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A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input

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Abstract

In this paper, numerical solutions of elliptic partial differential equations with both random input and interfaces are considered. The random coefficients are piecewise smooth in the physical space and moderately depend on a large number of random variables in the probability space. To relieve the curse of dimensionality, a sparse grid collocation algorithm based on the Smolyak construction is used. The numerical method consists of an immersed finite element discretization in the physical space and a Smolyak construction of the extreme of Chebyshev polynomials in the probability space, which leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. Numerical experiments on two-dimensional domains are also presented. Convergence is verified and compared with the Monte Carlo simulations.

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Acknowledgments

The first and third authors are partially supported by the National Natural Science Foundation of China Grants No. 11471166 and No. 11426134, Natural Science Foundation of Jiangsu Province grant No. BK20141443, the Innovation Project for Graduate Education of Jiangsu Province No. KYLX15_0718 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The second author is partially supported by the AFSOR grant FA9550-09-1-0520 and the NIH grant 5R01GM96195-2, NCSU RISF Fund, and CNSF Grants No. 11161036 and No. 11371199. The authors would like to thank two anonymous referees for their useful comments and suggestions which have helped to improve the paper greatly.

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Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, 210023, China

    Qian Zhang, Zhilin Li & Zhiyue Zhang

  2. Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    Zhilin Li

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  1. Qian Zhang
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Correspondence to Zhiyue Zhang.

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Zhang, Q., Li, Z. & Zhang, Z. A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input. J Sci Comput 67, 262–280 (2016). https://doi.org/10.1007/s10915-015-0080-x

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