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Stochastic finite element methods for partial differential equations with random input data*

Published online by Cambridge University Press:  12 May 2014

Max D. Gunzburger ,
Clayton G. Webster  and
Guannan Zhang
Max D. Gunzburger
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306, USA, E-mail: mgunzburger@fsu.edu, https://www.sc.fsu.edu/~gunzburg
Clayton G. Webster
Affiliation:
Department ofComputational and Applied Mathematics, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA, E-mail: webstercg@ornl.gov, http://www.csm.ornl.gov/~cgwebster
Guannan Zhang
Affiliation:
Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA, E-mail: zhangg@ornl.gov, http://www.csm.ornl.gov/~gz3
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Abstract

The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

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Type
Research Article
Information
Acta Numerica , Volume 23 , May 2014 , pp. 521 - 650
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

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