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

Published online by Cambridge University Press:  31 August 2012

Teemu Luostari ,
Tomi Huttunen  and
Peter Monk
Teemu Luostari
Affiliation:
Department of Applied Physics, University of Eastern Finland P.O. Box 1627, 70211 Kuopio, Finland. teemu.luostari@uef.fi; tomi.huttunen@uef.fi
Tomi Huttunen
Affiliation:
Department of Applied Physics, University of Eastern Finland P.O. Box 1627, 70211 Kuopio, Finland. teemu.luostari@uef.fi; tomi.huttunen@uef.fi
Peter Monk
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, 19716 DE, USA.; monk@math.udel.edu
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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.

Keywords

Ultra weak variational formulationerror estimatesplane wave basislinear elasticityupwind discontinuous Galerkin method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 183 - 211
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. Google Scholar
Barnett, A.H. and Betcke, T., An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons. SIAM J. Sci. Comput. 32 (2010) 1417–1441. Google Scholar
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. Springer (2008).
Buffa, A. and Monk, P., Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM : M2AN 42 (2008) 925–940. Google Scholar
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).
Cessenat, O. and Després, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255–299. Google Scholar
Cummings, P. and Feng, X., Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Mod. Methods Appl. Sci. 16 (2006) 139–160. Google Scholar
El Kacimi, A. and Laghrouche, O., 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. Google Scholar
Farhat, C., Harari, I. and Franca, L.P., A discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6455–6479. Google Scholar
Farhat, C., Harari, I. and Hetmaniuk, U., 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. Google Scholar
Gabard, G., Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225 (2007) 1961–1984. Google Scholar
R. Hardin, N. Sloane and W. Smith, Spherical coverings. Available on http://www.research.att.com/˜njas/coverings/index.html (1994).
Hiptmair, R., Moiola, A. and Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation : analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–284. Google Scholar
R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. In press.
Huttunen, T., Monk, P. and Kaipio, J.P., Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182 (2002) 27–46. Google Scholar
Huttunen, T., Monk, P., Collino, F. and Kaipio, J.P., The ultra weak variational formulation for elastic wave problems. SIAM J. Sci. Comput. 25 (2004) 1717–1742. Google Scholar
Huttunen, T., Monk, P. and Kaipio, J.P., The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation. Int. J. Numer. Methods Eng. 61 (2004) 1072–1092. Google Scholar
Huttunen, T., Malinen, M. and Monk, P., Solving Maxwell’s equations using the ultra weak variational formulation. J. Comput. Phys. 223 (2007) 731–758. Google Scholar
Huttunen, T., J.P. Kaipio and P. Monk,An ultra-weak method for acoustic fluid-solid interaction. J. Comput. Appl. Math. 213 (2008) 1667–1685. Google Scholar
V.D. Kupradze, Potential methods in the theory of elasticity. Israel Program for Scientific Translations (1965).
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 http://www.acoustics.asn.au.
Massimi, P., Tezaur, R. and Farhat, C., 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. Google Scholar
Melenk, M.M. and Babuška, I., The partition of unity finite element method : basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1996) 289–314. Google Scholar
A. Moiola, Trefftz-Discontinuous Galerkin Methods for Time-Harmonic Wave Problems. Ph.D. thesis, ETH Zürich (2011).
A. Moiola, Plane wave approximation in linear elasticity. To appear in Appl. Anal.
Moiola, A., Hiptmair, R. and Perugia, I., Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 65 (2011) 809–837. Google Scholar
Monk, P. and Wang, D.-Q., A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 175 (1999) 121–136. Google Scholar
Pao, Y.-H., Betti’s identity and transition matrix for elastic waves. J. Acoust. Soc. Am. 64 (1978) 302–310. Google Scholar
Perrey-Debain, E., Plane wave decomposition in the unit disc : convergence estimates and computational aspects. J. Comput. Appl. Math. 193 (2006) 140–156. Google Scholar
Sloan, I. and Womersley, R., Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21 (2004) 107–125. Google Scholar
Wang, D., Toivanen, J., 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. Google Scholar
R. Womersley and I. Sloan, Interpolation and cubature on the sphere. Available on http://web.maths.unsw.edu.au/˜rsw/Sphere.