Dual-based a Posteriori error estimate for stochastic finite element methods

Communications in Applied Mathematics and Computational Science

Volumn 2, Issue 1, 2007, Pages 83-115

  Mathelin, Lionel ^a   Le Maître, Olivier ^a  

^a CNRS   (France)

Author keywords

Error analysis; Refinement scheme; Stochastic finite element method; Uncertainty quantification

Indexed keywords

EID: 84973226358     PISSN: 15593940     EISSN: 21575452     Source Type: Journal    
DOI: 10.2140/camcos.2007.2.83     Document Type: Article

References (31)

1. M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, 1970
2. I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1-40. MR 88d:73036 Zbl 0593.65064
3. I. Babuška and W. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Meth. Engrg. 12 (1978), 1597-1615
4. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001), 1-102. MR 2004g:65147 Zbl 1047.76016
5. R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math. (2) 48 (1947), 385-392. MR 8,523a Zbl 0029.14302
6. P. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. R-2, 77-84. MR 53 #4569 Zbl 0368.65008
7. B. Debusschere, H. Najm, A. Matta, O. Knio, R. Ghanem, and O. Le Maître, Protein labelling reactions in electrochemical flow: numerical simulation and uncertainty propagation, Phys. Fluids 15 (2003), no. 8, 2238-2250
8. R. Ghanem and S. Dham, Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Transp. Porous Media 32 (1998), no. 3, 239-262. MR 2001e:76085
9. R. G. Ghanem, Probabilistic characterization of transport in heterogeneous media, Comp. Meth. App. Mech. Eng. 158 (1998), 199-220
10. R. G. Ghanem and P. D. Spanos, Stochastic finite elements: a spectral approach, Springer, New York, 1991. MR 91k:73102
11. V. Heuveline and R. Rannacher, Duality-based adaptivity in the hp-finite element method, J. Numer. Math. 11 (2003), no. 2, 95-113. MR 2004m:65196 Zbl 1050.65111
12. T. Hien and M. Kleiber, Stochastic finite element modeling in linear transcient heat transfer, Comp. Met. App. Mech. Eng. 144 (1997), 111-124
13. M. Kaminski and T. Hien, Stochastic finite element modeling of transcient heat transfer in layered composites, Int. Com. Heat and Mass Trans. 26 (1999), no. 6, 801-810
14. O. M. Knio and O. P. Le Maître, Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dynam. Res. 38 (2006), no. 9, 616-640. MR 2007f:76158
15. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys. 197 (2004), no. 1, 28-57. MR 2005a:76123
16. O. P. Le Maître, H. N. Najm, R. G. Ghanem, and O. M. Knio, Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys. 197 (2004), no. 2, 502-531. MR 2005b:65142
17. O. P. Le Maître, H. N. Najm, P. P. Pébay, R. G. Ghanem, and O. Knio, Multi-resolution analysis for uncertainty quantification in chemical systems, SIAM J. Sci. Comp. (2007), in press
18. O. Le Maître, M. T. Reagan, B. Debusschere, H. N. Najm, R. G. Ghanem, and O. M. Knio, Natural convection in a closed cavity under stochastic non-Boussinesq conditions, SIAM J. Sci. Comput. 26 (2004), no. 2, 375-394. MR 2005i:76094
19. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, A stochastic projection method for fluid flow, I: Basic formulation, J. Comput. Phys. 173 (2001), no. 2, 481-511. MR 2002k:76111
20. O. P. Le Maître, M. T. Reagan, H. N. Najm, R. G. Ghanem, and O. M. Knio, A stochastic projection method for fluid flow. II. Random process, J. Comput. Phys. 181 (2002), no. 1, 9-44. MR 2003g:76090
21. L. Mathelin, M. Y. Hussaini, and T. A. Zang, Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms 38 (2005), no. 1-3, 209-236. MR 2006a:76089 Zbl 02161195
22. S. Micheletti, S. Perotto, and M. Picasso, Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection-diffusion and the Stokes problems, SIAM J. Numer. Anal. 41 (2003), no. 3, 1131-1162. MR 2004g:65157 Zbl 1053.65089
23. J. Peraire, J. Peiró, and K. Morgan, Adaptive remeshing for three-dimensional compressible flow computations, J. Comp. Phys. 103 (1992), no. 2, 269-285
24. M. T. Reagan, H. N. Najm, R. G. Ghanem, and O. M. Knio, Uncertainty quantification in reacting flow simulations through non-intrusive spectral projection, Combust. Flames 132 (2003), 545-555
25. G. Schuëller, Computational stochastic mechanics: recent advances, Comput. Struct. 79 (2001), 2225-2234
26. R. Walters and L. Huyse, Uncertainty analysis for fluid mechanics with applications, Tech. report, NASA CR-2002-211449, 2002
27. X. Wan and G. E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys. 209 (2005), no. 2, 617-642. MR 2006e:65007 Zbl 1078.65008
28. N. Wiener, The homogeneous chaos, Amer. J. Math. 60 (1938), no. 4, 897-936. Zbl 0019.35406 JFM 64.0887.02
29. D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24 (2002), no. 2, 619-644. MR 2003m:60174 Zbl 1014.65004
30. D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys. 187 (2003), no. 1, 137-167. MR 2004c:76110 Zbl 1047.76111
31. D. Xiu and G. E. Karniadakis, Supersensitivity due to uncertain boundary conditions, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2114-2138. MR 2005f:76078 Zbl 1075.76623