Energy Minimizing Bases for Efficient Multiscale Modeling and Linear ...

SPE 119137 Energy Minimizing Bases for Efficient Multiscale Modeling and Linear Solvers in Reservoir Simulation Olivier Dubois, IMA, University of Minnesota; Ilya D. Mishev, SPE, Exxon Mobil Upstream Research Company; and Ludmil Zikatanov, Penn State University

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Reservoir Simulation Symposium held in The Woodlands, Texas, USA, 2–4 February 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract We demonstrate the applicability of energy minimizing bases for two-phase flow simulations, and we highlight several advantageous properties that they possess. We show how they can be implemented to obtain efficient serial (algebraic multigrid) and parallel (Additive Schwarz with coarse space correction) linear solvers for large-scale heterogeneous problems. We also confirm with experiments that using such bases for coarsening a problem produces a numerical solution of quality comparable to that of other available multiscale techniques. Finally, we also indicate how different coarsening levels can be used in different regions, in order to keep a finer resolution near certain areas of interest and significantly coarser resolution elsewhere. A multilevel approach is adopted to efficiently construct the basis functions for the coarser resolution. Introduction A computationally intensive component at the core of reservoir simulators consists of solving large sparse linear systems which arise from the discretization of partial differential equations in the model. Fast linear solvers are desired to reduce the computational time. Two popular classes of modern solvers are domain decomposition methods (for parallel implementations) and multigrid methods (for fast serial solvers). In some cases, coarsening (upscaling) the problem can reduce greatly the number of unknowns, at the cost of some accuracy. These strategies all involve a common feature: in each case, one needs to define a coarse version of the problem, typically achieved through the construction of a coarse grid and the definition of appropriate intergrid transfer operators. In general, the task is reduced to finding an interpolation operator from a suitably defined coarse space of unknowns back to the original fine space. Equivalently, one may think of defining this process by a set of coarse basis vectors: these vectors naturally induce an interpolation operator, and their span forms the coarse space. The convergence of domain decomposition and multigrid methods are highly dependent on the choice of coarse basis functions, especially for problems in heterogeneous media, when the permeability tensor exhibits large jumps and strong anisotropy. The classical method is to introduce a coarse grid and use linear interpolation from the coarse to the fine grid: this method does not depend on the coefficients of the matrix being involved. The main goal of our investigation is to improve the robustness of the solvers with respect to heterogeneous coefficients; there has been considerable research in this vein on algebraic multigrid and domain decomposition methods. In particular, some recent papers closely related to our work are (Vanek 1996), (Wan 2000), (Xu 2004) and (Graham 2007). With the same intention in mind, several multiscale methods have been developed for the accurate and efficient numerical solution of large, strongly heterogeneous problems. A popular approach consists of constructing multiscale basis functions on a coarse mesh (e.g. multiscale versions of finite elements (Hou 1997), mixed finite elements (Chen 2002), finite volumes (Jenny 2003), or mimetic finite differences (Aarnes 2008)). However, this requires building a geometric coarse mesh and choosing boundary conditions for the basis functions. Consequently, this approach cannot be applied to non-hierarchical meshes such as unstructured PEBI grids, and thus proves impractical for implementation in existing reservoir simulators without considerable effort. Recently, Wan, Chan and Smith (Wan 2000) introduced an energy minimizing method in the context of algebraic multigrid solvers. Later, Xu and Zikatanov (Xu 2004) realized that the basis functions are “discrete harmonic” and they provided efficient algorithms for computing the basis functions. These basis functions are constructed using a purely algebraic process (and therefore are suitable for PEBI grids and finite volume discretizations) and possess the desirable

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properties of the multiscale approaches. In this paper, we demonstrate the use of energy minimizing bases for solving an elliptic model problem relevant to two-phase flow simulations. The paper is organized as follows. We first briefly review the basic ideas behind multigrid and domain decomposition solvers. We then recall the definition of energy minimizing basis functions, explain how they can be computed, and emphasize some computational issues of the method when applied in the context of two-level domain decomposition preconditioners. We illustrate the performance of this method on 2D and 3D model problems. In a slightly different context of application, we compare the accuracy of the coarsening process with that of other methods, such as multiscale finite elements. Finally, we proceed to show, with a simple example, how energy minimizing basis functions can be defined and applied to the adaptive coarsening of a problem, where a finer resolution is kept near certain areas of interest and significantly coarser resolution is allowed elsewhere. This can be achieved without employing any information about the geometry of the underlying fine mesh. Model Problem and Iterative Solvers We consider the simple elliptic model problem

− ∇ ⋅ ( K∇ u ) + η u = f

in Ω,

(K∇u ) ⋅ n = 0 on ∂Ω, where we choose η > 0 to obtain a well-defined unique solution. We may think of u as the pressure solution in reservoir simulation. After discretizing this continuous problem, for example by introducing a PEBI grid and applying a cell-centered finite volume discretization (a two-point flux approximation) or by using a finite element method on a mesh, the problem becomes to solve a large linear system

Au = b . In this process, we often assume that the permeability tensor is constant over each grid cell or element. Let h denote the mesh size and N the dimension of the linear system. Next, we briefly outline the important components of multigrid and domain decomposition methods. Definition of a Coarse Problem If we choose a number N c of coarse basis vectors

ϕ1 ,ϕ2 ,…,ϕ N

c

, where

N c is smaller than N , then we can define

the coarse space as the span of these basis vectors. An interpolation operator from the coarse space to the fine space of unknowns can be constructed by simply appending the column vectors ϕ j next to each other to form the matrix

(

P = ϕ1 ϕ 2

ϕN

c

).

A restriction operator from the fine space to the coarse space is obtained through the transpose, of the matrix A is then given by

R = PT . A coarse version

A0 = PT AP . Multigrid Methodology A multigrid algorithm can be built from two main ingredients: a smoothing operation S and a set of coarse basis functions. Each iteration of multigrid will then consists of three steps, given a current approximation u : 1. Pre-smoothing: apply k steps of

u ← u + S (b − Au) . 2.

Coarse space correction step: −1

u ← u + PA0 PT (b − Au) . 3.

Post-smoothing: apply k steps of

u ← u + S (b − Au) . This method is a two-grid iteration if the coarse problem is solved exactly. Instead, by applying the multigrid solver −1

v , we get a general multigrid V-cycle. When the coarse space is not constructed from a geometric coarse mesh but algebraically directly from the matrix A , the algorithm is called an Algebraic Multigrid method (AMG).

recursively to solve A0

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Domain Decomposition Methods The aim of this class of methods is to obtain parallel efficiency of the linear solver by dividing the computational load across several processors. The basic idea is to decompose the domain Ω into M overlapping subdomains Ω j . Each subdomain is then assigned to a different processing unit of a parallel computer. The way this is often achieved in practice is to first partition the unknowns into non-overlapping subsets (as in Figure 1), and then extend each subdomain by layers of neighboring cells to get overlap.

Figure 1: Decomposition of a reservoir into 4 subdomains.

Let

R j denote the restriction operator that takes a global vector u and retains only the unknowns from the interior of

subdomain

Ω j . Using this operator and its transpose (which corresponds to a prolongation by zeros outside the subdomain),

we define local problems by

Aj = R j ARTj . The Additive Schwarz (AS) preconditioner (see (Toselli 2005) and references therein) is M

PAS−1 = ∑ RTj A−j 1R j . j =1

A−j 1 can all be computed in parallel by different processors. It is well-known that the convergence of the iterative method with the AS preconditioner is independent of the mesh size h when the width of the

The application of the local operators

overlapping region is kept proportional to the diameter of the subdomains. However, the convergence deteriorates linearly as we increase the number of subdomains, preventing the method to be scalable. To remedy this problem, a coarse component can be added to the preconditioner, by choosing a number of coarse basis functions proportional to the number of subdomains. This leads to a two-level Additive Schwarz (AS2) preconditioner M

PAS−12 = ∑ RTj A−j 1R j + PA0 PT . −1

j =1

This allows for a global mean of communication across all the subdomains. Such a two-level preconditioner is closely related to a multigrid (a two-grid) method, but in which the coarse space is generally much coarser compared to the fine grid. The performance of this type of preconditioner depends strongly on the decomposition of the domain (Usadi 2007); we do not discuss this particular issue in this paper. Choices of Coarse Spaces The choice of the interpolation operator P can affect greatly the convergence of multigrid and domain decomposition methods. The simplest example of an interpolation operator P is to pick a coarse partition of the unknowns, and for each subset, use a coarse basis function that is 1 for unknowns inside the subset ad 0 elsewhere. This leads to a constant interpolation. Another well-analyzed choice is linear interpolation: given, say, a finite element coarse mesh, the coarse basis functions can be chosen to be the piecewise linear hat functions on this coarse mesh, which will give a more accurate interpolation operator. Note that linear and constant interpolations do not depend at all on the matrix A . A method commonly implemented in multigrid and domain decomposition codes is smoothed aggregation (Vanek 1996; Toselli 2005). Starting with constant basis functions, an operator-dependent smoother is applied several times, for example a weighted Jacobi iteration or Gauss-Seidel. The resulting vectors will decay more rapidly in directions of low permeability and be smoother in directions where the permeability is large. Realizing that the coarse space should depend on the matrix A to improve robustness, the multiscale finite element method (MsFEM) can be viewed as a suitable modification of linear interpolation: given a finite element coarse mesh, we can

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choose basis functions that have value 1 at a coarse node and 0 at all others (as in a typical linear finite element hat function) but such that they are locally discrete A-harmonic inside each element (i.e. Aϕ j = 0 locally in the interior of an element). However, this requires choosing boundary conditions on each edge (or face in three dimensions) for solving inside each element; linear and so-called oscillatory Dirichlet boundary conditions were proposed (Hou 1997). When the permeability tensor K has very large jumps and strong anisotropy, the standard choices of coarse basis functions (such as linear interpolation from a coarse mesh) are not very effective. We are interested in finding choices of coarse basis functions that give a convergence that is much more robust with respect to the heterogeneity of the medium, without the necessity of exploiting specific geometrical mesh information. Energy Minimizing Basis (EM) An interesting motivation for the energy minimizing idea can be seen from a recent result of (Graham 2007). Suppose for the moment that we have a scalar permeability coefficient α and that we have chosen a coarse basis consisting of finite element functions. In the context of domain decomposition, a new estimate for the convergence of the AS2 preconditioner was established; in particular it depends heavily on the so-called coarse space robustness indicator Nc

⎧

⎫

γ (α ) = max ⎨1 + H 2j − d ∫ α ∇ϕ j dx ⎬ . j =1

⎩

2

⎭

Ω

This indicator essentially takes the maximum of the energy of each basis function. The upper bound in (Graham 2007) shows that, by minimizing γ (α ) we improve the estimate for the convergence of the AS2 preconditioner, hence leading naturally to the idea of minimizing the energy of the basis functions. The first step in constructing energy minimizing basis functions is to define a set of supports

~ Ω j for these functions,

such that

~ supp(ϕ j ) ⊆ Ω j . Usually, we wish the basis functions to have localized supports, to conserve the level of sparsity in the coarse matrix

A0 . For

the purpose of this paper, we will not consider this issue, although it is crucial to choose the supports carefully when the permeability tensor is strongly anisotropic (Brannick 2006). Thus, we will assume from now on that the supports have already been determined. Since each coarse basis function is forced to be 0 outside its support, we need only find the values inside, φ j := ϕ j |Ω~ . j

~ ~ We can define a local version of the global problem in each support Ω j as before: let R j be the restriction operator to the ~ ~ ~T T T ~ interior of the support, then A j := R j AR j . The energy of each basis function can be written locally as ϕ j Aϕ j = φ j A jφ j . The energy minimizing basis functions are constructed by solving the constrained optimization problem

min φj

1 Nc T ~ ∑ φ j A jφ j , such that 2 j =1

Nc

∑ϕ j =1

j

= e.

where e is a vector of ones. The constraint that the basis functions have to sum up to 1 (to form a partition of unity) is imposed in order for the coarse space to approximate the fine space to some degree. An explicit formula for the minimizers of this problem can be found using Lagrange multipliers. Namely, the energy minimizing basis functions are given by

~ ~ ~

ϕ j = R Tj A j−1R j λ where the Lagrange multiplier

λ is first computed by solving the linear system

Tλ = e

Nc

where

~ ~ ~ T := ∑ R Tj A j−1 R j . j =1

Computational Considerations Note that the construction of the basis functions is done as a preprocessing step, before the linear solver (multigrid or domain decomposition) is started. The major computational cost in this construction is solving for the Lagrange multiplier λ . The matrix T is of the same dimension as the global matrix A , however T is usually much easier to invert due to a

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lower condition number. Consider first the context of multigrid methods, in which the ratio H / h (diameter of the coarse supports divided by the fine mesh size) is relatively small. It is shown by Xu and Zikatanov (Xu 2004) that about 5 or 6 −6

iterations of Conjugate Gradient (CG) are sufficient to resolve λ to within a tolerance of 10 , by using a simple Jacobi preconditioner for T . Moreover, the convergence rate is independent of the jump in the coefficients. Observe that in this

~ Aj can be fully inverted, and thus the matrix T can

case, because the coarse basis supports are very small, the local matices

be assembled explicitely. However, in the context of the two-level Additive Schwarz preconditioners, the ratio

H / h is typically much larger: we ~ are coarsening from a very large number of unknowns down to a much smaller set. Thus, in that case the local matrices A j are too expensive to invert explicitly, and we cannot assemble T . The problem of finding an optimal preconditioner for the linear system Tλ = e also becomes a difficult task in this scenario. We have experimented with the two natural preconditioners Nc ~ ~ ~ P1−1 := A and P2−1 := ∑ R Tj A j R j , j =1

but neither proved to be robust with respect to h , H and the jump in the coefficients. In view of these observations, we propose an alternative way to compute a set of energy minimizing basis functions: instead of going directly from a large to a much smaller set of unknowns, we can go through several intermediate levels, minimizing the energy at each step. We can then impose that H / h remains relatively small at each step, thus avoiding the preconditioning problem mentioned above. If we use L different levels, we will then compute L interpolation matrices, and the compounded interpolation P is obtained by composing the ones from each step of coarsening,

P = P1P2

PL −1PL .

This strategy will not give the same coarse basis functions as in the original “one-step” method. In the experiments, we will denote this modified energy minimizing method as (Multilevel EM). Suppose we have an initial guess for the basis functions

{φ }

0 Nc j j =1 .

Then, an initial guess for the Lagrange multiplier can be

derived from the formula

⎛ N c ~T ~ ⎞ λ 0 = ⎜⎜ ∑ R j R j ⎟⎟ ⎝ j =1 ⎠

−1

Nc

~T ~ ~ 0 j Aj R j φ j .

∑R j =1

This helps in reducing the number of iterations needed by the CG method to solve Tλ = e . Note that the entire process outlined in this section to construct energy minimizing basis functions is purely algebraic: it can be applied by using only the information contained in the matrix A , without knowing anything about the underlying mesh or discretization method. An Algebraic Multiscale Method (AMs) Multiscale methods such as MsFEM, MsFV (multiscale finite volume) and MsMFD (multiscale mimetic finite differences) have two drawbacks: they require an explicit coarse mesh to work with, and they must set boundary conditions in order to solve for the coarse basis functions on each coarse element. We now introduce a fully algebraic variant, that we will refer to as (AMs). Suppose we have determined a set of supports

~ Ω j and also corresponding coarse nodes ~ x j (one per support). Then, algebraically, we can compute multiscale basis

functions that are locally discrete A-harmonic, satisfying

~ φ j = 1 at ~x j , and Ajφ j = 0 . This gives

N c uncoupled linear systems to solve. To obtain a partition a unity, the basis functions need to be rescaled to sum

up to 1. This “algebraic” multiscale method can be viewed as locally minimizing the energy of each basis function independently. Figure 2 illustrates different basis functions for a problem in 2D with a log-normal permeability coefficient, which is shown on the upper left on a log scale in base 10. Notice the significant difference between the basis functions obtained from the fully coupled energy minimization and from the algebraic multiscale method. In general, the EM basis functions appear to have a sharper peak compared to the AMs basis functions.

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Figure 2: Examples of basis functions obtained from different methods, for a problem with log-normal permeability.

Application to Two-Level Domain Decomposition In this section, we employ the energy minimizing basis to define a coarse space for a two-level Additive Schwarz preconditioner. We first begin by confirming the optimality of the preconditioner with respect to mesh size and the scalability of the convergence with the number of subdomains. For this purpose, let us consider the model problem with uniform isotropic permeability tensor K = I and the reaction term η = 0.1 on a unit cube. We decompose this domain uniformly −1

−1

−1

into H × H × H subdomains and construct a random vector for the right-hand side b . Recall that H denotes the coarse mesh size and δ is the width of the overlap. Table 1 shows the results: observe that the energy minimization coarse space yields a preconditioner which is effective when increasing the number of subdomains, while preserving the optimality with respect to the mesh size h . Now, as a model problem relevant to reservoir simulation, we consider the SPE 10 comparative solution project (Christie and Blunt 2001). This model reservoir is divided uniformly into 60x220x85 cells, for a total of 1,122,000 cells, see Figure 3.

Figure 3: The horizontal component of the permeability tensor for the SPE 10 model problem.

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AS (one-level)

Constant Energy Minimizing Interpolation Interpolation Increasing number of subdomains, while H / h remains constant, h = 1 / 64 52 43 30 = 1 / 2, δ = 8h

H H = 1 / 2, δ = 8h H = 1 / 8, δ = 2h

98

73

37

183

84

42

Increasing number of subdomains, fixed overlap width, 79 63 = 1 / 2, δ = h

H H = 1 / 4, δ = h H = 1 / 8, δ = h

47

117

81

44

172

71

35

Decreasing mesh size 44 2h

h = 1 / 16, δ = h = 1 / 32, δ = 4h h = 1 / 64, δ = 8h

h = 1 / 64

h , fixed overlap width, H = 1 / 2 38

27

50

43

31

52

43

30

Table 1: Number of iterations needed by the CG method, preconditioned with Additive Schwarz, to reach a tolerance of

10 −10 .

Let us first consider a two-dimensional version of this problem, posed on a single horizontal layer of the SPE 10 model. We choose the right-hand side of the problem to model four wells at the corners of the domain, and an injector at the center. In this case, to be able to compare with MsFEM, we use a bilinear finite element discretization. We decompose the domain into 3x11=33 subdomains, with overlap δ = 2h . This decomposition induces a natural coarse mesh, and thus we can define a linear interpolation, as well as MsFEM. In the case of AMs and EM, we choose the coarse basis supports to be the same as the supports of the linear basis functions. Finally, we solve the linear system using a preconditioned CG method. The numbers of iterations are shown in Table 2 for four different layers of the model. We observe that the energy minimizing method leads to the smallest number of iterations, and that it can provide significant improvement over all the other methods (e.g. for layer 40). Layer One-level preconditioner Linear Coarse Space MsFEM (linear) MsFEM (oscillating) Algebraic Multiscale Energy Minimizing

10 172

30 170

40 294

70 280

50

53

145

144

41

45

101

87

45

44

114

64

50

45

115

63

41

41

58

50

Table 2: Number of iterations for the preconditioned CG method to reach a tolerance of 10

−8

.

For the full 3D model problem, we use a cell-centered finite volume discretization, and choose a vector of random values between -1 and 1 as the right-hand side of the linear system. Again, we solve the problem using a preconditioned CG method. Here, the domain is decomposed uniformly into 3x5x5=75 subdomains, and we use a minimal overlap δ = h . The results from Table 3 again demonstrate that using EM can signicantly reduce the number of iterations need by the Additive Schwarz preconditioner. Method No. of iterations Additive Schwarz (one-level) 1036 Constant Interpolation 906 Smoothed Aggregation 740 Algebraic Multiscale 560 Energy Minimizing 476 Table 3: Number of iterations for preconditioned CG to reach a residual within a tolerance of

10 −8 .

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Application to Upscaling Instead of solving the global linear system coarse problem of smaller dimension

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Au = b precisely, suppose you wish to approximate the solution by solving a

u 0 = PA0−1 P T b A0 := P T AP as before using a given set of coarse basis vectors. How does the error u − u 0 ∞ behave as H decreases and approaches h , for different choices of coarse bases? To illustrate this, let us consider a simple problem on a where

2D square, with a periodic smooth coefficient as shown in Figure 4(a). Figure 4(b) presents a plot of the error as we decrease H , i.e. making the coarse problem closer and closer to the original fine problem. The linear basis functions and multiscale 2

finite element basis functions both give O ( H ) accuracy. On the other hand, the energy minimizing and the algebraic multiscale methods do not converge at the same rate; an explanation for this is provided by the fact that these bases will preserve constants (because they form partitions of unity), but they will not preserve linear functions, and therefore cannot achieve second order accuracy; this was noted in (Xu 2004). On the other hand, Multilevel EM performs surprisingly well. It has essentially the same error as MsFEM, which decays at the rate of

O( H 2 ) .

(a) (b) Figure 4: The scalar permeability is shown on left. The plot on the right is the error when coarsening from a grid of size h=1/128 to a coarse grid of size H, as we decrease H.

Adaptive Coarsening Here we investigate the applicability of the energy minimizing basis method for adaptive coarsening. We assume we can predict where the solution will vary sharply, and where it will stay smooth. We would like to be able to easily reduce the number of unknowns in regions where the solution should remain flat, while keeping all the unknowns in regions of interest to retain accuracy. In some sense, this is opposite to the idea of the Local Grid Refinement (LGR), which consists in adding a patch with refined mesh where better resolution is required. Both methods have to provide a “coupling” of coarse and fine mesh elements. LGR works very well for structured Cartesian grids (Ewing 1991). Unfortunately, the theory for unstructured grids is not well developed. Moreover, the implementation of accurate LGR methods in production simulators is very difficult and error prone. In this section, we demonstrate how this can be achieved at the algebraic level by using energy minimizing basis functions. Consider the cell-centered finite volume grid shown in Figure 5(a). Suppose we expect a sharp front in the solution localized in the region between x = 1 / 3 and x = 1 / 2 . Figure 5(b) proposes an adapted grid, in which we are coarsening away from the region if we expect the solution to be flat there. Such a grid would cause problems for certain types of discretizations (hanging nodes in finite elements, special interpolation for finite difference schemes to keep acceptable accuracy, etc.), but not when applying energy minimizing basis functions. First, we can define coarse nodes for the adapted grid, they are shown as circles in Figure 5(b). Note that in the region of interest, all fine nodes will be kept as coarse nodes; the black dots represent fine nodes that will be eliminated by the coarsening process. Secondly, we need to define the supports for the coarse basis functions. The red dots in Figure 5(b) give examples of supports we can choose for three different coarse nodes; basically the idea here is to extend the support, starting at the coarse node, until we reach (without including) the neighboring coarse nodes. Once these coarse supports are chosen, then the energy minimizing basis functions can be constructed as before.

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(a) (b) Figure 5: On the left, the original fine grid is shown. On the right, the grid has been adapted grid: coarsening is done away from a banded region. The red dots show examples of supports we pick for the coarse basis functions.

Figure 6 shows an explicit example for a small test case: the fine grid has 324 cells, the adapted coarse space has dimension 84 only. Notice that the coarse solution obtained through the EM method looks (visually) a lot closer to the exact fine grid solution, when compared to the coarse solution obtained from constant interpolation. The relative errors, in the ∞

norm, which are given in Table 4, confirm the superior accuracy obtained through the EM basis.

Figure 6: The permeability field (log scale in base 10), with the exact and approximate solution obtained from the adapted grid.

Choice of Coarse Basis Constant Interpolation Linear Interpolation (*) Algebraic Multiscale Energy Minimizing

Relative Error 0.0563 0.0294 0.0127 0.0093

Table 4: Relative error when compared to the solution of the global problem on the fine grid. (*) Here, linear interpolation is done on a uniform coarse grid of size

H = 3h , since it cannot be applied to the adapted grid.

Conclusions We demonstrated that energy minimizing bases can be very useful for the development of efficient linear solvers for reservoir simulation. We presented a novel application to two-level Additive Schwarz preconditioners: the energy minimizing coarse space was very robust in our tests and showed very good theoretical scalability. The performance of the Algebraic Multigrid method was investigated in (Wan 2000), and in (Xu 2004) for finite element problems. Our research confirms that energy minimizing AMG can be used also for finite volume discretizations. Because energy minimizing basis can be defined purely at the algebraic level without using any mesh information, they can be implemented in current reservoir simulators with relatively small programming efforts. In addition, we introduced an algebraic variant of a multiscale coarse space. While this method is not as effective as the energy minimizing coarse space method, it may prove computationally advantageous for time-dependent problems: if the permeability changes only in a small region, the multiscale basis functions can be efficiently updated locally. Finally, we provided a “proof of concept” for using energy minimizing basis functions for adaptive coarsening; the algebraic nature of the method allows us to perform adaptivity without relying on grid information. There is much more theoretical research necessary to better understand the properties of these methods, such as their accuracy. Acknowledgements We would like to thank the ExxonMobil Upstream Research Company for giving us the opportunity to publish this paper. We also acknowledge several helpful and enlightening discussions we had with Bret Beckner, Serguei Maliassov, Jason Shaw, Robert Shuttleworth, Klaus Wiegand and Hui Zhou.

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Nomenclature A = global discretization matrix A0 = coarse matrix (restricted to coarse space)

Aj ~ Aj

= local matrix restricted to subdomain j

AS AMG CG

= = = = = = =

h H K N Nc P Rj ~ Rj Ω Ωj ~ Ωj

δ

= local matrix restricted to coarse support j Additive Schwarz Algebraic Multigrid Method Conjugate Gradient algorithm fine mesh size diameter of coarse supports permeability tensor dimension of global linear system

= number of coarse basis functions (dimension of coarse space) = interpolation from coarse to fine space = restriction to the interior of subdomain j = restriction to the interior of coarse support j = domain of computation = subdomain j = support for coarse basis function j = measure of the width of the overlapping region

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