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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 18, Issue 2

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

Luoping Chen
/ Yanping Chen
• Corresponding author
• School of Mathematical Science, South China Normal University, Guangzhou 510631, P. R. China
• Email
• Other articles by this author:
/ Xiong Liu
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/cmam-2017-0025

## Abstract

In 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 ${\mathcal{𝒯}}_{H}\left(D\right)$ with small scale of sparse collocation points $\eta \left(L,N\right)$ and solve a linearized equations on a fine mesh ${\mathcal{𝒯}}_{h}\left(D\right)$ using large scale of sparse collocation points $\eta \left(\mathrm{\ell },N\right)$ (where $\eta \left(L,N\right),\eta \left(\mathrm{\ell },N\right)$ are the numbers of sparse grid with respect to different levels $L,\mathrm{\ell }$ 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 $h\approx {H}^{3},\eta \left(\mathrm{\ell },N\right)\approx {\left(\eta \left(L,N\right)\right)}^{3}$, the novel two-level discretization method achieves the same convergence accuracy in norm $\parallel \cdot {\parallel }_{{\mathcal{ℒ}}_{\rho }^{2}\left(\mathrm{\Gamma }\right)\otimes {\mathcal{ℒ}}^{2}\left(D\right)}$ (${\mathcal{ℒ}}_{\rho }^{2}\left(\mathrm{\Gamma }\right)$ 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 ${\mathcal{𝒯}}_{h}\left(D\right)$ and large scale collocation points $\eta \left(\mathrm{\ell },N\right)$ in random spaces.

MSC 2010: 65M10; 78A48

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Revised: 2017-05-22

Accepted: 2017-06-07

Published Online: 2017-07-28

Published in Print: 2018-04-01

Luoping Chen is supported by the National Natural Science Foundation of China (No. 11501473) and the Fundamental Research Funds for the Central Universities of China (No. 2682016CX108). Yanping Chen is supported National Natural Science Foundation of China (No. 11671157 and No. 91430104). Xiong Liu is supported by Lingnan Normal University general project (No. 2014YL1408).

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 2, Pages 165–179, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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