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 PDE, random coefficients, Wiener chaos, spectral finite elements
@article{M2AN_2010__44_5_1135_0,
author = {Lototsky, Sergey V. and Rozovskii, Boris L. and Wan, Xiaoliang},
title = {Elliptic equations of higher stochastic order},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1135--1153},
year = {2010},
publisher = {EDP Sciences},
volume = {44},
number = {5},
doi = {10.1051/m2an/2010055},
mrnumber = {2731406},
zbl = {1203.65020},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2010055/}
}
TY - JOUR AU - Lototsky, Sergey V. AU - Rozovskii, Boris L. AU - Wan, Xiaoliang TI - Elliptic equations of higher stochastic order JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 1135 EP - 1153 VL - 44 IS - 5 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2010055/ DO - 10.1051/m2an/2010055 LA - en ID - M2AN_2010__44_5_1135_0 ER -
%0 Journal Article %A Lototsky, Sergey V. %A Rozovskii, Boris L. %A Wan, Xiaoliang %T Elliptic equations of higher stochastic order %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 1135-1153 %V 44 %N 5 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2010055/ %R 10.1051/m2an/2010055 %G en %F M2AN_2010__44_5_1135_0
Lototsky, Sergey V.; Rozovskii, Boris L.; Wan, Xiaoliang. Elliptic equations of higher stochastic order. ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 1135-1153. doi: 10.1051/m2an/2010055
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