Abstract
In this note we propose a discontinuous Galerkin in space, continuous Galerkin in time method for a problem arising in elastodynamics with phase transition. We make use of a dispersion operator from (Bona et al., Math. Comput. 82(283), 1401–1432, 2013) [3] allowing us to construct a consistent scheme. We derive goal-oriented a posteriori error estimators for this scheme based on dual weighted residuals. We conclude by summarising extensive numerical experiments.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abeyaratne, R., Knowles, J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal. 114(2), 119–154 (1991)
Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)
Bona, J.L., Chen, H., Karakashian, O., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)
Braack, M., Prohl, A.: Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension. M2AN Math. Model. Numer. Anal. 47, 401–420 (2013)
Chalons, C., LeFloch, P.G.: High-order entropy-conservative schemes and kinetic relations for van der Waals fluids. J. Comput. Phys. 168(1), 184–206 (2001)
Cheng, Y., Shu, C.W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2008)
Diehl, D., Kremser, J., Kröner, D., Rohde, C.: Numerical solution of Navier–Stokes–Korteweg systems by local discontinuous Galerkin methods in multiple space dimensions. Appl. Math. Comput. 272, Part 2, 309–335 (2016)
Estep, D., French, D.: Global error control for the continuous Galerkin finite element method for ordinary differential equations. RAIRO Modél. Math. Anal. Numér. 28(7), 815–852 (1994)
Giesselmann, J.: Low mach asymptotic preserving scheme for the Euler-Korteweg model. IMA J. Numer. Anal. 32(2), 802–832 (2015)
Giesselmann, J.: Relative entropy in multi-phase models of 1d elastodynamics: convergence of a non-local to a local model. J. Differ. Equ. 258, 3589–3606 (2015)
Giesselmann, J., Makridakis, C., Pryer, T.: Energy consistent DG methods for the Navier-Stokes-Korteweg system. Math. Comput. 83, 2071–2099 (2014)
Giesselmann, J., Pryer, T.: Goal oriented a posteriori for an elastodynamics model with phase transition. In preparation
Giesselmann, J., Pryer, T.: Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics. IMA J. Numer. Anal. 36(4), 1685–1714 (2016)
Jamet, D., Lebaigue, O., Coutris, N., Delhaye, J.M.: The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169(2), 624–651 (2001)
Jamet, D., Torres, D., Brackbill, J.: On the theory and computation of surface tension: the elimination of parasitic currents through energy conservation in the second-gradient method. J. Comput. Phys 182, 262–276 (2002)
Karakashian, O., Makridakis, C.: A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation. Math. Comput. 84(293), 1145–1167 (2015)
Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)
Lakkis, O., Pryer, T.: Gradient recovery in adaptive finite-element methods for parabolic problems. IMA J. Numer. Anal. 32(1), 246–278 (2012)
Slemrod, M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81(4), 301–315 (1983)
Tian, L., Xu, Y., Kuerten, J.G.M., van der Vegt, J.J.W.: An \(h\)-adaptive local discontinuous Galerkin method for the Navier-Stokes-Korteweg equations. J. Comput. Phys. 319, 242–265 (2016)
Acknowledgements
T.P. gratefully acknowledges support of the EPSRC grant EP/P000835/1. J.G. gratefully acknowledges support of the Baden-Württemberg fundation for the project “Numerical Methods for Multi-Phase Flows with Strongly Varying Mach Numbers”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Giesselmann, J., Pryer, T. (2017). Goal-Oriented Error Analysis of a DG Scheme for a Second Gradient Elastodynamics Model. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects . FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-319-57397-7_39
Download citation
DOI: https://doi.org/10.1007/978-3-319-57397-7_39
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-57396-0
Online ISBN: 978-3-319-57397-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)