacji: EN

Abstrakty

EN

This paper deals with hp-type adaptation in the discontinuous Galerkin (DG) method. The DG method is formulated in this paper with a non-zero mesh skeleton width, which leads to a version of the method called in this paper the interface discontinuous Galerkin (IDG) method. In this formulation, the mesh skeleton has a finite volume and special finite elements are used for discretization. The skeleton spatial calculations are performed using the finite difference or mid-values formulas which are based on the shape functions of the neighbouring finite elements. The Dirichlet boundary conditions are applied using a nonzero width of the material between the outer boundary and a finite element aligned with the boundary. Next, the paper discusses the mesh refinement of hp type. In the IDG method, the mesh does not have to be conforming. The Zienkiewicz-Zhu (ZZ) error indicator is adapted in the IDG method for the purpose of mesh refinement. The paper is illustrated with two-dimensional examples, in which the mesh refinement for an elliptic problem is performed.

Słowa kluczowe

EN

discontinuous Galerkin method; hp refinement; Zienkiewicz-Zhu error estimation

PL

nieciągła metoda Galerkina; HP udoskonalenie; Zienkiewicz-Zhu szacowanie błędu

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Methods in Engineering and Science
• Rocznik: 2016
• Tom: Vol. 23, no. 1
• Strony: 43--67
• Opis fizyczny: Bibliogr. 57 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] M Ainsworth. A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM Journal on Numerical Analysis, 45(4): 1777–1798, 2007.
• [2] J.H. Argyris, I. Fried, D.W. Scharpf. The TUBA family of plate elements for the matrix displacement method. The Aeronautical Journal, 72(692): 701–709, 1968.
• [3] 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.
• [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] 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.
• [6] 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.
• [7] E. Burman, B. Stamm. Bubble stabilized discontinuous Galerkin methods on conforming and non-conforming meshes. Calcolo, 48(2): 189–209, 2011.
• [8] U.H. C¸ alık-Karaköse, H. Askes. A recovery-type a posteriori error estimator for gradient elasticity. Computers & Structures, 154: 204–209, 2015.
• [9] Y Chen, J. Huang, Y Huang, Y Xu. On the local discontinuous Galerkin method for linear elasticity. Mathematical Problems in Engineering, 2010: 20 pp., 2010.
• [10] 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.
• [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, 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.
• [13] 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, September 1989.
• [14] E. Creusé, S. Nicaise. A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods. Journal of Computational and Applied Mathematics, 234(10): 2903–2915, 2010.
• [15] C. Dawson, S. Sun, M.F. Wheeler. Compatible algorithms for coupled flow and transport. Computer Methods in Applied Mechanics and Engineering, 193(23–26): 2565–2580, 2004.
• [16] L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy. Toward a universal h-p adaptive finite element strategy. Part 1. Constrained approximation and data structure. Computer Methods in Applied Mechanics and Engineering, 77(1): 79–112, 1989.
• [17] L.F. Demkowicz, K. Banas, J. Kucwaj, W. Rachowicz. 2D hp adaptive package. Technical Report SAM Report 4/1992, Section of Applied Mathematics, Cracow University of Technology, September 1992.
• [18] 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.
• [19] Y. Epshteyn, B. Riviére. Analysis of discontinuous Galerkin methods for incompressible two-phase flow. Journal of Computational and Applied Mathematics, 225(2): 487–509, 2009.
• [20] A. Ern, J. Proft. A posteriori discontinuous Galerkin error estimates for transient convection-diffusion equations. Applied Mathematics Letters, 18(7): 833–841, 2005.
• [21] H. Fahs, S. Lanteri. A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics. Journal of Computational and Applied Mathematics, 234(4): 1088–1096, 2010. Proceedings of the Thirteenth International Congress on Computational and Applied Mathematics (ICCAM-2008), Ghent, Belgium, 7–11 July, 2008.
• [22] P.E. Farrell, S. Micheletti, S. Perotto. An anisotropic Zienkiewicz-Zhu-type error estimator for 3D applications. International Journal for Numerical Methods in Engineering, 85(6): 671–692, 2011.
• [23] 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.
• [24] G. Gassner, M. Staudenmaier, F. Hindenlang, M. Atak, C.-D. Munz. A space-time adaptive discontinuous Galerkin scheme. Computers & Fluids, 117: 247–261, 2015.
• [25] 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.
• [26] S. Giani, D. Schötzau, L. Zhu. An a-posteriori error estimate for hp-adaptive DG methods for convectiondiffusion problems on anisotropically refined meshes. Computers & Mathematics with Applications, High-order Finite Element Approximation for Partial Differential Equations, 67(4): 869–887, 2014.
• [27] G. Haikal, K.D. Hjelmstad. An enriched discontinuous Galerkin formulation for the coupling of non-conforming meshes. Finite Elements in Analysis and Design, The Twenty-First Annual Robert J. Melosh Competition, 46(6): 496–503, 2010.
• [28] 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.
• [29] P. Houston, B. Senior, E. Süli. hp-Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity. International Journal for Numerical Methods in Fluids, 40(1–2): 153–169, 2002.
• [30] J. Jaśkowiec, S. Milewski. The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions. Computers & Mathematics with Applications, 70(5): 962–979, 2015.
• [31] J. Jaśkowiec, P. Pluciński, J. Pamin. Thermo-mechanical XFEM-type modeling of laminated structure with thin inner layer. Engineering Structures, 100:511–521, 2015.
• [32] 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.
• [33] B. Karasözen, M. Uzunca, M. Manguoǧlu. Adaptive discontinuous Galerkin methods for nonlinear diffusionconvection-reaction equations. [In:] A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso [Eds.], Numerical Mathematics and Advanced Applications – ENUMATH 2013, vol. 103 of Lecture Notes in Computational Science and Engineering, pp. 85–93. Springer International Publishing, 2015.
• [34] H.H. Kim, E.T. Chung, C.-Y. Lam. Mortar formulation for a class of staggered discontinuous Galerkin methods. Computers & Mathematics with Applications, 71(8): 1568–1585, 2016.
• [35] S. Micheletti, S. Perotto. Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator. Computer Methods in Applied Mechanics and Engineering, 195(9–12): 799–835, 2006.
• [36] 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.
• [37] W.H. Reed, T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR- 73-479, Los Alamos Scientific Laboratory, Oct 1973.
• [38] B. Rivière. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics, 2008.
• [39] 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.
• [40] 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.
• [41] R. Schneider, Y. Xu, A. Zhou. An analysis of discontinuous Galerkin methods for elliptic problems. Advances in Computational Mathematics, 25(1–3): 259–286, 2006.
• [42] 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.
• [43] 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.
• [44] L. Song, Z. Zhang. Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method. Journal of Computational Physics, 299: 1004–1020, 2015.
• [45] F. Stan. Discontinuous Galerkin method for interface crack propagation. International Journal of Material Forming, 1(1): 1127–1130, 2008.
• [46] B.A. Szabó, I. Babuška. Finite Element Analysis. John Wiley & Sons Inc., New York, 1991.
• [47] M. Uzunca, B. Karasözen, M. Manguoǧlu. Adaptive discontinuous Galerkin methods for non-linear diffusionconvection-reaction equations. Computers & Chemical Engineering, 68: 24–37, 2014.
• [48] G.N. Wells, K. Garikipati, L. Molari. A discontinuous Galerkin formulation for a strain gradient-dependent damage model. Computer Methods in Applied Mechanics and Engineering, 193(33–35): 3633–3645, 2004.
• [49] M. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis, 15(1): 152–161, 1978.
• [50] G. Zboinski. Application of the three-dimensional triangular-prism hpq adaptive finite element to plate and shell analysis. Computers & Structures, 65(4): 497–514, 1997.
• [51] G. Zboiński. Hierarchical Modeling and Finite Element Approximation for Adaptive Analysis of Complex Structures [in Polish: Modelowanie hierarchiczne i metoda elementów skończonych do adaptacyjnej analizy struktur złożonych]. DSc Thesis, 520/1479/2001. Institute of Fluid Flow Machinery, Gdańsk, 2001.
• [52] 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.
• [53] T. Zhang, S. Yu. The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method. Applied Numerical Mathematics, 85: 128–141, 2014.
• [54] Z. Zhang. Ultraconvergence of the patch recovery technique II. Math. Comput., 69(229): 141–158, January 2000.
• [55] O.C. Zienkiewicz, J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. International Journal for Numerical Methods in Engineering, 24(2): 337–357, 1987.
• [56] O.C. Zienkiewicz, J.Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. International Journal for Numerical Methods in Engineering, 33(7): 1331–1364, 1992.
• [57] O.C. Zienkiewicz, J.Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33(7): 1365–1382, 1992.