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A multigrid solver for two-dimensional stochastic diffusion equations. (English) Zbl 1037.65007

Summary: Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the polynomial chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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[1] Abramowitz, M.; Stegun, L.A., Handbook of mathematical functions, (1970), Dover
[2] Babuska, I.; Chatzipantelidis, P., On solving elliptic stochastic partial differential equations, Comput. methods appl. mech. engrg., 191, 4093-4122, (2002) · Zbl 1019.65010
[3] I. Babuska, K.M. Liu, On solving stochastic initial-value differential equations, Technical report, TICAM Report 02-17, The University of Texas at Austin, 2002 · Zbl 1049.60051
[4] Bielewicz, E.; Górski, J., Shells with random geometric imperfections simulation-based approach, Int. J. nonlinear mech., 37, 777-784, (2002) · Zbl 1346.74119
[5] Cameron, R.H.; Martin, W.T., The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals, Ann. math., 48, 385-392, (1947) · Zbl 0029.14302
[6] McCormick, S.F., Multigrid methods, (1987), SIAM · Zbl 0483.65061
[7] Deb, M.K.; Babuska, I.M.; Oden, J.T., Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. methods appl. mech. engrg., 190, 6359-6372, (2001) · Zbl 1075.65006
[8] Debusschere, B.; Najm, H.; Matta, A.; Knio, O.; Ghanem, R.; Le Maı̂tre, O., Numerical simulation and quantitative uncertainty assessment of a reacting electrochemical microchannel, Phys. fluids, 15, 8, 2238-2250, (2003)
[9] B. Debusschere, H. Najm, A. Matta, T. Shu, O. Knio, R. Ghanem, O. Le Maı̂tre, Uncertainty quantification in a reacting electrochemical microchannel flow model, in: 5th Int. Conf. on Mod. and Sim. of Microsystems, 2002
[10] Ditlevsen, O.; Tarp-Johansen, N.J., Choice of input fields in stochastic finite-elements, Probabilist. engrg. mech., 14, 63-72, (1999)
[11] C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, vol. 1, Springer-Verlag, 1988 · Zbl 0706.76001
[12] Ghanem, R., Ingredients for a general purpose stochastic finite elements formulation, Comput. methods appl. mech. engrg., 168, 19-34, (1999) · Zbl 0943.65008
[13] Ghanem, R., The nonlinear Gaussian spectrum of lognormal stochastic processes and variables, ASME J. appl. mech., 66, 964-973, (1999)
[14] Ghanem, R.; Dham, S., Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Transport porous med., 32, 239-262, (1998)
[15] Ghanem, R.G., Probabilistic characterization of transport in heterogeneous media, Comput. methods appl. mech. engrg., 158, 199-220, (1998) · Zbl 0954.76079
[16] Ghanem, R.G.; Spanos, P.D., A spectral stochastic finite element formulation for reliability analysis, J. engrg. mech. ASCE, 117, 2351-2372, (1991)
[17] Ghanem, R.G.; Spanos, P.D., Stochastic finite elements: A spectral approach, (1991), Springer Verlag · Zbl 0953.74608
[18] Grudmann, H.; Waubke, H., Non-linear stochastic dynamics of systems with random properties, a spectral approach combined with statistical linearization, Int. J. nonlinear mech., 31, 5, 619-630, (1996) · Zbl 0906.70020
[19] Hien, T.D.; Kleiber, M., Stochastic finite element modelling in linear transient heat transfer, Comput. methods appl. mech. engrg., 144, 111-124, (1997) · Zbl 0890.73066
[20] Holden, H.; Oksendal, B.; Uboe, J.; Zhang, T., Stochastic partial differential equations, (1996), Birkhauser
[21] Kaminski, M.; Hien, T.D., Stochastic finite element modeling of tran-scient heat transfer in layered composites, Int. commun. heat mass, 26, 6, 801-810, (1999)
[22] O.M. Knio, R.G. Ghanem, Polynomial Chaos product and moment formulas: A user utility, Technical report, Johns Hopkins University, 2001
[23] Le Maı̂tre, O.P.; Knio, O.M.; Najm, H.N.; Ghanem, R.G., A stochastic projection method for fluid flow. I. basic formulation, J. comput. phys., 173, 481-511, (2001) · Zbl 1051.76056
[24] Le Maı̂tre, O.P.; Reagan, M.T.; Najm, H.N.; Ghanem, R.G.; Knio, O.M., A stochastic projection method for fluid flow. II. random process, J. comput. phys., 181, 9-44, (2002) · Zbl 1052.76057
[25] Liu, J.S., Monte Carlo strategies in scientific computing, (2001), Springer · Zbl 0991.65001
[26] Liu, N.; Hu, B.; Yu, Z.-W., Stochastic finite element method for random temperature in concrete structures, Int. J. solids struct., 38, 6965-6983, (2001) · Zbl 1075.74655
[27] Loève, M., Probability theory, (1977), Springer · Zbl 0359.60001
[28] Matthies, H.G.; Brenner, C.E.; Bucher, C.G.; Scares, C.G., Uncertainties in probabilistic numerical analysis of structures and solids–stochastic finite-elements, Struct. saf., 19, 3, 283-336, (1997)
[29] Matthies, H.G.; Bucher, C.G., Finite element for stochastic media problems, Comput. methods appl. mech. engrg., 168, 3-17, (1999) · Zbl 0953.74065
[30] H.G. Matthies, A. Keese, Multilevel solvers for the analysis of stochastic system, in: First MIT Conference on Computational Fluid and Solid Mechanics, Elsevier, 2001, pp. 1620-1622
[31] H.G. Matthies, A. Keese, Multilevel methods for stochastic systems, in: Proceedings of the 2nd European Conference on Computational Mechanics, Cracow, Poland, June 26, 2001 · Zbl 1435.65019
[32] Náprstek, J., Strongly non-linear stochastic response of a system with random initial imperfections, Probabilist. engrg. mech., 14, 141-148, (1999)
[33] Nicola, B.M.; Verlinden, B.; Beuselinck, A.; Jancsok, P.; Quenon, V.; Scheerlinck, N.; Verbosen, P.; de Baerdemaeker, J., Propagation of stochastic temperature fluctuations in refrigerated fruits, Int. J. refrig., 222, 81-90, (1999)
[34] Pellissetti, M.F.; Ghanem, R.G., Iterative solution of systems of linear equations arising in the context of stochastic finite elements, Adv. engrg. software, 31, 607-616, (2000) · Zbl 1003.68553
[35] Probstein, R.F., Physicochemical hydrodynamics, (1995), Wiley
[36] Schuëller, G.I., Computational stochastic mechanics–recent advances, Comput. struct., 79, 2225-2234, (2001)
[37] Sluzalec, A., Random heat flow with phase change, Int. J. heat mass transfer, 43, 2303-2312, (2000) · Zbl 0962.80002
[38] Sluzalec, A., Stochastic finite element analysis of two-dimensional eddy current problems, Appl. math. model., 24, 401-406, (2000) · Zbl 0974.78011
[39] Stoer, J.; Bulirsch, R., Introduction to numerical analysis, (1991), Springer · Zbl 1004.65001
[40] Theting, T.G., Solving Wick-stochastic boundary value problems using a finite element method, Stoch. stoch. rep., 70, 241-270, (2000) · Zbl 0974.65009
[41] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid, (2001), Academic Press
[42] Wiener, S., The homogeneous chaos, Amer. J. math., 60, 897-936, (1938) · Zbl 0019.35406
[43] D. Xiu, D. Lucor, G.E. Karniadakis, Modeling uncertainty in flow-structure interactions, in: First MIT conference on Computational Fluid and Solid Mechanics, Elsevier, 2001, pp. 1420-1423
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