Stochastic system for generalized polytropic filtration

Abstract

In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation. KEYWORDS generalized weak solution, monotone method, non-Newtonian polytropic filtration, stochastic partial differential equations, stochastic systems MSC CLASSIFICATION 60H15; 35D30; 35K59; 76A05; 93E03where Q T = (0, T) × D, the function u = u(t, x) is an unknown defined in Q T , and f (t, u) and G(t, u) are random external forces. Moreover, W is a cylindrical Wiener process evolving on L 2 (D), which enters the equation as an unknown, the function u 0 ∈ L 2 (D), and A is a nonlinear operator in the divergence form

Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients

“…For elliptic stochastic PDEs, there are many numerical methods, such as intrusive and non-intrusive methods. The stochastic collocation method and stochastic Galerkin method are important intrusive methods (see, e.g., [5][6][7][8][9]). The MC method is a non-intrusive method.…”