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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhipeng Yang ◽  
Xuejian Li ◽  
Xiaoming He ◽  
Ju Ming
Keyword(s):  
Hydraulic Conductivity ◽  
Collocation Method ◽  
Grid Method ◽  
Sparse Grids ◽  
Sparse Grid ◽  
Stochastic Collocation ◽  
Discrete Form ◽  
Darcy Model ◽  
Stochastic Collocation Method ◽  
Sparse Grid Method

<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>

A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 165-179
Author(s):  
Luoping Chen ◽  
Yanping Chen ◽  
Xiong Liu
Keyword(s):  
Collocation Method ◽  
Large Scale ◽  
Sparse Grid ◽  
Small Scale ◽  
Stochastic Collocation ◽  
Discretization Method ◽  
Coarse Mesh ◽  
Stochastic Collocation Method ◽  
Semilinear Equations ◽  
Collocation Points

AbstractIn this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh {\mathcal{T}_{H}(D)} with small scale of sparse collocation points {\eta(L,N)} and solve a linearized equations on a fine mesh {\mathcal{T}_{h}(D)} using large scale of sparse collocation points {\eta(\ell,N)} (where {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}}, the novel two-level discretization method achieves the same convergence accuracy in norm {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ({\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with {\mathcal{T}_{h}(D)} and large scale collocation points {\eta(\ell,N)} in random spaces.

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