A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

Abstract

<p style='text-indent:20px;'>In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface.</p>

Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model

“…In this model, the free fluid flow is described using the Stokes equation, and the flow in a porous medium is described by the Darcy equation. It is believed [1] that the joint Stokes-Darcy model is more adequate than each of the models considered separately.…”

“…The earliest work in this direction, in our opinion, is the paper of Kumar et al [2] published in 2018. Interest in this model has grown rapidly over the course of 3-4 years, as evidenced by numerous works published to this day [1,[3][4][5][6].…”

“…In [24], a stabilized mixed method for solving the problem is proposed that does not use Lagrange multipliers. In [1,25,26], a stochastic collocation method on sparse grids is developed for the Stokes-Darcy model with random hydraulic conductivity. In [27], an ensemble domain decomposition method is proposed, the peculiarity of which is that the solution of the problem is reduced to solving a system of linear algebraic equations with a common matrix of coefficients for each deterministic numerical model.…”