April  2022, 15(4): 893-912. doi: 10.3934/dcdss.2021104

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

1. 

Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100094, China

2. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

3. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Ju Ming

Received  March 2021 Revised  July 2021 Published  April 2022 Early access  September 2021

Fund Project: This work is partially supported by NSF grant DMS-1722647

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.

Citation: Zhipeng Yang, Xuejian Li, Xiaoming He, Ju Ming. A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 893-912. doi: 10.3934/dcdss.2021104
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Figure 1.  A sketch of the porous media domain $ \Omega_D $, the free-flow domain $ \Omega_S $, and the interface $ \Gamma $
Figure 2.  The convergence in $ L^2 $ norm for the expected value (left) and the variance (right) of velocity with $ N = 5 $ and $ L_{c} = 1/64 $
Figure 3.  The convergence in $ L^2 $ norm for the expected value (left) and the variance (right) of velocity with $ N = 10 $ and $ L_{c} = 1/64 $
Figure 4.  The convergence in $ L^2 $ norm for the expected value (left) and the variance (right) of velocity with $ N = 5 $, GQU method, and different correlation length $ L_{c} $
Figure 5.  The convergence in $ L^2 $ norm for the expected value (left) and the variance (right) of velocity with $ N = 10 $, GQU method, and different correlation length $ L_{c} $
Figure 6.  Numerical solutions of three samples of GQU with $ N = 10 $ and $ s = 6 $. The color represents the speed of flow and the streamlines show the direction of the flow
Figure 7.  Variance of the speed of samples of GQU with $ N = 10 $ and $ s = 6 $ in total domain. The color represents the variance of the speed
Table 1.  Number of sparse grid nodes with different accuracy level $ s $ when $ N = 5 $
$ s $ 2 3 4 5 6 7
KPU 11 51 151 391 903 1743
GQU 11 61 241 781 2203 5593
$ s $ 2 3 4 5 6 7
KPU 11 51 151 391 903 1743
GQU 11 61 241 781 2203 5593
Table 2.  Number of sparse grid nodes with different accuracy level $ s $ when $ N = 10 $
$ s $ 2 3 4 5 6
KPU 21 201 1201 5281 19105
GQU 21 221 1581 8761 40405
$ s $ 2 3 4 5 6
KPU 21 201 1201 5281 19105
GQU 21 221 1581 8761 40405
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