Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

See all formats and pricing
More options …
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:
  • De Gruyter OnlineGoogle Scholar
/ Xiong Liu
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/cmam-2017-0025


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 𝒯H(D) with small scale of sparse collocation points η(L,N) and solve a linearized equations on a fine mesh 𝒯h(D) using large scale of sparse collocation points η(,N) (where η(L,N),η(,N) are the numbers of sparse grid with respect to different levels L, 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 hH3,η(,N)(η(L,N))3, the novel two-level discretization method achieves the same convergence accuracy in norm ρ2(Γ)2(D) (ρ2(Γ) is the weighted 2 space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯h(D) and large scale collocation points η(,N) in random spaces.

Keywords: Stochastic Collocation Method; Finite Element Method; Semilinear Equation; Convergence Analysis

MSC 2010: 65M10; 78A48


  • [1]

    G. Adomian, Nonlinear Stochastic Systems: Theory and Application to Physics, Springer, Netherlands, 1989. Google Scholar

  • [2]

    A. Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier–Stokes equations, Numer. Math. 68 (1994), no. 2, 189–213. CrossrefGoogle Scholar

  • [3]

    I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev. 52 (2010), no. 2, 317–355. Web of ScienceCrossrefGoogle Scholar

  • [4]

    I. Babuška, R. Tempone and G. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42 (2004), no. 2, 800–825. CrossrefGoogle Scholar

  • [5]

    V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000), no. 4, 273–288. CrossrefGoogle Scholar

  • [6]

    N. Bellomo and R. Riganti, Nonlinear Stochastic Systems in Physics and Mechanics, World Scientific, Singapore, 1987. Google Scholar

  • [7]

    R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numer. 7 (1998), 1–49. CrossrefGoogle Scholar

  • [8]

    Y. Chen, Y. Huang and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations, Int. J. Numer. Meth. Eng. 57 (2003), no. 2, 193–209. CrossrefGoogle Scholar

  • [9]

    Y. Chen, H. Liu and S. Liu, Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods, Int. J. Numer. Meth. Eng. 69 (2007), no. 2, 408–422. Web of ScienceCrossrefGoogle Scholar

  • [10]

    C. Chien and B. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006), no. 4, 1287–1304. CrossrefGoogle Scholar

  • [11]

    C. Dawson and M. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. Math. 180 (1994), 191–191. CrossrefGoogle Scholar

  • [12]

    C. Dawson, M. Wheeler and C. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal. 35 (1998), no. 2, 435–452. CrossrefGoogle Scholar

  • [13]

    T. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms 18 (1998), no. 3, 209. CrossrefGoogle Scholar

  • [14]

    R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991. Google Scholar

  • [15]

    V. Girault and J. Lions, Two-grid finite-element schemes for the steady Navier–Stokes problem in polyhedra, Port. Math. 58 (2001), no. 1, 25–58. Google Scholar

  • [16]

    S. Hosder, R. Walters and R. Perez, A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations, Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exibit (Reno 2006), AIAA, Reston (2006), 1–19. Google Scholar

  • [17]

    A. Klimke, Sparse grid interpolation toolbox – User’s guide, IANS Report 2007/017, University of Stuttgart, 2007. Google Scholar

  • [18]

    O. Knio, H. Najm and R. Ghanem, A stochastic projection method for fluid flow: I. Basic formulation, J. Comput. Phys. 173 (2001), no. 2, 481–511. CrossrefGoogle Scholar

  • [19]

    W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier–Stokes equations, Appl. Math. Comput. 69 (1995), no. 2, 263–274. Google Scholar

  • [20]

    W. Layton, A. Meir and P. Schmidt, A two-level discretization method for the stationary MHD equations, Electron. Trans. Numer. Anal. 6 (1997), 198–210. Google Scholar

  • [21]

    W. Layton and L. Tobiska, A two-level method with backtracking for the Navier–Stokes equations, SIAM J. Numer. Anal. 35 (1998), no. 5, 2035–2054. CrossrefGoogle Scholar

  • [22]

    O. Le Maître, M. Reagan, H. Najm, R. Ghanem and O. Knio, A stochastic projection method for fluid flow: II. Random process, J. Comput. Phys. 181 (2002), no. 1, 9–44. CrossrefGoogle Scholar

  • [23]

    X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys. 228 (2009), no. 8, 3084–3113. Web of ScienceCrossrefGoogle Scholar

  • [24]

    X. Ma and N. Zabaras, A stochastic mixed finite element heterogeneous multiscale method for flow in porous media, J. Comput. Phys. 230 (2011), no. 12, 4696–4722. CrossrefWeb of ScienceGoogle Scholar

  • [25]

    H. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Math. 194 (2005), no. 12, 1295–1331. Google Scholar

  • [26]

    H. Najm, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Annu. Rev. Fluid Mech. 41 (2009), 35–52. CrossrefWeb of ScienceGoogle Scholar

  • [27]

    F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal 46 (2008), no. 5, 2309–2345. Web of ScienceCrossrefGoogle Scholar

  • [28]

    B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, 2003. Google Scholar

  • [29]

    M. Shinozuka and Y. Wen, Monte Carlo solution of nonlinear vibrations, AIAA J. 10 (1972), no. 1, 37–40. CrossrefGoogle Scholar

  • [30]

    S. A. Smoljak, Quadrature and interpolation formulae on tensor products of certain function classes, Dokl. Akad. Nauk 4 (1963), no. 5, 240–243. Google Scholar

  • [31]

    T. Utnes, Two-grid finite element formulations of the incompressible Navier–Stokes equations, Commun. Numer. Meth. Eng. 13 (1997), no. 8, 675–684. CrossrefGoogle Scholar

  • [32]

    D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput. 27 (2005), no. 3, 1118–1139. CrossrefGoogle Scholar

  • [33]

    D. Xiu and G. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24 (2002), no. 2, 619–644. CrossrefGoogle Scholar

  • [34]

    D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys. 187 (2003), no. 1, 137–167. CrossrefGoogle Scholar

  • [35]

    D. Xiu, D. Lucor, C. Su and G. Karniadakis, Stochastic modeling of flow-structure interactions using generalized polynomial chaos, ASME J. Fluid Engrg. 124 (2002), no. 1, 51–69. CrossrefGoogle Scholar

  • [36]

    J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231–237. CrossrefGoogle Scholar

  • [37]

    J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759–1777. CrossrefGoogle Scholar

  • [38]

    J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17–25. Google Scholar

  • [39]

    W. Yao and T. Lu, Numerical comparison of three stochastic methods for nonlinear PN junction problems, Front. Math. China 9 (2014), no. 3, 659–698. Web of ScienceCrossrefGoogle Scholar

  • [40]

    Q. Zhang, Z. Li and Z. Zhang, A sparse grid stochastic collocation method for elliptic interface problems with random input, J. Sci. Comput. 67 (2016), no. 1, 262–280. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2017-04-19

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, DOI: https://doi.org/10.1515/cmam-2017-0025.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in