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Elliptic equations of higherstochastic order

Published online by Cambridge University Press:  26 August 2010

Sergey V. Lototsky ,
Boris L. Rozovskii  and
Xiaoliang Wan
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.

Keywords

Elliptic PDErandom coefficientsWiener Chaosspectral finite elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 1135 - 1153
Copyright
© EDP Sciences, SMAI, 2010

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