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

Teemu Luostari; Tomi Huttunen; Peter Monk

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

## Access Full Article

top## Abstract

top## How to cite

topLuostari, Teemu, Huttunen, Tomi, and Monk, Peter. "Error estimates for the ultra weak variational formulation in linear elasticity." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 183-211. <http://eudml.org/doc/273271>.

@article{Luostari2013,

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 - Modélisation Mathématique et Analyse Numérique},

keywords = {ultra weak variational formulation; error estimates; plane wave basis; linear elasticity; upwind discontinuous Galerkin method},

language = {eng},

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/273271},

volume = {47},

year = {2013},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2013

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

UR - http://eudml.org/doc/273271

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. Zbl1008.65080MR1885715
- [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. Zbl1216.65151MR2652084
- [3] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. Springer (2008). Zbl1012.65115MR2373954
- [4] A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM : M2AN 42 (2008) 925–940. Zbl1155.65094MR2473314
- [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. Zbl0955.65081MR1618464
- [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. Zbl1134.35317MR2194984
- [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. Zbl1158.74485MR2502292
- [9] C. Farhat, I. Harari and L.P. Franca, A discontinuous enrichment method. Comput. Methods Appl. Mech. Eng.190 (2001) 6455–6479. Zbl1002.76065MR1870426
- [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. Zbl1027.76028MR1963058
- [11] G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys.225 (2007) 1961–1984. Zbl1123.65102MR2349691
- [12] R. Hardin, N. Sloane and W. Smith, Spherical coverings. Available on http://www.research.att.com/˜njas/coverings/index.html (1994).
- [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. Zbl1229.65215MR2783225
- [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. Zbl1269.78013
- [15] T. Huttunen, P. Monk and J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys.182 (2002) 27–46. Zbl1015.65064MR1936802
- [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. Zbl1093.74028MR2087333
- [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. Zbl1075.76648MR2094680
- [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. Zbl1117.78011MR2319231
- [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. Zbl1182.76949MR2382712
- [20] V.D. Kupradze, Potential methods in the theory of elasticity. Israel Program for Scientific Translations (1965). Zbl0188.56901MR223128
- [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 http://www.acoustics.asn.au. Zbl1093.74028
- [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. Zbl1195.74292MR2463186
- [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. Zbl0881.65099MR1426012
- [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. Zbl1319.74003MR3197936
- [26] A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys.65 (2011) 809–837. Zbl1263.35070MR2843918
- [27] P. Monk and D.-Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng.175 (1999) 121–136. Zbl0943.65127MR1692914
- [28] Y.-H. Pao, Betti’s identity and transition matrix for elastic waves. J. Acoust. Soc. Am.64 (1978) 302–310. Zbl0399.73038MR495457
- [29] E. Perrey-Debain, Plane wave decomposition in the unit disc : convergence estimates and computational aspects. J. Comput. Appl. Math.193 (2006) 140–156. Zbl1092.65092MR2228711
- [30] I. Sloan and R. Womersley, Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math.21 (2004) 107–125. Zbl1055.65038MR2065291
- [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. Zbl1242.76143MR2878822
- [32] R. Womersley and I. Sloan, Interpolation and cubature on the sphere. Available on http://web.maths.unsw.edu.au/˜rsw/Sphere. Zbl0952.65013

## NotesEmbed ?

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