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Elliptic equations of higherstochastic order

Published online by Cambridge University Press:  26 August 2010

Sergey V. Lototsky
Affiliation:
Department of Mathematics, USC, Los Angeles, CA 90089, USA. lototsky@math.usc.edu; http://www-rcf.usc.edu/~lototsky
Boris L. Rozovskii
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. rozovsky@dam.brown.edu
Xiaoliang Wan
Affiliation:
Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA. xlwan@math.lsu.edu
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Abstract

This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

Babuška, I. and Suri, M., The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994) 578632. CrossRef
Babuška, I., Tempone, R. and Zouraris, G.E., Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800825. CrossRef
Cameron, R.H. and Martin, W.T., The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functions. Ann. Math. 48 (1947) 385392. CrossRef
Cao, Y., On convergence rate of Wiener-Ito expansion for generalized random variables. Stochastics 78 (2006) 179187.
P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002).
Elliott, F.W., Jr., D.J. Horntrop and A.J. Majda, A Fourier-wavelet Monte Carlo method for fractal random fields. J. Comput. Phys. 132 (1997) 384408. CrossRef
T. Hida, H.-H. Kuo, J. Potthoff and L. Sreit, White noise. Kluwer Academic Publishers, Boston (1993).
H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic partial differential equations. Birkhäuser, Boston (1996).
Itô, K., Stochastic integral. Proc. Imp. Acad. Tokyo 20 (1944) 519524. CrossRef
G.E. Karniadakis and S.J. Sherwin, Spectral/hp element methods for computational fluid dynamics. Second edition, Numerical Mathematics and Scientific Computation, Oxford University Press, New York (2005).
Yu.G. Kondratiev, P. Leukert, J. Potthoff, L. Streit, W. Westerkamp, Generalized functionals in Gaussian spaces: the characterization theorem revisited. J. Funct. Anal. 141 (1996) 301318. CrossRef
H.-H. Kuo, White noise distribution theory. Probability and Stochastics Series, CRC Press, Boca Raton (1996).
M. Loève, Probability theory – I, Graduate Texts in Mathematics 45. Fourth edition, Springer-Verlag, New York (1977).
Lototsky, S.V. and Rozovskii, B.L., Stochastic differential equations driven by purely spatial noise. SIAM J. Math. Anal. 41 (2009) 12951322. CrossRef
D. Nualart, The Malliavin calculus and related topics. Second edition, Probability and its Applications (New York), Springer-Verlag, Berlin (2006).
Pilipović, S. and Seleši, D., Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007) 79110. CrossRef
Pilipović, S. and Seleši, D., On the generalized stochastic Dirichlet problem. Part I: The stochastic weak maximum principle. Potential Anal. 32 (2010) 363387. CrossRef
Ch. Schwab, p- and hp-finite element methods, Theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (1998).
Shinozuka, M. and Deodatis, G., Simulation of stochastic processes by spectral representation. AMR 44 (1991) 191204.
Theting, T.G., Solving Wick-stochastic boundary value problems using a finite element method. Stochastics Stochastics Rep. 70 (2000) 241270. CrossRef
Våge, G., Variational methods for PDEs applied to stochastic partial differential equations. Math. Scand. 82 (1998) 113137. CrossRef
Wan, X., Rozovskii, B. and Karniadakis, G.E., A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. Proc. Natl. Acad. Sci. USA 106 (2009) 1418914194. CrossRef