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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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References

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover Publications, Inc, New York (1965).
Babuška, I., Nobile, F. and Tempone, R., A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM J. Numer. Anal. 45 (2007) 1005–1034. Google Scholar
Babuška, I., Tempone, R. and Zouraris, G.E., Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations. SIAM J. Numer. Anal. 42 (2004) 800–825. Google Scholar
Babuška, I., Tempone, R. and Zouraris, G.E., Solving elliptic boundary-value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1251–1294. Google Scholar
Bieri, M., Andreev, R. and Schwab, C., Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31 (2009) 4281–4304. Google Scholar
Bieri, M. and Schwab, C., Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1149–1170. Google Scholar
A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005).
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed. Texts in Appl. Math., vol. 15. Springer-Verlag, New York (2002).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg (2006).
Cohen, A., DeVore, R. and Schwab, C., Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs. Foundations Comput. Math. 10 (2010) 615–646. Google Scholar
Deb, M. K., Babuška, I. and Oden, J.T., Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359–6372. Google Scholar
Eiermann, M., Ernst, O.G. and Ullmann, E., Computational aspects of the stochastic finite element method. Comput. Visualiz. Sci. 10 (2007) 3–15. Google Scholar
Ernst, O., Mugler, A., Ullmann, E. and Starkloff, H.J., On the convergence of generalized polynomial chaos. ESAIM: M2AN 46 (2012) 317–339. Google Scholar
R.V. Field and M. Grigoriu, Convergence Properties of Polynomial Chaos Approximations for L 2-Random Variables, Sandia Report SAND2007-1262 (2007).
Frauenfelder, P., Schwab, C. and Todor, R.A., Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205–228. Google Scholar
R.A. Freeze, A Stochastical-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media. Water Resources Research (1975) 725–741.
Galvis, J. and Sarkis, M., Approximating infinity–dimensional stochastic Darcy‘s Equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (2009) 3624–3651. Google Scholar
J. Galvis and M. Sarkis, Regularity results for the ordinary product stochastic pressure equation, to appear in SIAM J. Math. Anal. (preprint 2011) 1–31.
Ghanem, R., Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Engrg. 168 (1999) 19–34. Google Scholar
Ghanem, R., Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. J. Engrg. Mech. 125 (1999) 26–40. Google Scholar
Ghanem, R. and Dham, S., Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media. Transport in Porous Media 32 (1998) 239–262. Google Scholar
R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991).
Gittelson, C.J., Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237–263. Google Scholar
Godoy, E., Ronveaux, A., Zarzo, A. and Area, I., Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Computat. Appl. Math. 84 (1997) 257–275. Google Scholar
E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications. Amer. Math. Soc. 31 (1957).
O. Kallenberg, Foundations of modern probability. Springer-Verlag, Berlin (2001).
O.P. Le Maître and O.M. Knio, Spectral Methods for Uncertainty Quantification. Scientific Computation: With Applications to Computational Fluid Dynamics. Springer-Verlag (2010).
M. Loève, Probability Theory II. 4th Edition. Springer-Verlag, New York, Heidelberg, Berlin (1978).
Lucor, D., Su, C. H and Karniadakis, G.E., Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Engrg. 60 (2004) 571–596. Google Scholar
Maroni, P. and Rocha, Z., Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor. 47 (2008) 291–314. Google Scholar
Matthies, H.G. and Keese, A., Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295–1331. Google Scholar
Mugler, A. and Starkloff, H.J., On elliptic partial differential equations with random coefficients. Stud. Univ. Babes-Bolyai Math. 56 (2011) 473–487. Google Scholar
A. Narayan and J.S. Hesthaven, Computation of connection coefficients and measure modifications for orthogonal polynomials. BIT Numer. Math. (2011).
W. Nowak, Geostatistical Methods for the Identification of Flow and Transport Parameters in the Subsurface, Ph.D. Thesis. Universität Stuttgart (2005).
Schwab, C. and Gittelson, C.J., Sparse tensor discretizations of high–dimensional parametric and stochastic PDEs. Acta Numer. 20 (2011) 291–467. Google Scholar
Todor, R. A. and Schwab, C., Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232–261. Google Scholar
Umegaki, H. and Bharucha-Reid, A.T., Banach Space-Valued Random Variables and Tensor Products of Banach Spaces. J. Math. Anal. Appl. 31 (1970) 49–67. Google Scholar
Wan, X. and Karniadakis, G.E., Beyond Wiener Askey Expansions: Handling Arbitrary PDFs. J. Scient. Comput. 27 (2006) 455–464. Google Scholar
Wiener, N., Homogeneous Chaos. Amer. J. Math. 60 (1938) 897–936. Google Scholar
D. Xiu, Numerical methods for stochastic computations: A spectral method approach. Princeton Univ. Press, Princeton and NJ (2010).
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927–4948. Google Scholar
Xiu, D. and Karniadakis, G.E., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J. Sci. Comput. 24 (2002) 619–644. Google Scholar
Xiu, D. and Karniadakis, G.E., A new stochastic approach to transient heat conduction modeling with uncertainty. Inter. J. Heat and Mass Transfer 46 (2003) 4681–4693. Google Scholar
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137–167. Google Scholar
D. Xiu, D. Lucor, C. H Su and G.E. Karniadakis, Performance Evaluation of Generalized Polynomial Chaos, Computational Science – ICCS 2003, edited by P.M.A. Sloot, D. Abramson, A.V. Bogdanov, J.J. Dongarra, Albert Y. Zomaya and Y.E. Gorbachev, Lect. Notes Comput. Sci., vol. 2660. Springer Verlag (2003).
D. Zhang, Stochastic Methods for Flow in Porous Media. Coping with Uncertainties. Academic Press, San Diego, CA (2002).