Elliptic equations of higher stochastic order

Abstract

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.Mathematics Subject Classification. 35R60, 65L60, 60H15, 60H35.

A Discussion on Two Stochastic Elliptic Modeling Strategies

“…In this work, we consider a general case, where we assume that a(x,ω) is lognormal and the underlying Gaussian random process is homogeneous stationary and ergodicity is not required. For such a set-up, we refer to [1,2,[6][7][8]14] and references therein for theoretical and numerical studies for model I and [9][10][11][17][18][19] and references therein for model II. The difference between models I and II is twofold: a scaling factor induced by the way of applying the Wick product and the regularization induced by the Wick product itself.…”

Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

“…This model has been studied in the past, for example, by Wan et al [8], Theting [16] and Lototsky et al [17]. The main motivation was that the Wick product is consistent with the Skorokhod stochastic integral, and it is expected that the solution can be smoothed to some extent by the convolution in the probability space.…”

Numerical Approximation of Elliptic Problems With Log-Normal Random Coefficients

“…Theoretical difficulties of problem (1) are mainly related to the lack of uniform ellipticity, where the Lax-Milgram lemma is not applicable. The existence and uniqueness of the solution of problem (1) are usually established with respect to a weighted norm [11,20,29] or a weighted measure [24], or by using the Fernique theorem [7,33]. Considering the Wiener chaos approach and Galerkin projection [10,20], the difficulties of numerical approximation of problem (1) are twofold: First, if we start from the theoretical study [24,29], a different test space rather than L 2 (F; H 1 0 (D)) is required,,which may be not easy to construct.…”

“…The existence and uniqueness of the solution of problem (1) are usually established with respect to a weighted norm [11,20,29] or a weighted measure [24], or by using the Fernique theorem [7,33]. Considering the Wiener chaos approach and Galerkin projection [10,20], the difficulties of numerical approximation of problem (1) are twofold: First, if we start from the theoretical study [24,29], a different test space rather than L 2 (F; H 1 0 (D)) is required,,which may be not easy to construct. Here F := (Ω, F, P ) is the probability space for ω, detailed presentation of F is given in Section 2.…”