Article Contents
Article Contents

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

• * Corresponding author: Ju Ming

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.

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

 Citation:

• 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

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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