doi:10.1016/j.jcp.2006.02.026 
Copyright © 2006 Elsevier Inc. All rights reserved.
A stochastic variational multiscale method for diffusion in heterogeneous random media
aMaterials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, 188 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801, USA
Received 26 October 2005;
revised 24 February 2006;
accepted 27 February 2006.
Available online 18 April 2006.
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Abstract
A stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media. The exact solution of the governing equations is split into two components: a coarse-scale solution that can be captured on a coarse mesh and a subgrid solution. A localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse-scale solution. The coarse-scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model. The generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models. Finally, various numerical examples are considered for evaluating the method.
Keywords: Variational multiscale; Multiscale finite element; Operator upscaling; Stochastic diffusion; Subgrid modelling; Generalized polynomial chaos
Fig. 1. (A) Schematic of the time integration framework: Δt is the coarse-time step and t′ is the local time coordinate. The integration parameters A(t′) and B(t′) are shown in the figure. Also, 
and 
are identified as the coarse solution fields at the start and end of the coarse time step, respectively. (B) Schematic of a typical coarse element sub-domain: The coordinates normal and tangential to the element edges are denoted by the letters n and γ, respectively.
Fig. 2. Example I – decay of a sine hill (results at time = 0.05): (A)–(C) Coefficients u0, u1 and u2 obtained from the GPCE of the fully-resolved stochastic finite element solution. (D)–(F) Coefficients u0, u1 and u2 obtained from the GPCE of the fine-scale reconstruction of the VMS solution with a quasistatic subgrid assumption. (G)–(I) Coefficients u0, u1 and u2 obtained from the GPCE of the fine-scale reconstruction of the VMS solution with a dynamic subgrid assumption.
Fig. 3. Example I – decay of a sine hill (results at time = 0.2): (A)–(C) Coefficients u0, u1 and u2 obtained from the GPCE of the fully-resolved stochastic finite element solution. (D)–(F) Coefficients u0, u1 and u2 obtained from the GPCE of the fine-scale reconstruction of the VMS solution with a quasistatic subgrid assumption. (G)–(I) Coefficients u0, u1 and u2 obtained from the GPCE of the fine-scale reconstruction of the VMS solution with a dynamic subgrid assumption.
Fig. 4. Example I – decay of a sine hill: Plot of L2 error in GPCE coefficients vs time: (Quasistatic subgrid case). (A) For a 10 × 10 coarse-mesh with a 20 × 20 subgrid mesh. (B) For a 20 × 20 coarse-mesh with a 10 × 10 subgrid mesh. (Dynamic subgrid case) (C) For a 10 × 10 coarse-mesh with a 20 × 20 subgrid mesh. (D) For a 20 × 20 coarse-mesh with a 10 × 10 subgrid mesh.
Fig. 5. A gray scale plot of a two-phase (α–β) microstructure. The intensities are scaled to the interval [0, 1] with zero representing the pure α-phase and one representing the pure β-phase.
Fig. 6. Example II – diffusion in a microstructure (results at time 0.05): Coefficients u0, u1 and u2 in the GPCE of the solution for the following: (A)–(C) Fully-resolved stochastic finite element solution. (D)–(F) Fine-scale reconstruction of the VMS solution with a quasistatic subgrid assumption (20 × 20 coarse-mesh, 10 × 10 subgrid mesh). (G)–(I) Coarse-scale solution (20 × 20 coarse-mesh). (J)–(L) Coarse-scale solution (10 × 10 coarse-mesh).
Fig. 7. Example II – diffusion in a microstructure (results at time 0.2): Coefficients u0, u1 and u2 in the GPCE of the solution for the following: (A)–(C) Fully-resolved stochastic finite element solution. (D)–(F) Fine-scale reconstruction of the VMS solution with a quasistatic subgrid assumption (20 × 20 coarse-mesh, 10 × 10 subgrid mesh). (G)–(I) Coarse-scale solution (20 × 20 coarse-mesh). (J)–(L) Coarse-scale solution (10 × 10 coarse-mesh).
Fig. 8. Example II – (A)–(C) Fully-resolved stochastic FEM simulation: coefficients u3, u4 and u5 in the GPCE expansion of the solution. (D)–(F) Fine-scale reconstruction of the stochastic solution using a quasistatic subgrid assumption in the VMS simulation: Coefficients u3, u4 and u5 in the GPCE expansion of the solution obtained at time 0.2 (non-dimensional).
Table 1.
Computational parameters used in Example I
Table 2.
Computational parameters used in Example II
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