Elliptic equations of higher stochastic order
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 1135-1153

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.

DOI : https://doi.org/10.1051/m2an/2010055
Classification:  35R60,  65L60,  60H15,  60H35
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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     pages = {1135-1153},
     doi = {10.1051/m2an/2010055},
     zbl = {1203.65020},
     mrnumber = {2731406},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_5_1135_0}
}
Lototsky, Sergey V.; Rozovskii, Boris L.; Wan, Xiaoliang. Elliptic equations of higher stochastic order. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, pp. 1135-1153. doi : 10.1051/m2an/2010055. http://www.numdam.org/item/M2AN_2010__44_5_1135_0/

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