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

Antje Mugler; Hans-Jörg Starkloff

- Volume: 47, Issue: 5, page 1237-1263
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMugler, Antje, and Starkloff, Hans-Jörg. "On the convergence of the stochastic Galerkin method for random elliptic partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1237-1263. <http://eudml.org/doc/273157>.

@article{Mugler2013,

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.},

author = {Mugler, Antje, Starkloff, Hans-Jörg},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {equations with random data; stochastic Galerkin method; generalized polynomial chaos; spectral methods; elliptic partial differential equation with random coefficients; convergence; stochastic variational problem; finite element; random field; stochastic Petrov-Galerkin method},

language = {eng},

number = {5},

pages = {1237-1263},

publisher = {EDP-Sciences},

title = {On the convergence of the stochastic Galerkin method for random elliptic partial differential equations},

url = {http://eudml.org/doc/273157},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Mugler, Antje

AU - Starkloff, Hans-Jörg

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 5

SP - 1237

EP - 1263

AB - 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.

LA - eng

KW - equations with random data; stochastic Galerkin method; generalized polynomial chaos; spectral methods; elliptic partial differential equation with random coefficients; convergence; stochastic variational problem; finite element; random field; stochastic Petrov-Galerkin method

UR - http://eudml.org/doc/273157

ER -

## References

top- [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover Publications, Inc, New York (1965).
- [2] I. Babuška, F. Nobile and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM J. Numer. Anal.45 (2007) 1005–1034. Zbl1151.65008MR2318799
- [3] I. Babuška, R. Tempone and G.E. Zouraris, Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations. SIAM J. Numer. Anal.42 (2004) 800–825. Zbl1080.65003MR2084236
- [4] I. Babuška, R. Tempone and G.E. Zouraris, 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. Zbl1087.65004MR2121215
- [5] M. Bieri, R. Andreev and C. Schwab, Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput.31 (2009) 4281–4304. Zbl1205.35346MR2566594
- [6] M. Bieri and C. Schwab, Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Engrg.198 (2009) 1149–1170. Zbl1157.65481MR2500242
- [7] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005). Zbl1092.46001MR2176612
- [8] 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). Zbl1012.65115MR1894376
- [9] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg (2006). Zbl1121.76001MR2223552
- [10] A. Cohen, R. DeVore and C. Schwab, Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs. Foundations Comput. Math.10 (2010) 615–646. Zbl1206.60064MR2728424
- [11] M. K. Deb, I. Babuška and J.T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg.190 (2001) 6359–6372. Zbl1075.65006MR1870425
- [12] M. Eiermann, O.G. Ernst and E. Ullmann, Computational aspects of the stochastic finite element method. Comput. Visualiz. Sci.10 (2007) 3–15. Zbl1123.65004MR2295930
- [13] O. Ernst, A. Mugler, E. Ullmann and H.J. Starkloff, On the convergence of generalized polynomial chaos. ESAIM: M2AN 46 (2012) 317–339. Zbl1273.65012MR2855645
- [14] R.V. Field and M. Grigoriu, Convergence Properties of Polynomial Chaos Approximations for L2-Random Variables, Sandia Report SAND2007-1262 (2007).
- [15] P. Frauenfelder, C. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg.194 (2005) 205–228. Zbl1143.65392MR2105161
- [16] R.A. Freeze, A Stochastical-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media. Water Resources Research (1975) 725–741.
- [17] J. Galvis and M. Sarkis, Approximating infinity–dimensional stochastic Darcy‘s Equations without uniform ellipticity. SIAM J. Numer. Anal.47 (2009) 3624–3651. Zbl1205.60121MR2576514
- [18] 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. Zbl1258.60041MR3023390
- [19] R. Ghanem, Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Engrg.168 (1999) 19–34. Zbl0943.65008MR1666714
- [20] R. Ghanem, Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. J. Engrg. Mech.125 (1999) 26–40.
- [21] R. Ghanem and S. Dham, Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media. Transport in Porous Media32 (1998) 239–262. MR1776495
- [22] R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). Zbl0722.73080MR1083354
- [23] C.J. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci.20 (2010) 237–263. Zbl05684685MR2649152
- [24] E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Computat. Appl. Math.84 (1997) 257–275. Zbl0909.65008MR1475378
- [25] E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications. Amer. Math. Soc. 31 (1957). Zbl0078.10004MR89373
- [26] O. Kallenberg, Foundations of modern probability. Springer-Verlag, Berlin (2001). Zbl0892.60001MR1876169
- [27] O.P. Le Maître and O.M. Knio, Spectral Methods for Uncertainty Quantification. Scientific Computation: With Applications to Computational Fluid Dynamics. Springer-Verlag (2010). Zbl1193.76003MR2605529
- [28] M. Loève, Probability Theory II. 4th Edition. Springer-Verlag, New York, Heidelberg, Berlin (1978). Zbl0108.14202
- [29] D. Lucor, C. H Su and G.E. Karniadakis, Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Engrg. 60 (2004) 571–596. Zbl1060.70515MR2057526
- [30] P. Maroni and Z. Rocha, Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor.47 (2008) 291–314. Zbl1147.65020MR2385739
- [31] H.G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg.194 (2005) 1295–1331. Zbl1088.65002MR2121216
- [32] A. Mugler and H.J. Starkloff, On elliptic partial differential equations with random coefficients. Stud. Univ. Babes-Bolyai Math.56 (2011) 473–487. MR2843705
- [33] A. Narayan and J.S. Hesthaven, Computation of connection coefficients and measure modifications for orthogonal polynomials. BIT Numer. Math. (2011). Zbl1247.65026MR2931359
- [34] W. Nowak, Geostatistical Methods for the Identification of Flow and Transport Parameters in the Subsurface, Ph.D. Thesis. Universität Stuttgart (2005).
- [35] C. Schwab and C.J. Gittelson, Sparse tensor discretizations of high–dimensional parametric and stochastic PDEs. Acta Numer.20 (2011) 291–467. Zbl1269.65010MR2805155
- [36] R. A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal.27 (2007) 232–261. Zbl1120.65004MR2317004
- [37] H. Umegaki and A.T. Bharucha-Reid, Banach Space-Valued Random Variables and Tensor Products of Banach Spaces. J. Math. Anal. Appl.31 (1970) 49–67. Zbl0292.60008MR261650
- [38] X. Wan and G.E. Karniadakis, Beyond Wiener Askey Expansions: Handling Arbitrary PDFs. J. Scient. Comput.27 (2006) 455–464. Zbl1102.65006MR2285794
- [39] N. Wiener, Homogeneous Chaos. Amer. J. Math.60 (1938) 897–936. Zbl0019.35406MR1507356JFM64.0887.02
- [40] D. Xiu, Numerical methods for stochastic computations: A spectral method approach. Princeton Univ. Press, Princeton and NJ (2010). Zbl1210.65002MR2723020
- [41] D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg.191 (2002) 4927–4948. Zbl1016.65001MR1932024
- [42] D. Xiu and G.E. Karniadakis, The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J. Sci. Comput.24 (2002) 619–644. Zbl1014.65004MR1951058
- [43] D. Xiu and G.E. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty. Inter. J. Heat and Mass Transfer46 (2003) 4681–4693. Zbl1038.80003
- [44] D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys.187 (2003) 137–167. Zbl1047.76111MR1977783
- [45] 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). Zbl1188.60038MR2103735
- [46] D. Zhang, Stochastic Methods for Flow in Porous Media. Coping with Uncertainties. Academic Press, San Diego, CA (2002).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.