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International Journal for Uncertainty Quantification

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ISSN Druckformat: 2152-5080

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COMPUTING GREEN′S FUNCTIONS FOR FLOW IN HETEROGENEOUS COMPOSITE MEDIA

David A. Barajas-Solano

Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, Mail Code 0411, La Jolla, California 92093-0411, USA

Daniel M. Tartakovsky

Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, Mail Code 0411, La Jolla, California 92093-0411, USA

Abstract

Green′s functions lie at the foundation of many uncertainty quantification and uncertainty reduction techniques (e.g., the moment differential equation approach, parameter and/or source identification, and data assimilation). We discuss an accurate and numerically efficient approach to compute Green′s functions for transport processes in heterogeneous composite media. We focus on elliptic partial differential equations with (random) discontinuous coefficients. The approach relies on a regularization technique to obtain an associated regular problem, which can be solved using standard finite element methods. We perform numerical experiments to assess the performance of the regularization approach and to evaluate the effects of strong coefficient discontinuities on the Green′s function behavior.

KEYWORDS: uncertainty quantification, stochastic elliptical partial differential equations, moment differential equation, composite media


1. INTRODUCTION

Green′s functions are often used to quantify parametric uncertainty in physical systems described by partial differential equations (PDEs). They allow for direct analysis of the effects of uncertain forcings (source functions, initial and boundary conditions) whose effect is additive, and facilitate quantification of uncertainty in system parameters (conductivity, porosity, etc.) whose effect is multiplicative. Green′s functions were employed to handle random parameters in a variety of fields, including dispersion of passive scalars in turbulent flows [1], flow and reactive transport [2–8] in porous media, subsurface imaging [9], and parameter estimation and source identification [10].

Our analysis is motivated by the nonlocal formalism [2, 3], which employs moment differential equations (MDEs) to quantify uncertainty in predictions of steady-state flow in heterogeneous porous media with uncertain conductivity. The approach we propose relies on Green′s functions to represent the nonlocal nature of the ensemble statistics of a system′s response (i.e., hydraulic heads and fluxes). This methodology, coupled with asymptotic expansions in (small) variances of system parameters, was used to model linear [4], nonlinear [11], and free-surface [7, 12] flows, as well as transport of chemically inert [13, 14] and active [8] solutes in porous media with statistically homogeneous and inhomogeneous [6] uncertain (random) parameters.

A typical example of the use of Green′s functions in the context of uncertainty quantification is the computation of statistics of the hydraulic head h for steady saturated flow, governed by an elliptic equation [4]

(1)

where uncertain hydraulic conductivity k(x) and source function f(x) are modeled as random fields. Let Y(x) = ln K(x) be multivariate Gaussian, with mean 〈Y〉 and variance σ2Y, and correlation function CY (x, y). For mildly heterogeneous media with small variances σ2Y, the mean hydraulic head 〈h(x)〉 can be expanded into a perturbation series in powers of σ2Y [4],

(2)

The zeroth-order approximation satisfies

(3)

where K is the geometric mean of the field k(x). The first-order approximation can then be computed as

(4)

In this expression, G(y; x) is the Green′s function associated with (3) and (1)(x) is given by

(5)

Other statistics of h(x) can be expressed in terms of the Green′s function G as well.

When Green′s functions are derived analytically, such approaches can be computationally more efficient than Monte Carlo simulations and other numerical techniques for solving stochastic PDEs. They also provide physical insight into how uncertainty in parameters and/or driving forces affects predictive uncertainty. However, Green′s functions for many problems of practical significance cannot be obtained analytically. The need to compute Green′s functions numerically is by far the largest computational expense in the uncertainty quantification approaches that rely on them.

The presence of the Dirac delta function in a Green′s function PDE compromises the accuracy and convergence of regular finite element methods [15]. Further contributing to the loss of solution regularity are discontinuous coefficients in the governing PDEs [16] that describe, for example, flow in heterogeneous composite media [6]. The ability to compute Green′s functions efficiently in such a setting is of crucial importance to many uncertainty quantification efforts. Meeting this goal for uncertainty quantifications in elliptic PDEs is the major goal of the present analysis.

Green′s functions of some elliptic operators decay with the distance from the location of the Dirac forcing (point source). It is therefore possible to speak of a “support domain” for the Green′s function, defined as a portion of the computational domain in which the function is nonzero with a given degree of accuracy. Outside of the support domain the Green′s function is small enough not to affect a global quantity, e.g., an integral or another quantity dependent on the function in a weak sense. In such a case, homogeneous boundary conditions of a Green′s function PDE are transferred to the boundary of the support domain. Being able to solve a Green′s function PDE on the support domain, a (small) subset of the computational domain, leads to a significant reduction in computational time.

2. REGULARIZED FORMULATION OF GREEN′S FUNCTION PROBLEM

Let Ω ⊂ ℝd (d = 1, 2, 3) denote a convex domain with the Lipschitz-continuous boundary Γ. The domain Ω is composed of two disjoint units Ω1 and Ω2 such that Ω = Ω1 ∪ Ω2 with Γ12 being the boundary between them (Fig. 1). Consider an elliptic PDE for the Green′s function G(x; x0),

(6)

FIG. 1: Composite domain Ω.

subject to the boundary conditions

(7)
(8)

The coefficient K(x) is piecewise constant in Ω1 and Ω2, δ(xx0) is the Dirac delta function centered at point x0, and n is the unit normal vector pointing outward of Ω. The Dirichlet and Neumann boundary conditions are prescribed in the boundary segments ΓD and ΓN(Γ = ΓD ∩ ΓN), respectively. Along the interface Γ12, the Green′s function and its flux satisfy the continuity conditions

(9)

where [·] denotes the jump of an enclosed quantity across the interface, and n12 is the unit vector normal to Γ12, pointing from Ω1 to Ω2. The Green′s function G is singular at x = x0 due to the Dirac forcing.

Let H1(Ω) be the space of square-integrable functions with square-integrable weak derivatives up to the first order, and V its subspace given by

(10)

Multiplying both sides of (6) with vV and applying the Green′s theorem, one obtains the following weak formulation: Find GdV such that

(11)

The problem with this standard formulation is that GH1(Ω) due to the singularity at x0. This implies that the solution G of (6) is not globally smooth enough for Gd to converge to G in the limit of mesh refinement [15]. In order to overcome this issue, we follow the finite element strategy [15–17] for solving the Poisson-Boltzmann equation (6). The approach divides the solution into a singular part and a regular part [17]. The regular problem admits an unique solution H1 and thus can be computed using a standard finite element approach [16].

To illustrate the regularization methodology, consider a domain Ω such that the boundary Γ1 of Ω1 contains a portion of the domain boundary Γ, and let x0 ∈ Ω1. Following [16], the total solution G can be decomposed into three components Gs, Gh, and such that

(12)

We define Gs, the singular component, as a solution of

(13)

That is, Gs is the free-space Green′s function for Laplace′s equation with forcing δ/K1. For d = 2, Gs = −(2πK1)−1 ln(|xx0|) where | · | is the Euclidean norm. We also define the component Gh, the harmonic extension of the trace of the singular component Gs on Γ1 into Ω1, as a solution of

(14)

Substituting (12)–(14) into (6) and enforcing the boundary conditions (7) and (8), one obtains the following problem for the regular component :

(15)

It is proved in [16] that is unique and lies in H1(Ω), thus completing a methodology for approximating numerically the solution of (6).

2.1 Extensions of the Regularization Formulation

The approach described above can be generalized to more elaborate Green′s function problems as long as a singular component analogous to (13) can be computed in closed form. Consider for example the equation

(16)

which arises in the context of unsaturated flow in porous media [5]. For constant K = K1, the free-space solution of this problem is

where 𝒦0 is the zeroth-order modified Bessel function of the second kind. A harmonic expansion and a regular component can be defined in a fashion similar to (14) and (15).

Another possible generalization is the relaxation of the piecewise constant condition on K(x) to the weaker piecewise Lipschitz continuity [18]. Such K(x) fields capture small-scale variability inside each domain Ωi, superimposed on large-scale variability between domains. Consider a setting similar to (6), but with the piecewise Lipschitz continuous K(x). While K1 is no longer constant, we nevertheless define the singular component Gs as a solution of

(17)

where K* = K(x0). This solution is often available analytically, and from it one can define the harmonic extension in a weak sense as a solution GhV1 of

(18)

with V1 = {vH1(Ω) : v|Γ1 = 0}. The remaining regular component is defined in a manner analogous to (15).

3. NUMERICAL EXAMPLES

The regularized formulation of the Green′s function problem discussed in the previous section makes it explicit that differing values of K within subdomains Ωi (i = 1, 2) affect the global behavior of the Green′s function. This difference can be of one order of magnitude as in the Poisson-Boltzmann problem, or of many orders of magnitude as in subsurface applications [19]. To study the effect this difference has on a solution of the Green′s function, we solve (6) on the domain shown in Fig. 2, with K1 = 1.0 and K2 = 10.0 or 0.1. The solutions G for both cases, computed using linear triangular finite elements and a grid size h = 0.02, are shown in Fig. 3.

FIG. 2: Domain of example 1.

FIG. 3: Green′s functions of example 1 with K1 = 1 and (a) K2 = 10, (b) K2 = 0.1.

To assess the accuracy of the regularization approach compared to a standard discrete approximation, we computed solutions to the previous problem, using both the regularization approach and the discrete approximation Gd of (11), for different grid sizes h. For each value of h, let Ɛp and Ɛh denote the absolute difference between G and Gd at point (0.5, 0.5) and at a distance h in the x1 direction from the point source, respectively. Table 1 shows that Ɛp decays with h as h2, while Ɛh remains constant independent of grid size; this confirms that Gd does not converge uniformly to G in the limit of mesh refinement, as stated in [15]. This result is to be expected as it is not possible for an approximation in H1 to capture the behavior of the Green′s function in the vicinity of the singularity. The regularization approach is more accurate than the standard discrete approximation, especially for low-resolution (large grid size h) simulations in which the vicinity of the singularity can be rather large.

Next we evaluate the size of the support domains of the Green′s functions in Fig. 3. Figure 3(a) shows that if the Dirac impulse is located in the low-K region, then the support domain lies within Ω1 and a portion of Ω2 in the vicinity of Γ12. If the Dirac impulse lies within the high-K region, then the jump conditions (9) at the interface Γ12 can be approximated with the homogeneous Neumann (no-flow) boundary condition for the Green′s function PDE defined on Ω1 and the Green′s function PDE defined on Ω2 is subject to the continuity (Dirichlet) condition [G] = 0 for y ∈ Γ12. The two PDEs become decoupled, and the support domain of G becomes larger [Fig. 3(b)]. To illustrate these points further, we compute solutions of (6) defined on the L × L square domain shown in Fig. 4. Its internal geometry is reconstructed in [19] from synthetic geostatistical data by means of indicator kriging with K1 = exp(−0.1) and K2 = exp(7.0). The Green′s function, for the Dirac forcing placed in both subdomains, is presented in Fig. 5.

FIG. 4: Domain of example 2. K1 (blue) = exp(−0.1), K2 (red) = exp(7.0), L = 12.

FIG. 5: Green′s functions of example 2 for (a) x0 = (9, 6), (b) x = (5, 10).

As expected from the earlier discussion, Fig. 5(a) shows that the Green′s function with x0 in the low-K region has the support domain restricted to that subdomain. A sizable portion of the rest of the domain can then be disregarded, with the extent of that reduction depending on the level of accuracy required. The Dirac forcing located in the high-K region significantly extends the support domain, reducing the aforementioned numerical advantages.

Finally, we use the Green′s functions in Fig. 5 to solve a stochastic problem

(19)

TABLE 1: Absolute differences Ɛp and Ɛh for various grid sizes


hƐpƐh
K1 = 1, K2 = 10K1 = 1, K2 = 0.1K1 = 1, K2 = 10K1 = 1, K2 = 0.1
0.025.98 × 10−54.58 × 10−57.3 × 10−37.3 × 10−3
0.0259.39 × 10−57.17 × 10−57.3 × 10−37.3 × 10−3
0.053.96 × 10−43.02 × 10−47.3 × 10−37.1 × 10−3
0.101.89 × 10−31.44 × 10−37.3 × 10−37.7 × 10−3

in the composite domain Ω of Fig. 4 subjected to homogeneous Dirichlet boundary conditions. The source function f(x) is a stationary random field with unit mean and variance, and exponential correlation function Cf(x; y) = exp (−|xy|/l) with correlation length l = L/2. In terms of the Green′s function for problem (6), the mean and variance of h can be written as

(20)
(21)

Results computed using (20) and (21) are shown in Fig. 6.

FIG. 6: Mean (a) and variance (b) of hydraulic conductivity of example 3.

4. CONCLUSIONS

A regularization methodology for the numerical computation of Green′s functions of elliptic boundary value problems in heterogeneous composite media has been studied. Green′s functions are routinely used in uncertainty quantification, particularly in the moment differential equations approach to solving stochastic partial differential equations. Numerical experiments confirm that the regularization methodology allows for accurate and computationally efficient computation of Green′s functions compared to standard methods that do not take into account the lack of regularity of the solution stemming from the Dirac forcing. In the case of composite media with discontinuous coefficients, the strength of the jump and the location of the forcing have a damping or amplifying effect on the value of the Green′s function throughout the domain. If damping occurs, the opportunity arises for reducing the support domain of the Green′s function, thus reducing the problem size and cutting the computation cost.

ACKNOWLEDGMENTS

This research was supported by the DOE Office of Science Advanced Scientific Computing Research (ASCR) program in Applied Mathematical Sciences; and by research grant no. IS-4090-08R from BARD, the United States—Israel Binational Agricultural Research and Development Fund.

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