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On the convergence of the stochastic Galerkin method for random elliptic partial differential equations∗

Published online by Cambridge University Press:  09 July 2013

Antje Mugler  and
Hans-Jörg Starkloff
Antje Mugler
Affiliation:
Mathematisches Institut, Brandenburgische Technische Universität Cottbus, 03013 Cottbus, Germany.. mugler@math.tu-cottbus.de
Hans-Jörg Starkloff
Affiliation:
Fachgruppe Mathematik, Westsächsische Hochschule Zwickau, 08056 Zwickau, Germany.; hans.joerg.starkloff@fh-zwickau.de
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Abstract

In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random variables only and may have distributions different from a lognormal one. We show that in this case the standard stochastic Galerkin approach does not necessarily produce a sequence of approximate solutions that converges in the natural norm to the exact solution even in the case of a lognormal coefficient. By using weighted test function spaces we develop an alternative stochastic Galerkin approach and prove that the associated sequence of approximate solutions converges to the exact solution in the natural norm. Hereby, ideas for the case of lognormal coefficient fields from earlier work of Galvis, Sarkis and Gittelson are used and generalized to the case of positive random coefficient fields with basically arbitrary distributions.

Keywords

Equations with random datastochastic Galerkin methodgeneralized polynomial chaosspectral methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1237 - 1263
Copyright
© EDP Sciences, SMAI, 2013

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