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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 toelliptic equations with random coefficients which are generallynonlinear functions of white noise. Singularity issues are avoidedby using the Itô-Skorohod calculus to interpret the interactionsbetween the coefficients and the solution. The solution is constructedby means of the Wiener Chaos (Cameron-Martin) expansions. Theexistence and uniqueness of the solutions are established underrather weak assumptions, the main of which requires only that theexpectation of the highest order (differential) operator is anon-degenerate elliptic operator. The deterministic coefficientsof the Wiener Chaos expansion of the solution solve a lower-triangularsystem of linear elliptic equations (the propagator). This structureof the propagator insures linear complexity of the related numericalalgorithms. Using the lower triangular structure and linearity of thepropagator, the rate of convergence is derived for a spectral/hp finiteelement approximation. The results of related numerical experiments arepresented.

Keywords

Elliptic PDErandom coefficientsWiener Chaosspectral finite elements

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