| Issue |
ESAIM: M2AN
Volume 47, Number 1, January-February 2013
|
|
|---|---|---|
| Page(s) | 183 - 211 | |
| DOI | https://doi.org/10.1051/m2an/2012025 | |
| Published online | 31 August 2012 | |
Error estimates for the ultra weak variational formulation in linear elasticity∗
1
Department of Applied Physics, University of Eastern Finland P.O.
Box 1627, 70211
Kuopio,
Finland
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2 Department of Mathematical Sciences, University of Delaware,
Newark, 19716 DE, USA.
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Received:
6
January
2012
Revised:
2
May
2012
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.
Mathematics Subject Classification: 65N15 / 65N30 / 74J05 / 74S30
Key words: Ultra weak variational formulation / error estimates / plane wave basis / linear elasticity / upwind discontinuous Galerkin method
Authors acknowledge Finnish Doctoral Programme in Computational Sciences (FICS) and Academy of Finland (project 213476 Finnish Centre of Excellence in Inverse Problems Research). The research of P. Monk was partially supported by the grant FA9550-05-1-0127 from the US-AFOSR.
© EDP Sciences, SMAI, 2012
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