Error estimates for the ultra weak variational formulation in linear elasticity∗

Teemu Luostari; Tomi Huttunen; Peter Monk

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 47, Issue: 1, page 183-211
  • ISSN: 0764-583X

Abstract

top
We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier’s equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L2(Ω) norm in terms of the best approximation error. Our final result is an L2(Ω) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.

How to cite

top

Luostari, Teemu, Huttunen, Tomi, and Monk, Peter. "Error estimates for the ultra weak variational formulation in linear elasticity∗." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2012): 183-211. <http://eudml.org/doc/222157>.

@article{Luostari2012,
abstract = {We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier’s equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L2(Ω) norm in terms of the best approximation error. Our final result is an L2(Ω) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.},
author = {Luostari, Teemu, Huttunen, Tomi, Monk, Peter},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Ultra weak variational formulation; error estimates; plane wave basis; linear elasticity; upwind discontinuous Galerkin method; ultra weak variational formulation},
language = {eng},
month = {8},
number = {1},
pages = {183-211},
publisher = {EDP Sciences},
title = {Error estimates for the ultra weak variational formulation in linear elasticity∗},
url = {http://eudml.org/doc/222157},
volume = {47},
year = {2012},
}

TY - JOUR
AU - Luostari, Teemu
AU - Huttunen, Tomi
AU - Monk, Peter
TI - Error estimates for the ultra weak variational formulation in linear elasticity∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/8//
PB - EDP Sciences
VL - 47
IS - 1
SP - 183
EP - 211
AB - We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier’s equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L2(Ω) norm in terms of the best approximation error. Our final result is an L2(Ω) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.
LA - eng
KW - Ultra weak variational formulation; error estimates; plane wave basis; linear elasticity; upwind discontinuous Galerkin method; ultra weak variational formulation
UR - http://eudml.org/doc/222157
ER -

References

top
  1. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779.  Zbl1008.65080
  2. A.H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons. SIAM J. Sci. Comput.32 (2010) 1417–1441.  Zbl1216.65151
  3. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. Springer (2008).  Zbl1135.65042
  4. A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM : M2AN42 (2008) 925–940.  Zbl1155.65094
  5. O. Cessenat, Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques. Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine (1996).  
  6. O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal.35 (1998) 255–299.  Zbl0955.65081
  7. P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Mod. Methods Appl. Sci.16 (2006) 139–160.  Zbl1134.35317
  8. A. El Kacimi and O. Laghrouche, Numerical modeling of elastic wave scattering in frequency domain by partition of unity finite element method. Int. J. Numer. Methods Eng.77 (2009) 1646–1669.  Zbl1158.74485
  9. C. Farhat, I. Harari and L.P. Franca, A discontinuous enrichment method. Comput. Methods Appl. Mech. Eng.190 (2001) 6455–6479.  Zbl1002.76065
  10. C. Farhat, I. Harari and U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng.192 (2003) 1389–1429.  Zbl1027.76028
  11. G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys.225 (2007) 1961–1984.  Zbl1123.65102
  12. R. Hardin, N. Sloane and W. Smith, Spherical coverings. Available on (1994).  URIhttp://www.research.att.com/˜njas/coverings/index.html
  13. R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation : analysis of the p-version. SIAM J. Numer. Anal.49 (2011) 264–284.  
  14. R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. In press.  
  15. T. Huttunen, P. Monk and J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys.182 (2002) 27–46.  Zbl1015.65064
  16. T. Huttunen, P. Monk, F. Collino and J.P. Kaipio, The ultra weak variational formulation for elastic wave problems. SIAM J. Sci. Comput.25 (2004) 1717–1742.  Zbl1093.74028
  17. T. Huttunen, P. Monk and J.P. Kaipio, The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation. Int. J. Numer. Methods Eng.61 (2004) 1072–1092.  Zbl1075.76648
  18. T. Huttunen, M. Malinen and P. Monk, Solving Maxwell’s equations using the ultra weak variational formulation. J. Comput. Phys.223 (2007) 731–758.  Zbl1117.78011
  19. T. Huttunen, J.P. Kaipio and P. Monk,An ultra-weak method for acoustic fluid-solid interaction. J. Comput. Appl. Math.213 (2008) 1667–1685.  Zbl1182.76949
  20. V.D. Kupradze, Potential methods in the theory of elasticity. Israel Program for Scientific Translations (1965).  Zbl0188.56901
  21. T. Luostari, T. Huttunen and P. Monk, The ultra weak variational formulation for 3D elastic wave problems, in Proc. 20th International Congress on Acoustics, ICA (2010).Available in 2010 on  Zbl06183188URIhttp://www.acoustics.asn.au.
  22. P. Massimi, R. Tezaur and C. Farhat, A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media. Int. J. Numer. Methods Eng.76 (2008) 400–425.  Zbl1195.74292
  23. M.M. Melenk and I. Babuška, The partition of unity finite element method : basic theory and applications. Comput. Methods Appl. Mech. Eng.139 (1996) 289–314.  Zbl0881.65099
  24. A. Moiola, Trefftz-Discontinuous Galerkin Methods for Time-Harmonic Wave Problems. Ph.D. thesis, ETH Zürich (2011).  
  25. A. Moiola, Plane wave approximation in linear elasticity. To appear in Appl. Anal. Zbl1319.74003
  26. A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys.65 (2011) 809–837.  Zbl1263.35070
  27. P. Monk and D.-Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng.175 (1999) 121–136.  Zbl0943.65127
  28. Y.-H. Pao, Betti’s identity and transition matrix for elastic waves. J. Acoust. Soc. Am.64 (1978) 302–310.  Zbl0399.73038
  29. E. Perrey-Debain, Plane wave decomposition in the unit disc : convergence estimates and computational aspects. J. Comput. Appl. Math.193 (2006) 140–156.  Zbl1092.65092
  30. I. Sloan and R. Womersley, Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math.21 (2004) 107–125.  Zbl1055.65038
  31. D. Wang, J. Toivanen, R. Tezaur and C. Farhat,Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons. Int. J. Numer. Methods Eng.89 (2012) 403–417.  Zbl1242.76143
  32. R. Womersley and I. Sloan, Interpolation and cubature on the sphere. Available on  Zbl0952.65013URIhttp://web.maths.unsw.edu.au/˜rsw/Sphere.

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.