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

References

[1] G. Adomian, Nonlinear Stochastic Systems: Theory and Application to Physics, Springer, Netherlands, 1989. 10.1007/978-94-009-2569-4Suche in 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. 10.1007/s002110050056Suche in Google 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. 10.1137/100786356Suche in Google 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. 10.1137/S0036142902418680Suche in Google 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. 10.1023/A:1018977404843Suche in Google Scholar

[6] N. Bellomo and R. Riganti, Nonlinear Stochastic Systems in Physics and Mechanics, World Scientific, Singapore, 1987. 10.1142/0387Suche in Google Scholar

[7] R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numer. 7 (1998), 1–49. 10.1017/S0962492900002804Suche in Google 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. 10.1002/nme.668Suche in Google 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. 10.1002/nme.1775Suche in Google 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. 10.1137/030602447Suche in Google 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. 10.1090/conm/180/01971Suche in Google 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. 10.1137/S0036142995293493Suche in Google Scholar

[13] T. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms 18 (1998), no. 3, 209. 10.1023/A:1019129717644Suche in Google Scholar

[14] R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991. 10.1007/978-1-4612-3094-6Suche in 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. Suche in 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. 10.2514/6.2006-891Suche in Google Scholar

[17] A. Klimke, Sparse grid interpolation toolbox – User’s guide, IANS Report 2007/017, University of Stuttgart, 2007. Suche in 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. 10.1006/jcph.2001.6889Suche in Google 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. 10.1016/0096-3003(94)00134-PSuche in 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. Suche in 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. 10.1137/S003614299630230XSuche in Google 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. 10.1006/jcph.2002.7104Suche in Google 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. 10.1016/j.jcp.2009.01.006Suche in Google 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. 10.1016/j.jcp.2011.03.001Suche in Google 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. 10.1016/j.cma.2004.05.027Suche in Google Scholar

[26] H. Najm, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Annu. Rev. Fluid Mech. 41 (2009), 35–52. 10.1146/annurev.fluid.010908.165248Suche in Google 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. 10.1137/060663660Suche in Google Scholar

[28] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, 2003. 10.1007/978-3-642-14394-6Suche in Google Scholar

[29] M. Shinozuka and Y. Wen, Monte Carlo solution of nonlinear vibrations, AIAA J. 10 (1972), no. 1, 37–40. 10.2514/6.1971-213Suche in Google 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. Suche in 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. 10.1002/(SICI)1099-0887(199708)13:8<675::AID-CNM98>3.0.CO;2-NSuche in Google 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. 10.1137/040615201Suche in Google 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. 10.21236/ADA460654Suche in Google 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. 10.21236/ADA461813Suche in Google 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. 10.21236/ADA461832Suche in Google Scholar

[36] J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231–237. 10.1137/0915016Suche in Google Scholar

[37] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759–1777. 10.1137/S0036142992232949Suche in Google Scholar

[38] J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17–25. 10.1090/S0025-5718-99-01180-1Suche in 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. 10.1007/s11464-013-0327-5Suche in Google 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. 10.1007/s10915-015-0080-xSuche in Google Scholar