New analytical tools for HDG in elasticity, with applications to elastodynamics

Du, Shukai; Sayas, Francisco-Javier

doi:10.1090/mcom/3499

New analytical tools for HDG in elasticity, with applications to elastodynamics
HTML articles powered by AMS MathViewer

by Shukai Du and Francisco-Javier Sayas;

Math. Comp. 89 (2020), 1745-1782

DOI: https://doi.org/10.1090/mcom/3499

Published electronically: December 30, 2019

HTML | PDF | Request permission

Abstract:

We present some new analytical tools for the error analysis of hybridizable discontinuous Galerkin (HDG) methods for linear elasticity. These tools allow us to analyze more variants of the HDG method using the projection-based approach, which renders the error analysis simple and concise. The key result is a tailored projection for the Lehrenfeld–Schöberl type HDG (HDG+ for simplicity) methods. By using the projection we recover the error estimates of HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We also present a semidiscrete (in space) HDG+ method for transient elastic waves and prove it is uniformly-in-time optimal convergent by using the projection-based error analysis. Numerical experiments supporting our analysis are presented at the end.

References

Lehel Banjai, Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments, SIAM J. Sci. Comput. 32 (2010), no. 5, 2964–2994. MR 2729447, DOI 10.1137/090775981
M. Bercovier and E. Livne, A $4$ CST quadrilateral element for incompressible materials and nearly incompressible materials, Calcolo 16 (1979), no. 1, 5–19. MR 555452, DOI 10.1007/BF02575759
Brandon Chabaud and Bernardo Cockburn, Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp. 81 (2012), no. 277, 107–129. MR 2833489, DOI 10.1090/S0025-5718-2011-02525-1
Bernardo Cockburn and Guosheng Fu, Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by $M$-decompositions, IMA J. Numer. Anal. 38 (2018), no. 2, 566–604. MR 3800033, DOI 10.1093/imanum/drx025
Bernardo Cockburn, Guosheng Fu, and Francisco Javier Sayas, Superconvergence by $M$-decompositions. Part I: General theory for HDG methods for diffusion, Math. Comp. 86 (2017), no. 306, 1609–1641. MR 3626530, DOI 10.1090/mcom/3140
Bernardo Cockburn, Zhixing Fu, Allan Hungria, Liangyue Ji, Manuel A. Sánchez, and Francisco-Javier Sayas, Stormer-Numerov HDG methods for acoustic waves, J. Sci. Comput. 75 (2018), no. 2, 597–624. MR 3780780, DOI 10.1007/s10915-017-0547-z
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn and Kassem Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math. 130 (2015), no. 2, 293–314. MR 3343926, DOI 10.1007/s00211-014-0661-x
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327–1353. MR 2904581, DOI 10.1090/S0025-5718-2011-02550-0
Bernardo Cockburn and Vincent Quenneville-Bélair, Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation, Math. Comp. 83 (2014), no. 285, 65–85. MR 3120582, DOI 10.1090/S0025-5718-2013-02743-3
Bernardo Cockburn and Ke Shi, Superconvergent HDG methods for linear elasticity with weakly symmetric stresses, IMA J. Numer. Anal. 33 (2013), no. 3, 747–770. MR 3081483, DOI 10.1093/imanum/drs020
Shukai Du and Francisco-Javier Sayas, An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method, SpringerBriefs in Mathematics, 2019.
G. Fu, B. Cockburn, and H. Stolarski, Analysis of an HDG method for linear elasticity, Internat. J. Numer. Methods Engrg. 102 (2015), no. 3-4, 551–575. MR 3340089, DOI 10.1002/nme.4781
Roland Griesmaier and Peter Monk, Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Sci. Comput. 49 (2011), no. 3, 291–310. MR 2853152, DOI 10.1007/s10915-011-9460-z
Allan Hungria, Daniele Prada, and Francisco-Javier Sayas, HDG methods for elastodynamics, Comput. Math. Appl. 74 (2017), no. 11, 2671–2690. MR 3725823, DOI 10.1016/j.camwa.2017.08.016
Christoph Lehrenfeld, Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems, Rheinisch-Westfalischen Technischen Hochschule Aachen, 2010.
C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. MR 923707, DOI 10.1007/BF01398686
Issei Oikawa, A hybridized discontinuous Galerkin method with reduced stabilization, J. Sci. Comput. 65 (2015), no. 1, 327–340. MR 3394448, DOI 10.1007/s10915-014-9962-6
Weifeng Qiu, Jiguang Shen, and Ke Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comp. 87 (2018), no. 309, 69–93. MR 3716189, DOI 10.1090/mcom/3249
Weifeng Qiu and Ke Shi, An HDG method for convection diffusion equation, J. Sci. Comput. 66 (2016), no. 1, 346–357. MR 3440284, DOI 10.1007/s10915-015-0024-5
Weifeng Qiu and Ke Shi, A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes, IMA J. Numer. Anal. 36 (2016), no. 4, 1943–1967. MR 3556409, DOI 10.1093/imanum/drv067
Francisco-Javier Sayas, From Raviart-Thomas to HDG, arXiv:1307.2491 (2013).
S.-C. Soon, B. Cockburn, and Henryk K. Stolarski, A hybridizable discontinuous Galerkin method for linear elasticity, Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. MR 2589528, DOI 10.1002/nme.2646