Robust a posteriori error estimation for parameter-dependent linear elasticity equations

Khan, Arbaz; Bespalov, Alex; Powell, Catherine; Silvester, David

doi:10.1090/mcom/3572

Robust a posteriori error estimation for parameter-dependent linear elasticity equations
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by Arbaz Khan, Alex Bespalov, Catherine E. Powell and David J. Silvester HTML | PDF

Math. Comp. 90 (2021), 613-636 Request permission

Abstract:

The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field partial differential equation model with the Young modulus represented as an affine function of a countable set of parameters. We introduce a weak formulation, establish its stability with respect to a weighted norm and discuss its approximation using stochastic Galerkin mixed finite element methods. We motivate an a posteriori error estimation scheme and establish upper and lower bounds for the approximation error. The constants in the bounds are independent of the Poisson ratio as well as the spatial and parametric discretisation parameters. We also discuss proxies for the error reduction associated with enrichments of the approximation spaces and we develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. The error reduction proxies are shown to be reliable and efficient in the incompressible limit case. Numerical results are presented to supplement the theory. All experiments were performed using open source (IFISS) software that is available online.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Markus Bachmayr, Albert Cohen, Dinh Dũng, and Christoph Schwab, Fully discrete approximation of parametric and stochastic elliptic PDEs, SIAM J. Numer. Anal. 55 (2017), no. 5, 2151–2186. MR 3694016, DOI 10.1137/17M111626X
Randolph E. Bank and R. Kent Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993), no. 4, 921–935. MR 1231320, DOI 10.1137/0730048
A. Bespalov, L. Rocchi, and D. Silvester, T–IFISS: a toolbox for adaptive FEM computation, Comput. Math. Appl. (2020), https://doi.org/10.1016/j.camwa.2020.03.005.
Alex Bespalov and Leonardo Rocchi, Efficient adaptive algorithms for elliptic PDEs with random data, SIAM/ASA J. Uncertain. Quantif. 6 (2018), no. 1, 243–272. MR 3771406, DOI 10.1137/17M1139928
Alex Bespalov and David Silvester, Efficient adaptive stochastic Galerkin methods for parametric operator equations, SIAM J. Sci. Comput. 38 (2016), no. 4, A2118–A2140. MR 3519560, DOI 10.1137/15M1027048
Alex Bespalov, Catherine E. Powell, and David Silvester, Energy norm a posteriori error estimation for parametric operator equations, SIAM J. Sci. Comput. 36 (2014), no. 2, A339–A363. MR 3177362, DOI 10.1137/130916849
Daniele Boffi and Rolf Stenberg, A remark on finite element schemes for nearly incompressible elasticity, Comput. Math. Appl. 74 (2017), no. 9, 2047–2055. MR 3715319, DOI 10.1016/j.camwa.2017.06.006
Albert Cohen and Ronald DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer. 24 (2015), 1–159. MR 3349307, DOI 10.1017/S0962492915000033
Adam J. Crowder and Catherine E. Powell, CBS constants & their role in error estimation for stochastic Galerkin finite element methods, J. Sci. Comput. 77 (2018), no. 2, 1030–1054. MR 3860199, DOI 10.1007/s10915-018-0736-4
Adam J. Crowder, Catherine E. Powell, and Alex Bespalov, Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation, SIAM J. Sci. Comput. 41 (2019), no. 3, A1681–A1705. MR 3952679, DOI 10.1137/18M1194420
Martin Eigel, Claude Jeffrey Gittelson, Christoph Schwab, and Elmar Zander, Adaptive stochastic Galerkin FEM, Comput. Methods Appl. Mech. Engrg. 270 (2014), 247–269. MR 3154028, DOI 10.1016/j.cma.2013.11.015
Victor Eijkhout and Panayot Vassilevski, The role of the strengthened Cauchy-Buniakowskiĭ-Schwarz inequality in multilevel methods, SIAM Rev. 33 (1991), no. 3, 405–419. MR 1124360, DOI 10.1137/1033098
Howard C. Elman, Alison Ramage, and David J. Silvester, IFISS: a computational laboratory for investigating incompressible flow problems, SIAM Rev. 56 (2014), no. 2, 261–273. MR 3201182, DOI 10.1137/120891393
Howard C. Elman, David J. Silvester, and Andrew J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2014. MR 3235759, DOI 10.1093/acprof:oso/9780199678792.001.0001
Roger G. Ghanem and Pol D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, New York, 1991. MR 1083354, DOI 10.1007/978-1-4612-3094-6
Diane Guignard and Fabio Nobile, A posteriori error estimation for the stochastic collocation finite element method, SIAM J. Numer. Anal. 56 (2018), no. 5, 3121–3143. MR 3867619, DOI 10.1137/17M1155454
Leonard R. Herrmann, Elasticity equations of incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3 (1965), 1896–1900. MR 184477, DOI 10.2514/3.3277
Viet Ha Hoang, Thanh Chung Nguyen, and Bingxing Xia, Polynomial approximations of a class of stochastic multiscale elasticity problems, Z. Angew. Math. Phys. 67 (2016), no. 3, Art. 78, 28. MR 3508141, DOI 10.1007/s00033-016-0669-4
Paul Houston, Dominik Schötzau, and Thomas P. Wihler, An $hp$-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25-28, 3224–3246. MR 2220917, DOI 10.1016/j.cma.2005.06.012
Arbaz Khan, Catherine E. Powell, and David J. Silvester, Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity, Internat. J. Numer. Methods Engrg. 119 (2019), no. 1, 18–37. MR 3973678, DOI 10.1002/nme.6040
Arbaz Khan, Catherine E. Powell, and David J. Silvester, Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible elasticity equations, SIAM J. Sci. Comput. 41 (2019), no. 1, A402–A421. MR 3907930, DOI 10.1137/18M117385X
Reijo Kouhia and Rolf Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg. 124 (1995), no. 3, 195–212. MR 1343077, DOI 10.1016/0045-7825(95)00829-P
H. G. Matthies, C. Brenner, C. Bucher, and C. Guedes Soares, Uncertainties in probabilistic numerical analysis of structures and solid–stochastic finite elements, Struct. Safety 19 (1997), 283–336.
Ricardo H. Nochetto and Andreas Veeser, Primer of adaptive finite element methods, Multiscale and adaptivity: modeling, numerics and applications, Lecture Notes in Math., vol. 2040, Springer, Heidelberg, 2012, pp. 125–225. MR 3076038, DOI 10.1007/978-3-642-24079-9
S. Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg. 176 (1999), no. 1-4, 313–331. New advances in computational methods (Cachan, 1997). MR 1665351, DOI 10.1016/S0045-7825(98)00343-0
Shen Shang and Gun Jin Yun, Stochastic finite element with material uncertainties: implementation in a general purpose simulation program, Finite Elem. Anal. Des. 64 (2013), 65–78. MR 3002394, DOI 10.1016/j.finel.2012.10.001
D. J. Silvester, A. Bespalov, Q. Liao, and L. Rocchi, Triangular IFISS (TIFISS) version 1.2., December 2018, http://www.manchester.ac.uk/ifiss/tifiss.
D. J. Silvester, A. Bespalov, and C. E. Powell, Stochastic IFISS (S-IFISS) version 1.04, June 2017, http://www.manchester.ac.uk/ifiss/sifiss.
Bingxing Xia and Viet Ha Hoang, Best $N$-term GPC approximations for a class of stochastic linear elasticity equations, Math. Models Methods Appl. Sci. 24 (2014), no. 3, 513–552. MR 3153066, DOI 10.1142/S0218202513500589