Solution verification, goal-oriented adaptive methods for stochastic advection–diffusion problems

Type
Article

Authors
Almeida, Regina C.
Oden, J. Tinsley

Date
2010-08

Abstract
A goal-oriented analysis of linear, stochastic advection-diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated. A class of model problems with random coefficients and source terms is cast in a variational setting. Specific quantities of interest are specified which are also random variables. A stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived. These are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error. Numerical examples are given to demonstrate the methodology for a specific model problem. © 2010 Elsevier B.V.

Citation
Almeida RC, Oden JT (2010) Solution verification, goal-oriented adaptive methods for stochastic advection–diffusion problems. Computer Methods in Applied Mechanics and Engineering 199: 2472–2486. Available: http://dx.doi.org/10.1016/j.cma.2010.04.001.

Acknowledgements
This research is partially supported by the Brazilian Government, through the Agency CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), under grant # 0858/08-0. The first author also would like to acknowledge the support of the J.T. Oden Faculty Fellowship Research Program at ICES. The support of the work of JTO under DOE contract DE-FC52-08NA28615 in connection with the Predictive Science Academic Alliance Program is gratefully acknowledged. Additionally, support of JTO under research grant KAUST U.S. Limited: US 00003 is gratefully acknowledged.

Publisher
Elsevier BV

Journal
Computer Methods in Applied Mechanics and Engineering

DOI
10.1016/j.cma.2010.04.001

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