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 data, stochastic Galerkin method, generalized polynomial chaos, spectral methods
@article{M2AN_2013__47_5_1237_0, author = {Mugler, Antje and Starkloff, Hans-J\"org}, title = {On the convergence of the stochastic {Galerkin} method for random elliptic partial differential equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1237--1263}, publisher = {EDP-Sciences}, volume = {47}, number = {5}, year = {2013}, doi = {10.1051/m2an/2013066}, mrnumber = {3100762}, zbl = {1297.65010}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013066/} }
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 PY - 2013 SP - 1237 EP - 1263 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013066/ DO - 10.1051/m2an/2013066 LA - en ID - M2AN_2013__47_5_1237_0 ER -
%0 Journal Article %A Mugler, Antje %A Starkloff, Hans-Jörg %T On the convergence of the stochastic Galerkin method for random elliptic partial differential equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1237-1263 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013066/ %R 10.1051/m2an/2013066 %G en %F M2AN_2013__47_5_1237_0
Mugler, Antje; Starkloff, Hans-Jörg. On the convergence of the stochastic Galerkin method for random elliptic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1237-1263. doi : 10.1051/m2an/2013066. http://www.numdam.org/articles/10.1051/m2an/2013066/
[1] Handbook of Mathematical Functions. Dover Publications, Inc, New York (1965).
and ,[2] A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM J. Numer. Anal. 45 (2007) 1005-1034. | MR | Zbl
, and ,[3] Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations. SIAM J. Numer. Anal. 42 (2004) 800-825. | MR | Zbl
, and ,[4] 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. | MR | Zbl
, and ,[5] Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31 (2009) 4281-4304. | MR | Zbl
, and ,[6] Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1149-1170. | MR | Zbl
and ,[7] Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005). | MR | Zbl
,[8] The Mathematical Theory of Finite Element Methods, 2nd ed. Texts in Appl. Math., vol. 15. Springer-Verlag, New York (2002). | MR | Zbl
and ,[9] Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg (2006). | MR | Zbl
, , and ,[10] Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs. Foundations Comput. Math. 10 (2010) 615-646. | MR | Zbl
, and ,[11] Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359-6372. | MR | Zbl
, and ,[12] Computational aspects of the stochastic finite element method. Comput. Visualiz. Sci. 10 (2007) 3-15. | MR | Zbl
, and ,[13] On the convergence of generalized polynomial chaos. ESAIM: M2AN 46 (2012) 317-339. | Numdam | MR | Zbl
, , and ,[14] Convergence Properties of Polynomial Chaos Approximations for L2-Random Variables, Sandia Report SAND2007-1262 (2007).
and ,[15] Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205-228. | MR | Zbl
, and ,[16] A Stochastical-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media. Water Resources Research (1975) 725-741.
,[17] Approximating infinity-dimensional stochastic Darcy‘s Equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (2009) 3624-3651. | MR | Zbl
and ,[18] Regularity results for the ordinary product stochastic pressure equation, to appear in SIAM J. Math. Anal. (preprint 2011) 1-31. | MR | Zbl
and ,[19] Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Engrg. 168 (1999) 19-34. | MR | Zbl
,[20] Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. J. Engrg. Mech. 125 (1999) 26-40.
,[21] Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media. Transport in Porous Media 32 (1998) 239-262. | MR
and ,[22] Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). | MR | Zbl
and ,[23] Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237-263. | MR
,[24] Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Computat. Appl. Math. 84 (1997) 257-275. | MR | Zbl
, , and ,[25] Functional Analysis and Semi-Groups, Colloquium Publications. Amer. Math. Soc. 31 (1957). | MR | Zbl
and ,[26] Foundations of modern probability. Springer-Verlag, Berlin (2001). | MR | Zbl
,[27] Spectral Methods for Uncertainty Quantification. Scientific Computation: With Applications to Computational Fluid Dynamics. Springer-Verlag (2010). | MR | Zbl
and ,[28] Probability Theory II. 4th Edition. Springer-Verlag, New York, Heidelberg, Berlin (1978). | Zbl
,[29] C. H Su and G.E. Karniadakis, Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Engrg. 60 (2004) 571-596. | MR | Zbl
,[30] Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor. 47 (2008) 291-314. | MR | Zbl
and ,[31] Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295-1331. | MR | Zbl
and ,[32] On elliptic partial differential equations with random coefficients. Stud. Univ. Babes-Bolyai Math. 56 (2011) 473-487. | MR
and ,[33] Computation of connection coefficients and measure modifications for orthogonal polynomials. BIT Numer. Math. (2011). | MR | Zbl
and ,[34] Geostatistical Methods for the Identification of Flow and Transport Parameters in the Subsurface, Ph.D. Thesis. Universität Stuttgart (2005).
,[35] Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20 (2011) 291-467. | MR | Zbl
and ,[36] Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232-261. | MR | Zbl
and ,[37] Banach Space-Valued Random Variables and Tensor Products of Banach Spaces. J. Math. Anal. Appl. 31 (1970) 49-67. | MR | Zbl
and ,[38] Beyond Wiener Askey Expansions: Handling Arbitrary PDFs. J. Scient. Comput. 27 (2006) 455-464. | MR | Zbl
and ,[39] Homogeneous Chaos. Amer. J. Math. 60 (1938) 897-936. | JFM | MR
,[40] Numerical methods for stochastic computations: A spectral method approach. Princeton Univ. Press, Princeton and NJ (2010). | MR | Zbl
,[41] Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927-4948. | MR | Zbl
and ,[42] The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J. Sci. Comput. 24 (2002) 619-644. | MR | Zbl
and ,[43] A new stochastic approach to transient heat conduction modeling with uncertainty. Inter. J. Heat and Mass Transfer 46 (2003) 4681-4693. | Zbl
and ,[44] Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137-167. | MR | Zbl
and ,[45] 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). | MR | Zbl
, ,[46] Stochastic Methods for Flow in Porous Media. Coping with Uncertainties. Academic Press, San Diego, CA (2002).
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