umer. nal c vol. 47, no. 1, pp. 740–761 for multiscale...

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SIAM J. NUMER. ANAL. c© 2009 Society for Industrial and Applied MathematicsVol. 47, No. 1, pp. 740–761

A POSTERIORI ANALYSIS AND ADAPTIVE ERROR CONTROLFOR MULTISCALE OPERATOR DECOMPOSITION SOLUTION OF

ELLIPTIC SYSTEMS I: TRIANGULAR SYSTEMS∗

V. CAREY† , D. ESTEP‡ , AND S. TAVENER†

Abstract. In this paper, we perform an a posteriori error analysis of a multiscale operatordecomposition finite element method for the solution of a system of coupled elliptic problems. Thegoal is to compute accurate error estimates that account for the effects arising from multiscalediscretization via operator decomposition. Our approach to error estimation is based on a well-knowna posteriori analysis involving variational analysis, residuals, and the generalized Green’s function.Our method utilizes adjoint problems to deal with several new features arising from the multiscaleoperator decomposition. In part I of this paper, we focus on the propagation of errors arisingfrom the solution of one component to another and the transfer of information between differentrepresentations of solution components. We also devise an adaptive discretization strategy based onthe error estimates that specifically controls the effects arising from operator decomposition. In partII of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.

Key words. a posteriori error analysis, adjoint problem, elliptic system, generalized Green’sfunction, goal-oriented error estimates, multiscale methods, operator decomposition, projection error

AMS subject classifications. 65N15, 65N30, 65N50

DOI. 10.1137/070689917

1. Introduction. Multiscale operator decomposition is a widely used techniquefor solving multiphysics, multiscale problems [14, 15]. The general approach is to de-compose the multiphysics problem into components involving simpler physics over arelatively limited range of scales and then to seek the solution of the entire systemthrough some sort of iterative procedure involving solutions of the individual compo-nents. This approach is appealing because there is generally a good understanding ofhow to solve a broad spectrum of single physics problems accurately and efficiently,and because it provides an alternative to accommodating multiple scales in one dis-cretization. However, multiscale operator decomposition presents an entirely new setof accuracy and stability issues, some of which are obvious and some subtle, and allof which are difficult to correct.

We motivate multiscale operator decomposition for elliptic systems by consideringa model of a thermal actuator. A thermal actuator is a microelectronic mechanicalswitch device (see Figure 1.1). A contact rests on thin braces composed of a con-ducting material. When a current is passed through the braces, they heat up andconsequently expand to close the contact. The system is modeled by a system ofthree coupled equations, each representing a distinct physical process. They are an

∗Received by the editors April 30, 2007; accepted for publication (in revised form) June 6, 2008;published electronically February 4, 2009.

http://www.siam.org/journals/sinum/47-1/68991.html†Department of Mathematics, Colorado State University, Fort Collins, CO 80523 (carey@math.

colostate.edu, [email protected]). The work of these authors was supported in part by theDepartment of Energy (DE-FG02-04ER25620).

‡Department of Mathematics and Department of Statistics, Colorado State University, FortCollins, CO 80523 ([email protected]). This author’s work was supported in part by theDepartment of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909), theNational Aeronautics and Space Administration (NNG04GH63G), the National Science Founda-tion (DMS-0107832, DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559), IdahoNational Laboratory (00069249), and the Sandia Corporation (PO299784).

740


OPERATOR DECOMPOSITION FOR ELLIPTIC SYSTEMS 741

Vsig

Vswitch

Heat Conduction

Electrostatics

Elasticity

Fig. 1.1. Sketch of a thermal actuator.

electrostatic current equation

(1.1) ∇ · (σ∇V ) = 0,

governing potential V (where current J = −σ∇V ), a steady-state energy equation

(1.2) ∇ · (κ(T )∇T ) = σ(∇V · ∇V ),

governing temperature T , and a linear elasticity equation giving the steady-state dis-placement d,

(1.3) ∇ · (λ tr(E)I + 2μE − β(T − Tref )I)

= 0, E =(∇d+ ∇d�)/2.

Using multiscale operator decomposition, the complete system (1.1–1.3) is de-composed into three components, each of which is solved with a code specialized tothe particular type of physics. Notice that the electric potential V can be calculatedindependently of T and d. The temperature T can be calculated once the electric po-tential V is known, while the calculation of displacement d requires prior knowledgeof T and therefore of V .

In general, we can write a coupled elliptic system on a domain Ω in the form

(1.4)

⎧⎪⎪⎨⎪⎪⎩L1(x, u1, Du1, . . . , un, Dun) = 0,

...Ln(x, u1, Du1, . . . , un, Dun) = 0.

x ∈ Ω.

A natural form of operator decomposition is to split the global multiphysics prob-lem into n “single-physics” components that are solved individually. In general, thesolution of each component requires knowledge of the solutions of all the other com-ponents; the full problem requires some form of iteration to obtain the solution.

It is possible to impose conditions on the system, the components, and the cou-pling that allow for an a priori convergence analysis. However, operator decompositionis problematic in practice because it is very difficult to verify such conditions and of-ten impractical to satisfy them. Indeed, numerical solutions obtained via operatordecomposition are affected significantly by the specific choice of decomposition. In


742 V. CAREY, D. ESTEP, AND S. TAVENER

this paper, we perform an a posteriori error analysis of a multiscale operator de-composition finite element method for the solution of a system of coupled ellipticproblems. The components of the problem are solved in sequence using independentdiscretizations. The goal is to compute accurate computational error estimates thatspecifically account for the effects arising from operator decomposition. We also de-vise an adaptive discretization strategy based on the error estimates that controls theeffects arising from multiscale operator decomposition.

The a posteriori analysis in this paper is based on a well-known approach involv-ing variational analysis, residuals, and the generalized Green’s function solving anadjoint problem [1, 2, 5, 6, 7, 8, 9, 12]. However, we modify this approach to accommo-date several new features arising from the operator decomposition. Three importantissues addressed here are as follows: (1) Errors in the solution of each componentpropagate into the solutions of the other components; (2) Transferring informationbetween different discretization representations potentially introduces new error; and(3) The adjoint operators associated with the fully coupled system and an operatordecomposition version are not generally equal. In addition, the analysis stays withinthe “single physics paradigm” by only requiring the solution of adjoint problems as-sociated with the individual components. These issues are characteristic of a broadrange of operator decomposition discretizations, e.g., [10, 13], and generally requireextensions to the usual a posteriori analysis techniques.

In this paper, we focus attention on analyzing the effects of transferring informa-tion between components, which is necessitated by operator decomposition. In orderto do so, we consider a “triangular” or one-way coupled system

(1.5)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

L1(x, u1, Du1) = 0,L2(x, u1, Du1, u2, Du2) = 0,L3(x, u1, Du1, u2, Du2, u3, Du3) = 0,

...Ln(x, u1, Du1, u2, Du2, u3, Du3, . . . , un, Dun) = 0.

x ∈ Ω.

This system can be solved by a finite sequence of component solutions by consideringthe n problems for L1, L2, . . . ,Ln sequentially. Such systems are important in prac-tice, e.g., the thermal actuator (1.1)–(1.3) has this form. In part II [3], we consideradditional sources of error arising from the iterative procedure required when solvinga fully-coupled system via operator decomposition.

We capture the essential features of (1.5) in a two component “one-way” coupledsystem of the form

(1.6)

⎧⎪⎨⎪⎩−∇ · a1∇u1 + b1 · ∇u1 + c1u1 = f1(x), x ∈ Ω,−∇ · a2∇u2 + b2 · ∇u2 + c2u2 = f2(x, u1, Du1), x ∈ Ω,u1 = u2 = 0, x ∈ ∂Ω,

where ai, bi, ci, fi are smooth functions on a bounded domain Ω in RN with boundary∂Ω and the coupling occurs through f2. We later generalize to coupling through thecoefficients of the elliptic operator for u2.

In section 2, we illustrate the main idea by applying the analysis to a linearalgebraic system. We perform the transfer error analysis in section 3 and presentcomputational examples when the corresponding discretizations are “related” in thesense that either both computational meshes are identical, or one mesh is generated by



a sequence of mesh refinements on the other mesh. In section 4, we consider the effectof using distinct discretizations for the two components and analyze the additionalerrors caused by using projections between the components. Additionally, we discussthe use of Monte Carlo integration to estimate these projection errors. We presentthe full adaptive algorithm in section 5, which we illustrate with several numericalexamples.

2. A linear algebra example. We introduce the notation and ideas in thecontext of a lower triangular linear system of equations. Let U be an approximatesolution of the linear system Au = b. We wish to compute a quantity of interest givenby a linear functional (ψ, u). The error e = u − U is not computable, but we cancompute the residual R = b − AU = Ae. Using the solution φ of the correspondingadjoint equation A�φ = ψ, the error representation for a linear functional of thesolution is

(ψ, u

)− (ψ,U) =(ψ, e

)=(A�φ, e

)=(φ,Ae

)=(φ,R

).

Now consider the triangular system

(2.1) Au =(

A11 0A21 A22

)(u1

u2

)=(b1b2

)= b ,

with approximate solution

U =(U1

U2

)≈(u1

u2

)= u.

We estimate the error in a quantity of interest in u2 only, given by the linear functional

(ψ(1), u

)=(ψ

(1)2 , u2

), where ψ =

(0ψ

(1)2

).

We employ the superscript (1), since we later pose additional auxiliary adjoint prob-lems. Clearly, estimates on linear functionals of u1 are independent of u2. The lowertriangular structure of A yields

A11u1 = b1 ,

A22u2 = b2 − A21u1 ,

and the corresponding residuals are

R1 = b1 − A11U1 ,

R2 = (b2 − A21U1) − A22U2 .

The residual R2 depends upon the solution of the first component, and any attemptto decrease this residual requires a consideration of the accuracy of U1. The adjointproblem to (2.1) is

(A�

11 A�21

0 A�22

)(φ

(1)1

φ(1)2

)=(

0ψ

(1)2

),



and the resulting error representation is

(2.2)

(ψ(1), e) =

(ψ

(1)2 , e2

)=(A�

22φ(1)2 , e2

)=(φ

(1)2 ,A22u2

)− (φ(1)2 ,A22U2

)=(φ

(1)2 , b2 − A21u1

)− (φ(1)2 ,A22U2

)=(φ

(1)2 , b2 − A21U1 − A22U2

)− (φ(1)2 ,A21e1

)=(φ

(1)2 , R2

)− (φ(1)2 ,A21e1

).

The first term of the error representation requires only U2 and φ(1)2 . Since the

adjoint system is upper triangular and

φ(1)2 =

(A�

22

)−1

ψ(1)2

is independent of the first component, the calculation of(φ

(1)2 , R2

)remains within

the “single physics paradigm.” The second term(φ

(1)2 ,A21e1

)represents the effect of

errors in U1 on the solution U2. At first glance, this term is uncomputable, but wenote that it is a linear functional of e1 since(

φ(1)2 ,A21e1

)=(A�

21φ(1)2 , e1

).

We therefore form the adjoint problem for the transfer error(

A�11 A�

21

0 A�22

)(φ

(2)1

φ(2)2

)=(ψ

(2)1

0

)=(

A�21φ

(1)2

0

).

The upper triangular block structure of A� immediately yields φ(2)2 = 0. As noted

earlier, error estimates of u1 should be independent of u2. Thus, A�11φ

(2)1 = ψ

(2)1 =

A�21φ

(1)2 , so that, once again, we can solve for φ(2) in the “single physics paradigm.”

Given φ(2), we obtain the secondary error representation

(2.3)(ψ(2), e

)=(ψ

(2)1 , e1

)=(A�

21φ(1)2 , e1

)=(A�

11φ(2)1 , e1

)=(φ

(2)1 , R1

).

Combining the first term of (2.2) with (2.3) yields the complete error representation

(2.4)(ψ(1), e

)=(φ

(1)2 , R2

)− (φ(2)1 , R1

),

which is a sum of the inner products of “single physics” residuals and adjoint solutionscomputed using the “single physics” paradigm.

3. Analysis of the discretization error. The corresponding weak form of(1.6) reads as follows: find ui ∈ W 1

2 (Ω) satisfying

(3.1)

{A1(u1, v1) = (f1, v1),A2(u2, v2) = (f2(x, u1, Du1), v2)

∀vi ∈ W 12 (Ω),

where

A1(u1, v1) = A1(u1, v1) ≡ (a1∇u1,∇v1) + (b1(x) · ∇u1, v1) + (c1u1, v1),A2(u2, v2) = A2(u2, v2) ≡ (a2∇u2,∇v2) + (b2(x) · ∇u2, v2) + (c2u2, v2)



are assumed to be coercive bilinear forms on Ω and Wmp (Ω) is the subspace of Wm

p (Ω)with zero trace on ∂Ω. We suppress the “cross” dependence on the other solutionsexcept in a few remarks below. After introducing (conforming) discretizations Sh,i(Ω),we solve the discretized system

(3.2)

{A1(U1, χ1) = (f1, χ1),A2(U2, χ2) = (f2(x, U1, DU1), χ2)

∀χi ∈ Sh,i(Ω).

In general, however, Sh,1 � Sh,2 (or vice-versa) on Ω, and we may be forced towork with either Π1→2f2(U1) or more generally with f2(x,Π1→2U1,Π1→2DU1), whereΠi→j is some projection from Sh,i to Sh,j . If the projection is to Sh,i from W 1

2 (Ωi),then we simply write the projection as Πi. The resulting discrete system becomes

(3.3)

{A1(U1, χ1) = (f1, χ1),A2(U2, χ2) = (f2(x,Π1→2U1,Π1→2DU1), χ2)

∀χi ∈ Sh,i(Ω).

Primary adjoint problem. We seek the error in a quantity of interest repre-sentable by a linear functional of the error e2, where ui−Ui = ei denotes the pointwiseerrors. Note that a quantity of interest involving only u1 can be computed withoutsolving for u2, hence, there is no loss of generality. The global adjoint problem, definedrelative to the quantity of interest, is{

−∇ · a1∇φ(1)1 − div(b1φ

(1)1 ) + c1φ

(1)1 + Lf2(u1)φ

(1)2 = 0,

−∇ · a2∇φ(1)2 − div(b2φ

(1)2 ) + c2φ

(1)2 = ψ

(1)2 ,

where

Lf2(u1)(u1 − U1) =∫ 1

0

∂f2∂u1

(u1s+ U1(1 − s)) ds

is a linearization of f2 and φ(1)1 and φ

(1)2 satisfy homogeneous Dirichlet boundary

conditions. The corresponding weak formulation is

(3.4)

{A∗

1(φ(1)1 , v1) + (Lf2(u1)φ

(1)2 , v1) = 0,

A∗2(φ

(1)2 , v2) = (ψ(1)

2 , v2)∀vi ∈ W 1

2 (Ω),

where

(3.5)

{A∗

1(φ(1)1 , v1) = (a1∇φ(1)

1 ,∇v1) − (div(b1φ(1)1 ), v1) + (c1φ

(1)1 , v1),

A∗2(φ

(1)2 , v2) = (a2∇φ(1)

2 ,∇v2) − (div(b2φ(1)2 ), v2) + (c2φ

(1)2 , v2).

Using the standard argument, we have the following error representation formula:

(3.6)(ψ(1), e

)=(ψ

(1)2 , e2

)= A∗

2

(φ

(1)2 , e2

)=(f2(x, u1, Du1), φ

(1)2

)−A2

(U2, φ

(1)2

).

Observe that φ(1)1 does not appear in the error representation formula. We define the

primary adjoint problem as

A∗2(φ

(1)2 , v2) = (ψ(1)

2 , v2) ∀v2 ∈ W 12 (Ω).

Remark 3.1. At first glance, it appears that we need only to solve the secondadjoint equation and thus do not need to construct the linearization Lf2. However,as seen in the linear algebra example, the analysis takes into account the transfer



of error from the solution of the first component. Estimating this transferred erroruses a nonlinear functional of the error to form the right-hand sides in “transferadjoint problems”(2.3) and (3.11). We approximate this nonlinear functional usingthe linearization Lf2. We evaluate the linearization at the computed solution U ,which can be justified by using Taylor’s theorem and assuming that the error u − Uis sufficiently small.

Adding and subtracting the projection of φ(1)2 onto the primal approximation

space (Π2φ(1)2 ) in (3.6) yields

(3.7) (ψ(1)2 , e2) = (f2(x, u1, Du1), (I − Π2)φ

(1)2 ) −A2(U2, (I − Π2)φ

(1)2 )

+ (f2(x, u1, Du1),Π2φ(1)2 ) −A2(U2,Π2φ

(1)2 ).

To simplify later constructions, we introduce the notion of the weak residual of asolution component, namely,

Ri(Ui, χ; ν) = (fi(ν), χ) −Ai(Ui, χ; ν)

and using this notation write (3.6) as

(ψ(1), e) = R2(U2, φ(1)2 ;u1),

indicating that this estimate depends on the solution u1.

3.1. Transfer error analysis. Error representation (3.7) is not computable,since u1 is unknown. We add and subtract

(f2(x, U1, DU1), (I − Π2)φ

(1)2

)from error

representation formula (3.7) and use the definition of approximate weak statement(3.2) to obtain(3.8)

(ψ(1)2 , e2) =

(f2(x, U1, DU1), (I − Π2)φ

(1)2

)−A2(U2, (I − Π2)φ(1)2 )

+(f2(x, u1, Du1) − f2(x, U1, DU1), φ

(1)2

)= R2

(U2, (I − Π2)φ

(1)2 ;U1

)+(f2(x, u1, Du1) − f2(x, U1, DU1), φ

(1)2

).

The first term on the right of (3.8) is a traditional dual-weighted residual expressionfor the error arising from discretization of the second component, while the remainingdifference represents the transfer error that arises from using an approximation of u1

in defining the coefficients in the equation for u2. The goal now is to estimate thistransfer error and its effect on the quantity of interest.

As with the linear algebra example in section 2, we recognize the transfer errorexpression as a functional of error in u1 and define

(f2(x, u1, Du1) − f2(x, U1, DU1), φ(1)2 )

as a new quantity of interest. Then, we construct a secondary adjoint problem tocompute the transfer error. In order to obtain a linear functional when f2 is nonlinearin u1, we linearize f2(u1) ≈ f2(U1) + Df2(U1) × (u1 − U1), where Df is the Frechetderivative of f2 at U1. The transfer error term becomes

(3.9)(Df2(U1) × e1, φ

(1)2

),

which is a linear functional of the error e1 that describes the effect of errors in U1

on the quantity of interest. Note that the Riesz representation theorem guaranteesthe existence of a ψ(2)

1 such that (ψ(2)1 , e1) equals (3.9), though ψ(2)

1 is not needed toevaluate the functional or compute the corresponding adjoint solution.



Transfer error adjoint problem. To estimate the new quantity of interest, wedefine

{(a1∇φ(2)

1 ,∇v1)− (div(b1φ

(2)1 ), v1

)+(c1φ

(2)1 , v1) + (Lf2(u1)φ

(2)2 , v1

)= ψ

(2)1 ,(

a2∇φ(2)2 ,∇v2

)− (div(b2φ(2)2 ), v2

)+(c2φ

(2)2 , v2

)= 0,

∀vi ∈ W 12 (Ω).

The second equation has the trivial solution, and the secondary adjoint problemreduces to the “transfer error adjoint problem”

(3.10)(a1∇φ(2)

1 ,∇v1)− (div(b1φ

(2)1 ), v1

)+(c1φ

(2)1 , v1

)=(ψ

(2)1 , v1

) ∀v1 ∈ W 12 (Ω).

The transfer error representation formula is given by

(3.11)

(ψ

(2)1 , e1

)= A∗

1(φ(2)1 , e1) = A1(e1, φ

(2)1 )

=(f1, (I − Π1)φ

(2)1

)−A1(U1, (I − Π1)φ(2)1 ),

where we have used Galerkin orthogonality to introduce the projection of φ onto thediscretization space (as f1 does not depend on u). Inserting (3.11) into (3.8) yields

(3.12)(ψ, e

)=(f2(x, U1, DU1), (I − Π2)φ

(1)2

)−A2(U2, (I − Π2)φ(1)2 )

+(f1, (I − Π1)φ

(2)1

)−A1(U1, (I − Π1)φ(2)1 )

or

(ψ, e

)= R2(U2, (I − Π2)φ

(1)2 ;U1) + R1(U1, (I − Π1)φ

(2)1 ).

Remark 3.2. If the model problem includes coupling in the coefficients of thesecond differential operator, i.e.,(3.13)⎧⎪⎨⎪⎩−∇ · a1(x)∇u1 + b1(x) · ∇u1 + c1(x)u1 = f1(x), x ∈ Ω,−∇ · a2(x, u1)∇u2 + b2(x, u1) · ∇u2 + c2(x, u1)u2 = f2(x, u1, Du1), x ∈ Ω,u1 = u2 = 0, x ∈ ∂Ω,

then the error representation formula for a quantity of interest that depends on u2

alone is

(ψ, e

)= R2(U2, (I − Π2)φ

(1)2 ;u1).

Since this is not computable, we replace each term in the weak residual with the sameterm evaluated at U1, yielding

(ψ, e

)= R2(U2, (I − Π2)φ

(1)2 ;U1) +

(f2(u1) − f2(U1), φ

(1)2

)− ((a2(u1) − a2(U1))U2, φ

(1))− ((b2(u1) − b2(U1)) · ∇U2, φ

(1))

− ((c2(u1) − c2(U1))U2, φ(1)).



We linearize f2, a2, b2, and c2 around U1 to obtain an approximate transfer errorterm(Df2(e1), φ

(1)2

)+(Da2(e1)∇U2,∇φ(1)

2

)+(Db2(e1) · ∇U2, φ

(1)2

)+(Dc2(e1)U2, φ

(1)2

).

This is a linear functional on L2(Ω), which we use as data to define the “transfer”error adjoint problem and derive a corresponding a posteriori error representation.For details on how to compute a quantity of interest that depends on u1 and u2(sothat the choice of linearizations for the coefficients in the equation for u2 enter directlyinto the “primary” error contribution), see [9].

Remark 3.3. For a “lower triangular” one-way coupled system of N ellipticequations and a quantity of interest based on the Nth component, we solve N total“single physics” adjoint problems and construct the error representation

(ψN , eN

)=

N∑i=1

RN−i+1

(UN−i+1, φ

(i);U).

We then solve a sequence of adjoint problems, as the corresponding linear functionalfor the ith adjoint problem (i > 1) can be defined recursively (assuming the couplingoccurs only through the right-hand side) as

i−1∑j=1

(DfN+1−j

Dui

∣∣∣∣U

(ei), φ(j)

).

This extends to coupling in all of the coefficients as above.

3.2. Numerical examples. The following three numerical examples highlightthe features of the analysis and the importance of accounting for the transfer error.In the following computations, we approximately solve all adjoint problems using con-tinuous, piecewise quadratic elements in order to be able to evaluate the interpolantsarising from Galerkin orthogonality. We denote these approximate adjoints solutionsby Φ and use them in place of φ in error representation (3.12). For adaptive meshrefinement, we write the estimate as a sum of element contributions and derive abound by introducing norms. We base the adaptive mesh refinement on the standardoptimization approach using the principle of equidistribution [6] applied to the bound.We refine elements whose element contribution to the error bound is greater than halfa standard deviation from the mean error contribution or refine a fixed fraction ofthe elements with the greatest element contributions, whichever criterion yields thegreater refinement. We do not do any mesh coarsening, smoothing, or edge flips.

Example 3.1. This example demonstrates the fact that the transfer error can besignificant even if the individual components u1 and u2 are well resolved. We considera simple system

(3.14)

⎧⎪⎨⎪⎩−Δu1 = sin(4πx) sin(πy), (x, y) ∈ Ω,−Δu2 = b · ∇u1, (x, y) ∈ Ω,u = 0, (x, y) ∈ ∂Ω,

where

b =2π

(25 sin(4πx)

sin(πx)

), f(u) =

(sin(4πx) sin(πy)

b · ∇u1

), Ω = ([0, 1], [0, 1]).



0 0.2 0.4 0.6 0.8 1

0

0.5

1−1.5

−1−0.5

00.5

1

(a) U1

0 0.2 0.4 0.6 0.8 1

0

0.5

1−0.8−0.6−0.4−0.2

00.20.4

(b) U2

0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.5

1

1.5

2

(c) Φ(1)2

0 0.2 0.4 0.6 0.8 1

0

0.5

1−0.1

−0.050

0.050.10.150.2

(d) Φ(2)1

Fig. 3.1. Example 3.1. Primary and (nonzero) adjoint solutions computed on uniformly finemeshes. The adjoint solutions are largest near the region of the quantity of interest u2(.25, .25).

The quantity of interest is the solution value of u2 at (.25, .25), which we estimate usinga smooth delta function approximation with localized support. The correspondingglobal adjoint problem is

(3.15)

⎧⎪⎨⎪⎩−Δφ(1)

1 + Lf(u1)φ(1)2 = 0, (x, y) ∈ Ω,

−Δφ(1)2 = δregx , (x, y) ∈ Ω,

φ = 0, (x, y) ∈ ∂Ω,

where δregx is a regularized delta function and x = (.25, .25). Our primary adjointproblem is

−Δφ(1)2 = δregx , (x, y) ∈ Ω, φ = 0, (x, y) ∈ ∂Ω.

The secondary adjoint problem is

(3.16)

{Δφ(2)

1 = ∇ · (bφ(1)2 ), (x, y) ∈ Ω,

φ(2) = 0, (x, y) ∈ ∂Ω.

The primal system was solved using identical standard continuous piecewise linearfinite element discretizations for u1 and u2. We plot the results in Figure 3.1 and show



Table 3.1

Error contributions for Example 3.1.

Primary error Transfer error0.0042 0.0006

0 0.2 0.4 0.6 0.8 10

0.5

1−0.6

−0.4

−0.2

0

0.2

0.4

(a) U1

0 0.2 0.4 0.6 0.8 1

0

0.5

1−2

−1.5

−1

−0.5

0

0.5

(b) U2

Fig. 3.2. Example 3.2. Adaptivity based on the standard discretization error estimate for the“primary” error, ignoring the “transfer” error. Only the mesh for U2 is refined.

Table 3.2

Error contributions for Example 3.2.

Primary error Transfer error0.00005 0.110

the error contributions in Table 3.1. While the adjoint solution Φ(2)1 in Figure 3.1(d)

is concentrated near the location of the quantity of interest, it has nontrivial spatialstructure, and the transfer error represents 14% of the total error.

Example 3.2. This example illustrates the importance of computing the transfererror, since, for this problem, simply forcing the “primary” error contribution to besmall (by refining the second mesh only) does not provide any accuracy in the desiredquantity of interest. We reconsider (3.14) but with quantity of interest equal to theaverage value of u2 over the whole domain. The exact solution has zero averagevalue on Ω. We solve both components of the primary problem on an identical coarseinitial mesh, but adapt and refine only the mesh for u2 using the traditional weightedresidual, the first “primary” error term in (3.12), while neglecting the second “transfererror” term in (3.12). We show the results in Figure 3.2 and Table 3.2.

Ignoring the transfer error and the implied need to refine the first component pro-duces a completely unsuitable adaptive procedure. It is clear from Figure 3.2 that theaverage value of the second component is far from zero, and the actual computationalvalue is −0.2245. The estimated transfer error of 0.1 is, in fact, an underestimatesince Φ(2)

1 is based on the highly inaccurate solution U1, which is computed on a verycoarse mesh. The transfer error dominates the computation, and this error cannot bereduced without refining the mesh for u1.

Example 3.3. The third example shows that an “optimal” adaptive mesh for thequantity of interest that depends only on u2 may actually involve a richer discretiza-



0 0.2 0.4 0.6 0.8 1

0

0.5

1−1

−0.5

0

0.5

1

(a) U1

0 0.2 0.4 0.6 0.8 100.5

1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

(b) U2

Fig. 3.3. Example 3.3. Adaptivity based on the full estimate that accounts for “primary” and“transfer” errors. The quantity of interest U2(.25, .25) is more sensitive to errors in U1 than U2.

tion of u1 than u2. We consider system (3.14) with the quantity of interest equal tothe average value of u2 over the whole domain and initial coarse meshes as in theprevious example, but we use the transfer error contribution to adapt the mesh for u1

and the primary contribution to adapt the mesh for u2 so that the total error is lessthan 10−4. The resulting meshes are shown in Figure 3.3 and illustrate that despitethe fact that the quantity of interest involves only u2, the error inherited from u1 isthe most important contribution to consider. In this problem, the strong influenceof the transfer error is a result of the dependence of u2 on the gradient of u1, whicha priori has lower order accuracy. Similar behavior could also arise when u2 justdepends on u1.

4. Interpolation error analysis. We use a multiscale discretization for the“fully” adaptive Example 3.3, i.e., the components u1 and u2 were computed on dif-ferent meshes; see Figure 3.3. This raises the issue of understanding the effect oftranslating one component onto the mesh of the other component when performingthe integration necessary to form the discrete equations. Integration involving func-tions defined on different meshes can cause problems because these quantities may becomplicated, as illustrated in Figure 4.1.

In particular, traditional quadrature formulae based on sets of specific pointsmay not preserve the accuracy required for effective computation because a functiondefined on a different mesh is generally not sufficiently smooth. For example, the

Mesh for U1

Mesh for U2

Fig. 4.1. The problem of translation between meshes. Finite element functions on one meshare generally not smooth on another mesh.



integrand (f2, χ) is piecewise discontinuous on every element τi of mesh 2 in Example3.3, as b ·∇U1 is continuous only within elements of the mesh for U1. In general, if themeshes are not congruent, the integrand is C0 at best. Using a “traditional” higherorder quadrature rule will not necessarily lead to the expected increase in accuracyas the integrand (f2, χ) does not have sufficient regularity. Possible solutions includeeither the determination of local intersections of simplices and/or hexahedra or theconstruction of a global union mesh. However, both solutions are computationallyexpensive, and the global solution often requires several times more memory than thestorage of the two individual meshes, especially for three-dimensional problems.

4.1. Projections from mesh 1 to mesh 2. Instead of constructing a unionmesh, we use a projection Π1→2 from S1,h to S2,h and solve the discrete system givenby (3.3). This introduces additional sources of error. Starting from error representa-tion formula (3.6), we add and subtract

f2(x,Π1→2U1,Π1→2DU1, (I − Π2)φ(1)2 ),

yielding

(ψ(1), e) = (ψ(1)2 , e2)

=(f2(x,Π1→2U1,Π1→2DU1), (I − Π2)φ

(1)2

)−A2

(U2, (I − Π2)φ

(1)2

)+(f2(x, u1, Du1) − f2(x,Π1→2U1,Π1→2DU1), φ

(1)2

).

Adding and subtracting(f2(x, U1, DU1), φ

(1)2

)produces

(ψ(1), e) =(f2(x,Π1→2U1,Π1→2DU1), (I − Π2)φ

(1)2

)−A2

(U2, (I − Π2)φ

(1)2

)+(f2(x, u1, Du1) − f2(x, U1, DU1), φ

(1)2

)+(f2(x, U1, DU1) − f2(x,Π1→2U1,Π1→2DU1), φ

(1)2

).

The first two terms on the right represent the primary discretization error for a func-tional of the the second component, the third term on the right represents transfererror (3.11), and the fourth term is a new expression that represents the error fromthe projection Π1→2. The projection error can be decomposed as

(4.1)(Π1→2f2(x, U1, DU1) − f2(x,Π1→2U1,Π1→2DU1), φ

(1)2

)+ (I − Π1→2)

(f2(x, U1, DU1), φ

(1)2

).

The first inner product in (4.1) can be computed (with some effort) on Ω2,h. However,computing the second term raises the same numerical issues that caused the adoptionof the projection Π1→2 in the first place! We handle this term using the Monte Carlotechniques described in section 4.3.

4.2. Projections from mesh 2 to mesh 1. Complications from the use of pro-jections also arise in computations with the solution of the secondary adjoint problem.The secondary adjoint problem domain is Ω1,h, but φ(1)

2 is computed naturally on Ω2,h.



The new error representation formula for the transfer error becomes

(Df2(U1) × e1,Π2→1φ

(1)2

)+(Df2(U1) × e1, (I − Π2→1)φ

(1)2

),

which is the error contribution arising from the transfer as well as an additional termthat is large when the approximation spaces are significantly different. For example,this term is important when the original system is multiscale. The implicit ψ(2) forthe transfer error adjoint is now

(f2(u1) − f2(U1), φ

(1)2

)=(Df2(U1) × e1,Π2→1φ

(1)2

)=(ψ

(2)1 , e1

).

The additional term(Df2(U1) × e1, (I − Π2→1)φ

(1)2

)is a linear functional, so we

may define an additional “tertiary” adjoint problem to estimate this quantity.

Projection (“tertiary”) error adjoint problem. This problem has the sameform as transfer error adjoint (3.10), but with data ψ(3)

1 satisfying

(ψ

(3)1 , e1

)=(Df2(U1) × e1, (I − Π2→1)φ

(1)2

).

The resulting error representation formula is

(4.2)(ψ

(3)1 , e1

)=(f1, (I − Π1)φ

(3)1 ) −A1(U1, (I − Π1)φ

(3)1

)=(R1, (I − Π2→1)φ

(3)1

).

The error representation is therefore

(4.3)(ψ(1)

2 , e2) = R2(U2, (I − Π2)φ(1)2 ;U1) + R1(U1, (I − Π1)(φ

(2)1 + φ

(3)1 ))

+(Π1→2f2(U1) − f2(Π1→2U1), φ

(1)2

)+((I − Π1→2)f2(U1), φ

(1)2

).

Remark 4.1. Traditional simplex-based numerical integration methods that in-terrogate U1 at cubature points can be thought of as projecting the integrand f(U1)χ2

into a specific polynomial space Pτ defined on each simplex τ of the mesh for U2 andthen integrating exactly. We may express this “cubature error” as a projection errorand construct a corresponding error representation formula in a similar manner. Cu-bature error resulting from the fact that integration was not performed on a “union”mesh of two piecewise polynomial spaces may always be viewed as projection error.

Remark 4.2. In this discussion, we assume that the adjoint problems are solvedusing approximation spaces that are compatible with the corresponding primal ap-proximation space, e.g., using higher order Lagrange elements on the same mesh. Inpractice, different meshes may be used for the primal and adjoint solves. However,this introduces new projection operators between the corresponding approximationspaces as well as the additional terms due to the loss of Galerkin orthogonality. Weconfine ourselves to merely alluding to the notational complexities and length of theresulting error representation.

4.3. Monte Carlo Integration. Interpolation-based projections suffer frommesh-aliasing difficulties. An extreme example is given in Figure 4.2. For a morepractical example, we construct two quasi-uniform, unstructured meshes 1 and 2,



0 0.2 0.4 0.6 0.8 1−1

0

1 U1

Π1→2U1

Fig. 4.2. Interpolation errors for two meshes on Ω = [0, 1].

Table 4.1

Errors in various approximations of Ie.

|Ie − I1| |I1 − Igauss| |I1 − IΠ| |I1 − ISamp|0.000187 0.000246 0.0060 0.00041

both of size h on Ω = [0, 1] × [0, 1] and take the piecewise linear interpolant fI1 ofthe function f = sin(20hx) sin(20hy) on mesh 1. We first compute Ie =

∫Ω f dx and

I1 =∫ΩfI1 dx exactly and then construct three different approximations Igauss, IΠ,

and ISamp as follows:1. Igauss. Using a third order, four-point quadrature rule [16] on the triangles

of mesh 2 by interpolating fI1 at the corresponding quadrature points.2. IΠ. Projecting fI1 onto mesh 2 by interpolating fI1 at the nodes of mesh 2

and then using exact integration.3. ISamp. Performing the integration via a uniform weight quadrature rule us-

ing the quadrature points corresponding to the four-point quadrature ruleemployed by Igauss.

We show the accuracy in Table 4.1. Note that the work for all three methods isroughly the same. The smallest of the projection errors |I1 − Igauss| is larger thanthe interpolation error |Ie − I1|. The error in |I1 − IΠ| is a factor of 10 larger than|I1 − Igauss| and |I1 − ISamp|, which, for this problem, amounts to a factor of h−1.Note that the four-point Gauss quadrature rule is only slightly more accurate thanthe sampling rule ISamp.

Motivated by the example, we employ pseudorandom Monte Carlo integration us-ing p random uniformly distributed sample points on the reference element. The maindifficulty (and computational expense) when integrating on Ω2,h is the evaluation ofU1 at each random sample point, since this involves locating the point in the appro-priate element in Ω1,h. Nominally, this process requires (O(N)) operations per samplepoint, where N is the number of degrees of freedom for U1, hence O(MN) operationsfor the integration, where M is the number of degrees of freedom for U2. However,this approach may be greatly accelerated by using a geometric implementation of theassembly and point search algorithms.

We illustrate the search algorithm in Figure 4.3. We generate a random inte-gration point p1

1 in τ1 ∈ Ω2,h and determine the containing element of Ω1,h. Thiscould potentially involve a full search of Ω1,h, but as this is the initial element, a goodstarting guess for element location could be provided as an input. Once a match-ing simplex is found in Ω1,h, the computation is performed, and the next integrationpoint p1

2 is generated. Moreover, the last matching simplex is stored, so the geomet-ric search using edge/face neighbors and barycentric coordinates to guide neighbor



τi ∈ Ω2,h

u1,h

pi2

pi+11

pi1

Fig. 4.3. Monte Carlo integration point search.

selection for the next point is very fast. When the integration is finished, we selectthe next element to be an edge/face neighbor. Now when we generate p2

1, we have agood starting point, namely, the last match in p1

S which should be “close” to the realelement containing p2

1. The assembly routine keeps selecting edge neighbors until ithas looped over all elements recursively.

This algorithm works even with a primitive data structure as long as recursion isemployed. If the number of mesh elements is large, however, this may not be practicaldue to recursion limits. A nonrecursive algorithm could lead to termination beforeall element contributions for the mesh were calculated, as the next element returnedby the search could have all edge/face neighbors whose element contributions hadalready been calculated. The algorithm would have to “restart” from an element thathas not been computed. On quasi-uniform meshes with no fine scale features in thegeometry, the number of “restarts” also grows logarithmically with the number ofelements. Of course, with a more sophisticated data structure, either octree based or,for example, a mesh where the elements had been ordered by the use of a space-fillingcurve, the need for restarting would be eliminated.

When the meshes for Ω1,h and Ω2,h are both quasi-uniform on Ω, the number ofelements tested in Ω is bounded by some h-independent constant for each integrationpoint. Obviously, this is not the case for general adapted or anisotropic meshes, butin practice, the number of searches grows at most logarithmically with the numbersof degrees of freedom in u1. The convergence of this Monte Carlo integration schemefollows from standard results (see [11]) as the integrand can always be defined as thesum of integrals of continuous functions on individual simplices of the union mesh ofΩ1,h and Ω2,h.

4.4. Numerical examples. We demonstrate the significance of the projectionerrors with two examples.

Example 4.1. The first example illustrates how the projection error can influencea typical computation. We consider a system defined by (3.14), with two randomlygenerated initial meshes for u1 and u2. The initial mesh for u1 is finer than for u2 inorder to reduce the transfer error. The quantity of interest in this computation is theaverage value of u2. We show the results in Figure 4.4 and Table 4.2.

We use a local projector Π1→2,τ given by interpolation at the Gauss points (third-order three-point simplex rule) of simplices τ in Sh,2. Use of this projector wouldintegrate (U1, U2) exactly if the meshes were identical. The solution using this pro-jector is given by Figure 4.4(b). This is compared against a 16-point Monte Carlocomputation illustrated by Figure 4.4(c).



00.2

0.40.6

0.81

0

0.5

1−1

−0.5

0

0.5

1

(a) U1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

−0.5

0

0.5

1

1.5

2

(b) U2 computed with Π1→2

0 0.2 0.4 0.6 0.8 1

0

0.5

1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(c) U2 computed with high-sample Monte Carlointegration

Fig. 4.4. Example 4.1. The role of projection errors in nonaligned meshes. Note that themagnitude and oscillation of U2 computed with Π1→2,τ , shown in (b) are incorrect (a fine scale U2

is given by Figure 3.1).

Table 4.2

Example 4.1. Error contributions for computation shown in Figure 4.4.

Primary error Transfer error Projection error0.003533 0.021589 0.007908

As discussed in section 4.2, projection from S2h,2 to S2

h,1 can also lead to significantinaccuracies in computing the transfer error, necessitating the computation of tertiaryadjoint problem (4.2).

Example 4.2. As discussed in section 4.2, projection from S2h,2 to S2

h,1 can alsolead to significant inaccuracies in computing the transfer error, necessitating the com-putation of tertiary adjoint problem (4.2). This example shows that computationswith significant differences in mesh scale can contribute significantly to the error. Weagain use the system in Example 3.1 with the quantity of interest point value at(.15, 15), starting with a coarse identical initial mesh for u1 and u2 but refining onlythe mesh for u2. There is no projection error as Sh,2 ⊆ Sh,1. However, when wecompute the transfer error, we ignore the fact that a natural choice of decompositionfor the computation is integration over the simplices of Sh,2. Instead, we use the



Table 4.3

Example 4.2. Error contributions for the computation shown in Figure 4.5.

Primary error Transfer error Projection error Tertiary error0.000713 0.0905 0 0.0325

0 0.2 0.4 0.6 0.8 1 0

0.5

1

−1

−0.5

0

0.5

1

Fig. 4.5. Example 4.2. Tertiary adjoint solution Φ(3)1 which estimates the projection error in

computing the transfer error.

interpolation of φ(1)2 at the quadrature points at the simplices of Sh,1. To compute

(I − Π), we employ the actual nesting of the two meshes to perform an accurate (upto quadrature error on the fine scale mesh) computation of φ(3)

1 . We show the resultsin Table 4.3 and Figure 4.5.

5. An adaptive algorithm for the operator decomposition finite elementmethod. Given tolerance TOL on the error in the quantity of interest, an adaptivealgorithm that takes into account all the possible sources of error is given below.

while (the total error is less than TOL) doCompute U1 using standard integration.Compute U2 using 16-point M.C. integration for the coupling term.Compute Φ(1)

2 using standard integration.Compute Φ(2)

1 for given adjoint data using 16-point M.C. integration.if (the sum of two error contributions is greater than TOL) then

Refine both meshes based on the primary error contributions for U2 and thetransfer error contributions for U1.

elseCompute the projection error by comparing with a 64-point M.C. integration.Compute Φ(3)

1 .if (the total error is greater than TOL) then

Refine both meshes based on the primary and projection error contributionsfor U2 and the transfer and tertiary error contributions for U1.

end ifend if

end while.The algorithm drives the primary and transfer error contributions to within a

specified error tolerance and then checks for projection error by using 64-point MonteCarlo integration as an approximation to the identity operator I in (4.3) and at-



tempts to correct the projection error by refinement as well. Any projector could besubstituted for the M.C. integration used in computing U2 and Φ(2)

1 .We select the use of 16 sample points for the Monte Carlo integration based on

our experience from a series of numerical experiments where different functions wereinterpolated on a quasi-uniform mesh, and then integrated. This interpolant was thenintegrated using Monte Carlo with 2N sample points per simplex on a different quasi-uniform mesh (with the same approximate h); N = 4 gave the best tradeoff betweenspeed and accuracy.

5.1. Examples. We describe two applications of the algorithm to one-way cou-pled systems using different meshes for each solution component. In both examples,we start with identical coarse initial meshes (quasi-uniform with h ≈ .125) and adapteach mesh until both the primary and transfer error formulas are less than 10−4. Wecontrol projection error using Monte Carlo integration.

Example 5.1. In the first example, we approximate the value of u2 at (.25, .25),where (u1, u2) solves⎧⎪⎨⎪⎩−Δu1 = 64π2 sin 4π(x− .75 + |x− .75|) sin 4π(y − .75 + |y − .75|), (x, y) ∈ Ω,−Δu2 = u1, (x, y) ∈ Ω,u1 = u2 = 0, (x, y) ∈ ∂Ω,

with Ω = ([0, 1], [0, 1]). The corresponding adjoint problem is{−Δφ(1)

2 = δreg(x0), (x, y) ∈ Ω,φ

(1)2 = 0, (x, y) ∈ ∂Ω,

with x0 = (.25, .25). The transfer error adjoint problem is{−Δφ(2)

1 = φ(1)2 , (x, y) ∈ Ω,

φ(2)1 = 0, (x, y) ∈ ∂Ω.

The accurate computation of the quantity of interest u2(0.25, 0.25) does not re-quire the fine scale features of u1 near (0.75, 0.75) to be resolved. However, a quantityof interest equal to the value of u2 at (0.9, 0.9) near the localized features of u1 requiresbetter resolution of the details of u1. The adapted solutions U1 for both quantities ofinterest are given in Figure 5.1(c) and Figure 5.1(d), respectively.

Example 5.2. We now consider an example where convection in component u1

creates the need for refinement in u1 remote from the the goal-oriented refinement inu2.

(5.1)

⎧⎪⎨⎪⎩−Δu1 − b · ∇u1 = 103e−100‖x−x0‖2

, x ∈ Ω,−Δu2 = 103e−100‖x−x1‖2

u1, x ∈ Ω,u1 = u2 = 0, x ∈ ∂Ω,

where b = (100 40)�, x0 = (.75, .75), x1 = (.1, .5), and the quantity of interest is thepoint value u2(x2), x2 = (.2, .5). The corresponding adjoint problem for the primaryerror contribution is {

−Δφ(1)2 = δreg(x2), x ∈ Ω,

φ(1)2 = 0, x ∈ ∂Ω,



00.5100.20.40.60.81

−0.5

0

0.5

1

(a) U1 computed using a uniformly fine mesh

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.5

1

(b) U2 for the quantity of interest u2(0.25, 0.25)

00.20.40.60.81

0

0.2

0.4

0.6

0.8

1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c) U1 for the quantity of interest u2(0.25, 0.25)

0 0.2 0.4 0.6 0.8 10

0.5

1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

(d) U1 for the quantity of interest u2(0.9, 0.9)

Fig. 5.1. Example 5.1. Example of computational efficiency: U1 may be computed on a coarsediscretization, yet U2 may be determined with sufficient accuracy.

while the corresponding transfer error adjoint problem is

⎧⎨⎩−Δφ(2)

1 + b · ∇φ(2)1 = 103e−100‖x−x1‖2

φ(1)2 , x ∈ Ω,

φ(2)1 = 0, x ∈ ∂Ω.

The adjoint solution Φ(2)1 in Figure 5.2(c) shows the influence of the convection

term in the equation for u1. When the quantity of interest is a value of u2 in theconvective region of influence of the localized source term in the equation for u1, thesolution for u1 is resolved “upstream” of the location of the quantity of interest asshown in Figure 5.2(a).

When the quantity of interest is a value of u2 away from the convective region ofinfluence of the localized source term in the equation for u1, the adjoint solution φ(2)

1

has a similar structure to that shown in Figure 5.2(c) but has much smaller magni-tude. The resulting mesh for U1 need not even be detailed enough to eliminate thenumerical oscillation (from not satisfying the corresponding Peclet mesh condition).This situation is illustrated by Figure 5.2(d), where the choice of quantity of interestis u2 at (0.15, 0.15).



0 0.2 0.4 0.6 0.8 1

0

0.5

1−0.5

0

0.5

1

1.5

(a) U1 for quantity of interest u2(.2, .5)

0 0.2 0.4 0.6 0.8 1

0

0.5

10

1

2

3

4

5

(b) U2 for quantity of interest u2(.2, .5)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1−0.5

0

0.5

(c) φ(2)1 for quantity of interest u2(.2, .5)

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

1.5

(d) U1 for quantity of interest u2(.15, .15)

Fig. 5.2. The role of convection in Example 5.2. Note that altering the location of the quantityof interest alters the density and location of the resulting adapted meshes (same adaptive criteria).

6. Conclusion. In this paper, we perform an a posteriori error analysis of amultiscale operator decomposition finite element method for the solution of a systemof one-way coupled elliptic problems. The analysis specifically accounts for the effectsarising from multiscale operator decomposition, including the following issues: (1)Errors in the solution of each component propagate into the solutions of the othercomponents; and (2) Transferring information between different representations po-tentially introduces new error. We estimate the various sources of errors by definingauxiliary adjoint problems whose data are related to errors in the information passedbetween components. Through a series of examples, we demonstrate the importanceof accounting for the contributions to the error arising from multiscale operator de-composition. We also devise an adaptive discretization strategy based on the errorestimates that specifically controls the effects arising from operator decomposition.Finally, we demonstrate the usefulness of Monte Carlo integration methods for dealingwith a mismatch between discretizations of different components.

We extend this analysis to a “fully coupled” system in the form of (1.4) in part IIof this paper [3]. We address the important issue that the adjoint operator associatedwith the fully coupled system and an operator decomposition solution are not generallyequal. This difference requires additional strategies for error control. We consider theuse of noninterpolatory projectors based on averaging to reduce both transfer andprojection error in [4].



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[3] V. Carey, D. Estep, and S. Tavener, A posteriori analysis and adaptive error control foroperator decomposition methods for elliptic systems II: Fully coupled systems, Internat. J.Numer. Methods Engrg., submitted.

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