Volume 44, Number 5, September-October 2010Special Issue on Probabilistic methods and their applications
|Page(s)||1135 - 1153|
|Published online||26 August 2010|
Elliptic equations of higher stochastic order
Department of Mathematics, USC,
Los Angeles, CA 90089, USA. email@example.com; http://www-rcf.usc.edu/~lototsky
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. firstname.lastname@example.org
3 Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA. email@example.com
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.
Mathematics Subject Classification: 35R60 / 65L60 / 60H15 / 60H35
Key words: Elliptic PDE / random coefficients / Wiener Chaos / spectral finite elements
© EDP Sciences, SMAI, 2010
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