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

Zhipeng Yang; Xuejian Li; Xiaoming He; Ju Ming

doi:10.3934/dcdss.2021104

Article Contents

2022, Volume 15, Issue 4: 893-912. Doi: 10.3934/dcdss.2021104

This issue Previous Article Analytic continuation of noisy data using Adams Bashforth residual neural network Next Article An out-of-distribution-aware autoencoder model for reduced chemical kinetics

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
^* Corresponding author: Ju Ming

Received: February 28, 2021

Revised: June 30, 2021

Early access: September 2021

Published: April 2022

This work is partially supported by NSF grant DMS-1722647

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Abstract

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.

Keywords:

Stokes-Darcy flow,
stochastic partial differential equation,
Karhunen-Loève expansion,
sparse grid,
stochastic collocation method,
finite elements.

Mathematics Subject Classification: 76S05, 35R60, 65D40, 65N35, 65M60.

Citation:

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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 $

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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 $

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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 $

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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} $

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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} $

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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

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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

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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

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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

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