A posteriori analysis and adaptive error control for ...tavener/manuscripts/carey/coupled-semi... · In this paper, we develop an a ... model for the Belousov–Zhabotinsky reaction

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2013; 94:826–849Published online 4 April 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4482

A posteriori analysis and adaptive error control for operatordecomposition solution of coupled semilinear elliptic systems

V. Carey1, D. Estep2,*,† and S. Tavener1

1Department of Mathematics, Colorado State University, Fort Collins, CO 805232Department of Statistics, Colorado State University, Fort Collins, CO 80523

SUMMARY

In this paper, we develop an a posteriori error analysis for operator decomposition iteration methodsapplied to systems of coupled semilinear elliptic problems. The goal is to compute accurate error estimatesthat account for the combined effects arising from numerical approximation (discretization) and operatordecomposition iteration. In an earlier paper, we considered ‘triangular’ systems that can be solved with-out iteration. In contrast, operator decomposition iterative methods for fully coupled systems involve aniterative solution technique. We construct an error estimate for the numerical approximation error thatspecifically addresses the propagation of error between iterates and provide a computable estimate forthe iteration error arising because of the decomposition of the operator. Finally, we develop an adaptivediscretization strategy to systematically reduce the discretization error. Copyright © 2013 John Wiley &Sons, Ltd.

Received 25 November 2011; Revised 2 November 2012; Accepted 16 January 2013

KEY WORDS: a posteriori error estimates; adjoint problem; dual problem; error estimates; finiteelement method; generalized Green’s function; operator splitting; operator decomposition;coupled problems

1. INTRODUCTION

We develop an a posteriori error analysis framework for operator decomposition iteration methodsapplied to systems of coupled semilinear elliptic problems of the form,

L1.x,u1,Du1,D2u1/D f1.x,u1,u2,Du2,u3,Du3, � � � ,un,Dun/, x 2�,

L2.x,u2,Du2,D2u2/D f2.x,u1,Du1,u2,u3,Du3, � � � ,un,Dun/, x 2�,

...

Ln.x,un,Dun,D2un/D fn.x,u1,Du1,u2,Du2, � � � ,un�1,Dun�1,un/, x 2�,

(1.1)

where Dv and D2v are the first and second-order derivative operators, ¹Li , i D 1, � � � ,nº isa collection of linear uniformly coercive, elliptic differential operators, ¹fi , i D 1, � � � ,nº is acollection of differentiable functions, � is a convex polygonal domain with boundary @�, andEquation (1.1) is supplied with suitable boundary conditions on @�. We note that the coupling inthe system occurs through the right-hand sides only. We assume that Equation (1.1) satisfies suitableconditions to guarantee a solution in QW 1

2 .�/ in weak form, for example, generic conditions involvea uniform bound on the derivatives of f . An extension of our analysis to fully nonlinear ellipticsystems is straightforward but tedious in detail, for example, involving a messy linearization of thediffusion operator.

*Correspondence to: D. Estep, Department of Statistics, Colorado State University, Fort Collins, CO 80523.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

MULTISCALE OPERATOR DECOMPOSITION SOLUTION OF ELLIPTIC SYSTEMS 827

Interest in coupled physics problems and their solution arises in many fields. The Oregonatormodel for the Belousov–Zhabotinsky reaction system,

�@u1

@t� �D1r

2u1 D u1 � u21 � f u2

u1 � q

u1C q,

@u2

@t�D2r

2u2 D u1 � u2,

is an example of an important time-dependent coupled semilinear system. To consider wavestraveling with permanent form with velocity c in the x-direction, we make the ansatz, ui .t , x,y/Dui .�,y/, i D 1, 2, where �D x� ct . Upon substitution, we obtain the stationary coupled semilinearelliptic system,

��D1r2�u1 D �c

@u1

@�C u1 � u

21 � f u2

u1 � q

u1C q,

�D2r2�u2 D c

@u2

@�C u1 � u2,

where r2�.�/D @2.�/=@�2C @2.�/=@y2.

In many practical situations, coupled systems of partial differential equations are decomposedinto individual physics components, each of which is solved with a code specialized to theparticular type of physics, whereas the solution is obtained by various forms of iteration and/oroperator decomposition. Such approaches introduce new forms of instability and new sources oferror that must be included in an a posteriori error analysis, see [2] for an overview.

In this paper, we assume that the system (1.1) is solved by an operator decomposition approachthat involves iteratively solving the ordered sequence of problems,

Li�x, ui, Dui, D2ui

�D fi .x, Ou1,D Ou1, Ou2,D Ou2, � � � , Oui�1,D Oui�1, ui, OuiC1,D OuiC1, � � � , Oun,D Oun/ ,

which are obtained by substituting solutions Ouj for j ¤ i computed in a previous step in theequations in Equation (1.1) and then solving for ui . Theoretically, the sequence is iterated untilconvergence (if it does converge), whereas in practice, a finite number of iterations is used.

In [1], we present an a posteriori error analysis for operator decomposition methods applied tosystems of elliptic equations that had an ‘upper triangular’ form, so that one iteration through thesystem produces the solution. That analysis accounts for the following:

� errors arising from the discretization of each component elliptic problem,� the transfer of error between the component elliptic problems,� errors resulting from using different discretizations for the component elliptic problems.

In a fully coupled system (1.1), we need to iterate through the components until, hopefully, theiteration converges. In addition to the errors affecting the solution of a triangular system, the aposteriori error analysis now must also account for the following:

� the effects of finite iteration,� the transfer of errors between iterations.

These two effects are the focus of the analysis in this paper. The results in this paper can be combinedwith the full analysis of [1] to treat all five of these effects in one estimate.

We let U ¹kº be a numerical approximation obtained by iterating numerical discretizations of thedifferential equations k times. To carry out the a posteriori error analysis, we decompose the erroras

u�U ¹kº D u� u¹kº„ ƒ‚ …analytic iteration error

Cu¹kº �U ¹kº„ ƒ‚ …numerical error

D E¹kºC e¹kº, (1.2)

where u¹kº is the analytic solution obtained at iteration k by solving the sequence of iterateddifferential equations exactly. We estimate these two components separately. This decomposition

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2013; 94:826–849DOI: 10.1002/nme

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

is motivated by the observation that the iterative discrete approximation is a consistent numericalsolution of the analytic iterative problem. One consequence is that this simplifies the definitionof an appropriate adjoint for the error analysis. Namely, we use the adjoint associated with thesolution operator of the sequence of iterated component problems. This is a complex operator thatcan itself be defined through an iterated sequence of adjoint problems to the individual components.In practical terms, we can compute the resulting a posteriori error estimate without forming andsolving the adjoint for the fully coupled system. Solving the full adjoint of the coupled system iscomputationally impractical in situations in which operator decomposition iteration is used to solvethe forward problem.

The main focus of this paper is a posteriori error estimation of the numerical error e¹kº. At thekth step of an iterative solution process and for a given quantity of interest, the analysis accountsfor the following:

� the numerical errors made at the current iteration,� the numerical errors made at all previous iterations,� the error due to the iterative approximation.

For clarity of exposition, we do not include treatment of the errors arising from the use of differentdiscretizations for different components. Such effects are already treated in the earlier paper [1], andthose results can be combined with the results in this paper in a straightforward way.

To obtain a full a posteriori error estimate of the error u � U ¹kº, we have to also estimate theanalytic iteration error E¹kº. The difficulty is that this involves the true solution and an ‘iterative’true solution, both of which are unknown. However, for a fixed space mesh, we can adapt the clas-sical asymptotic estimator for the error in an iterative approximation to this situation. Lacking suchan estimate, the a posteriori error estimate of e¹kº that provides an estimate of the full error providedthe numerical error that dominates the iteration error, that is, in the limit of increasing iterations.

When we have estimates for E¹kº and e¹kº, we can then derive a generalized adaptive algorithmthat adjusts both the number of iterations and the level of discretization in each component to achievea desired accuracy with relative computational efficiency.

This work can be differentiated from a number of previous analyses of nonlinear problems, forexample, see references in [3–9], in concentrating on new issues arising in fully coupled systemsand dealing explicitly with the effects of operator decomposition and finite iteration. Ignoring thecoupling involved in treating a system, for example, simply estimating the error in each compo-nent in isolation, can lead to catastrophic failure of the error estimates. See Example 3.2 of [1] andExample 4.1 as follows.

The outline of the paper is as follows. To explain the ideas behind the definition of an appropriateadjoint operator for the iterated system and the a posteriori error analysis, we begin by derivingan a posteriori error estimate for the iterated solution of a finite dimensional algebraic systems inSection 2. This derivation contains the main ideas without the complications of differential equationdiscretization errors and moreover makes it easy to construct several illuminating examples. Wethen turn to the analysis of coupled semilinear elliptic problems in Section 3. We present results forthree particular classes of operator decomposition iteration techniques, namely, block Jacobi, blockGauss–Seidel, and relaxed block Jacobi. In Section 4, we give numerical examples of differentaspects of the error estimation framework, concluding with an adaptive algorithm that adaptivelyrefines both the computational mesh and the operator decomposition iteration to converge to anaccurate solution.

2. PRELIMINARY EXAMPLE: ITERATIVE SOLUTION OF ALGEBRAIC SYSTEMS

To explain the construction of an appropriate adjoint operator for an iterated solution approach andthe idea that the numerical error of the current iterate is affected by errors introduced at all previousiterations, we first present the analysis in the context of finite dimensional algebraic systems. Webegin with a linear problem and then treat a nonlinear problem.



2.1. Estimating the numerical error for linear algebraic systems

We consider the solution of

Aw D b.

We construct an iterative solution method using a matrix decomposition of A of the form

AD DCC,

where we assume that D is invertible. The solution procedure uses the observation that

.DCC/w D b) Dw D b �Cw

and solves

w¹0º D 0,Dw¹iº D b �Cw¹i�1º, i D 1, 2, : : : , k

³. (2.1)

We assume that the iterative scheme converges, which depends on the spectral radius of D�1C.At each stage of the iterative process, we compute a numerical solution W ¹iº � w¹iº. Our goal

is to estimate the error in a quantity of interest that is representable as a linear functional of thesolution, that is, a quantity of interest of the form .w, /, at any iterate i . We write this error as

.w �W ¹iº, /D .w �w¹iº, /„ ƒ‚ …analytic iteration error

C .w¹iº �W ¹iº, /„ ƒ‚ …numerical error

D E¹iºC e¹iº.

Here, a superscript in braces ¹iº indicates variables corresponding to forward iteration i . Let k bethe total number of forward iterations performed. Later, we use a superscript in square brackets Œj �

to denote the adjoint problem corresponding to forward iteration i D k � j C 1.

2.1.1. Estimation of the numerical error e¹iº. Because the previous iterate enters as data, the numer-ical error e¹iº depends not only upon the numerical error made at the i th iteration, but also onthe numerical errors made at previous iterations. Hence, we need to estimate the effects of these‘inherited errors’.

Given W ¹0º D 0, we compute the sequence of approximate solutions W ¹iº of Equation (2.1),i D 1, : : : , k, as

DW ¹iº D b �CW ¹i�1º. (2.2)

Theorem 2.1The error in the quantity of interest can be estimated as,

�e¹kº,

�D

kXjD1

R�W ¹k�jC1º,�Œj �IW ¹k�j º

�, (2.3)

where the adjoint problems are defined,

D>�Œ1� D , (2.4)

D>�Œj � D�C>�Œj�1�, j D 2, : : : , k, (2.5)

and

R.W ¹kº,�Œj �IW ¹mº/D .b �CW ¹mº �DW ¹kº,�Œj �/. (2.6)



Note that the adjoint problems (2.4)–(2.5) involve the simpler matrix associated with the iterativescheme, not the full adjoint of the original matrix A. To emphasize the role of the error in the mostrecent (kth) iteration, we may write

.e¹kº, /D .W ¹kº,�Œ1�IW ¹k�1º/CkXjD2

R.W ¹k�jC1º,�Œj �IW ¹k�j º/. (2.7)

Also, note that the sequence of coupled adjoint problems (2.5) yields the natural adjoint to the solu-tion operator associated with the iterative method. We emphasize this point by numbering the adjointproblems in the reverse order to the forward iteration number.

ProofThe estimates of the numerical error after k iterations is

.e¹kº, /D .w¹kº �W ¹kº, D>�Œ1�/D .D.w¹kº �W ¹kº/,�Œ1�/

D .b �Cw¹k�1º �DW ¹kº,�Œ1�/

D .b �CW ¹k�1º �DW ¹kº,�Œ1�/� .C.w¹k�1º �W ¹k�1º/,�Œ1�/

DR.W ¹kº,�Œ1�IW ¹k�1º/� .C.w¹k�1º �W ¹k�1º/,�Œ1�/. (2.8)

The first term on the right of Equation (2.8) is the usual error estimate depending on the com-putable residual of W ¹kº and the corresponding adjoint solution. The second term on the right ofEquation (2.8) estimates the contribution to the error ofW ¹kº resulting from the error inherited fromthe approximation W ¹k�1º to w¹k�1º. This inherited error term can be expressed as the error in anew quantity of interest at the previous iteration by noting that

�.C.w¹k�1º �W ¹k�1º/,�Œ1�/D .w¹k�1º �W ¹k�1º,�C>�Œ1�/D .e¹k�1º,�C>�Œ1�/.

To estimate the error in this new quantity of interest, we solve the new adjoint problem

D>�Œ2� D�C>�Œ1�,

to obtain

.e¹k�1º,�C>�Œ1�/D .w¹k�1º �W ¹k�1º,�C>�Œ1�/D .w¹k�1º �W ¹k�1º, D>�Œ2�/

D .b �Cw¹k�2º �DW ¹k�1º,�Œ2�/

D .b �CW ¹k�2º �DW ¹k�1º,�Œ2�/� .C.w¹k�2º �W ¹k�2º/,�Œ2�/

DR.W ¹k�1º,�Œ2�IW ¹k�2º/� .C.w¹k�2º �W ¹k�2º/,�Œ2�/.

Continuing in this fashion, we see that the desired estimate of the error in the quantity of interest.e¹kº, / after k iterations requires the solution of Equation (2.4) and the recursive solution of anordered sequence of .k � 1/ adjoint problems

D>�Œj � D�C>�Œj�1�, j D 2, : : : , k,

and

.e¹kº, /DkXjD1

R.W ¹k�jC1º,�Œj �IW ¹k�j º/.

�



2.1.2. The decay of influence of contributions from early iterations. To estimate the numerical errorin the quantity of interest after k iterations, we nominally need to solve k adjoint problems. (Oneof the form (2.4) and k � 1 of the form (2.5)). Each of these is solvable, but the number can besignificant. The sequence of adjoint problems has the form

�ŒjC1� D�.D�>C>/j�Œ1�, j D 1, : : : , k � 1.

By noting that

D�1.CD�1/DD D�1C,

we see that D�1C, CD�1, and D�>C> all have the same spectral radius. For the forward iterationto converge the matrix product, D�1C must have a spectral radius smaller than one. This means thatwe expect that

�Œm�lC1� D�.D�>C>/m�l�Œ1�! 0

for fixed l as m!1. This suggests that we might obtain a reasonable approximation

.e¹kº, /�lX

jD1

R.W ¹k�jC1º,�Œj �IW ¹k�j º/ (2.9)

for some small l > 1. The more rapid the convergence of the forward iteration (the smaller thespectral radius of D�1C), the smaller the value of l required.

2.1.3. Estimating the iterative error E¹kº. We begin with a classic estimate for the iteration erroron the basis of extrapolation. We define J .�/ D .�, / and denote E¹kº D w � w¹kº. Assuming alinearly convergent sequence J .E¹iº/, the classic asymptotic argument yields,

J .E¹kº/�� .J .w¹kº/�J .w¹k�1º//2J .w¹kº/�J .w¹k�1º/� .J .w¹k�1º/�J .w¹k�2º//

.

This estimate is not directly computable, however, the right-hand side can be further estimated as,

J .E¹kº/�

�.J .e¹kº/�J .e¹k�1º/CJ .W ¹kº/�J .W ¹k�1º//2

J .e¹kº/� 2J .e¹k�1º/CJ .e¹k�2º/CJ .W ¹kº/� 2J .W ¹k�1º/CJ .W ¹k�2º/. (2.10)

The expression (2.10) can be estimated using the a posteriori error estimate, but this is expensive.We may derive another estimate by assuming the numerical error is higher-order than the iterationerror, to obtain

J .E¹kº/� �.J .W ¹kº/�J .W ¹k�1º//2J .W ¹kº/� 2J .W ¹k�1º/CJ .W ¹k�2º/

. (2.11)

The approximation (2.11) may, however, lead to inaccurate estimates when the iteration andnumerical errors are of comparable size.

2.1.4. Numerical example. We construct a diagonally dominant, symmetric 20 � 20 matrix A D20I C R, where R is a random 20 � 20 matrix with Frobenius norm less than or equal to one.We decompose A into two matrices D and C, A D D C C, where D contains only the diagonalentries of A. We then solve Aw D b via operator decomposition iteration with the iteration given byEquation (2.1) where the solution at each iteration is obtained using conjugate gradient solver witha tolerance of 10�4. The iteration terminates when kW ¹iº �W ¹i�1ºk < 10�8 that is accomplishedin 11 iterations. We then solve (to within round-off) the sequence of 11 adjoint problems definedrecursively by Equations (2.4) and (2.5) for quantity of interest u20, that is, with D�Œ1� D D e20(where e20 is the unit vector with a single nonzero entry in the 20th row) and compute each of the



Table I. Error components for W ¹11º for example 2.1.4.

Total error Iteration error Numerical error Primary numerical errorjw¹20º �W ¹20ºj jw �w¹11ºj jw¹11º �W ¹11ºj jR.W ¹11º,�Œ1�IW ¹10º/j

7.43823� 10�5 2.2234� 10�11 7.4382� 10�5 9.465� 10�5

terms in the error representation formula (2.3) for k D 1, : : : , 11. We call the first term in this sumthe primary numerical error (Table I).

The expected decay of the contributions to the error e¹11º20 that are inherited from previousiterations is illustrated in Figure 1.

2.2. Estimating the numerical error for nonlinear algebraic systems

Nonlinearity introduces additional complexities for defining an adjoint problem. We solve thenonlinear equation

Aw D f .w/ (2.12)

by successive approximation

Aw¹iº D f .w¹i�1º/, i D 1, : : : , k (2.13)

with W ¹0º D 0. Given W ¹0º D 0, we compute the sequence of numerical approximations W ¹iº fori D 1, : : : , k as

AW ¹iº D f .W ¹i�1º/. (2.14)

Theorem 2.2The error in a quantity of interest is estimated as,

.e¹kº, /DkXjD1

R.W ¹k�jC1º,�Œj �IW ¹k�j º/, (2.15)

where the adjoint problems are defined,

A>�Œ1� D ,A>�Œj � D Lf .w¹k�j º,W ¹k�j º/>�Œj�1�, j D 2, : : : , k

³, (2.16)

0 2 4 6 8 10 12−13

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

log(

hist

. err

)

History Term (j)

Figure 1. Individual ‘history’ terms R.W ¹11�jC1º,�Œj �IW ¹11�jº/ for example 2.1.4 illustrating theexpected decay in contributions to e¹11º

20with index j .



using the linearization Lf .v,V / defined,

Lf .v,V /DZ 1

0

J.svC .1� s/V / ds,

where J.�/ is the Jacobian of f .

ProofThe steps are similar to the previous arguments,

.e¹kº, /D .e¹kº, A>�Œ1�/

D .Aw¹kº �AW ¹kº,�Œ1�/

D .f .w¹k�1º/�AW ¹kº,�Œ1�/

D .f .W ¹k�1º/�AW ¹kº,�Œ1�/C .f .w¹k�1º/� f .W ¹k�1º/,�Œ1�/

DR.W ¹kº,�Œ1�IW ¹k�1º/C .Lf .w¹k�1º,W ¹k�1º/e¹k�1º,�Œ1�/DR.W ¹kº,�Œ1�IW ¹k�1º/C .e¹k�1º,Lf .w¹k�1º,W ¹k�1º/>�Œ1�/DR.W ¹kº,�Œ1�IW ¹k�1º/C .e¹k�1º, A>�Œ2�/...

D

kXjD1

R.W ¹k�jC1º,�Œj �IW ¹k�j º/.

�

2.2.1. Linearization and adjoints for nonlinear adjoints. In general, there are multiple ways todefine an adjoint to a nonlinear operator [10]. Choosing a suitable definition is highly dependent onthe purpose for which the adjoint is intended. For a perturbation or error analysis, one systematicway to define an adjoint is based on linearization. Consider an approximate solution W � w ofEquation (2.12) computed without iteration, and define the residual

R.W /D AW � f .W /.

The standard analysis begins

A.w �W /� .f .w/� f .W //D�R.W /.

The integral mean value theorem yields

f .w/� f .W /D Lf .w �W /D

Z 1

0

J.swC .1� s/W / ds .w �W /.

Introducing the formal adjoint problem

.A�Lf /�� D

leads to the a posteriori error estimate

.w �W , /D .�R.W /,�/.

In practice, the linearization Lf cannot be computed and is approximated as,

Lf .v,V /DZ 1

0

J.svC .1� s/V / ds �Z 1

0

J.sV C .1� s/V / ds D J.V /. (2.17)

The approximation in the linearization may certainly affect the accuracy of the a posteriori errorestimate. However, the effect can be bounded a priori under regularity assumptions on the prob-lem, that is, the second derivatives of f are uniformly bounded in a compact region containing



the true solution, analytic iterative solutions, and iterative numerical solutions, and assuming thatthe iteration converges and the approximations are sufficiently close to the true solution. In partic-ular, W should be sufficiently close to w and R.W / should be sufficiently small. In practice, theapproximation (2.17) works well in many situations in the sense that the linearization error has rel-atively insignificant effect on accuracy of the estimate when the numerical solutions are reasonablyaccurate. Perhaps, more importantly, catastrophic failure of the a posteriori error estimate, that is anestimate that is low when the actual error is large, is relatively difficult to manage. See [8] for a dis-cussion of this point for systems of reaction-diffusion equations and an example where catastrophicfailure is created.

However, this simple approach to defining an adjoint operator when operator decomposition iter-ation is used can fail. The difficulty is that an iterative solution W ¹iº is actually solving a problemthat is significantly different than the original problem. See [2] for a discussion of this point.

The approach to define an adjoint for Theorem 2.2 avoids this issue by constructing a globaladjoint for the iterative solution operator via a sequence of coupled adjoints for each individual com-ponent problem obtained by linearization between the iterative analytic w¹iº and iterative numericW ¹iº solutions. The effects of this ‘local’ linearization can be controlled under local regularity andconvergence assumptions as described earlier. In practice,

A>�Œj � D Lf .w¹k�j º,W ¹k�j º/>�Œj�1�

is approximated by

A> Q�Œj � D J.W ¹k�j º/> Q�Œj�1� (2.18)

for j D 2, : : : , k.The price of the indicated approach to defining an adjoint for the iterated solution operator is that

the a posteriori error estimates accounts only for the numerical error e¹iº and leaves the analyticiterative error E¹iº remaining to be estimated. We adapt the estimate discussed in Section 2.1.3.

2.2.2. Numerical examples. We now consider three new situations that may arise for nonlinearlycoupled problems. Let

Aw D f .w/,

where

AD

0B@

3 �1 0 0

�1 3 0 0

0 0 3 �10 0 �1 3

1CA and f .w/D ˛

0BB@

exp.�ˇ.w3 � 0.4/2/exp.�ˇ.w4 � 0.6/2/exp.�ˇ.w1 � 0.5/2/exp.�ˇ.w2 � 0.3/2/

1CCA ,

and let the quantity of interest be the value of w2. In all cases, a high accuracy reference solution wwas generated using Newton’s method with a tolerance of 10�12, whereas we also computed highaccuracy reference iterative solutions w¹iº to full precision. To compare with estimates employingthe adjoint problem associated with the original system, the ‘global’ adjoint corresponding to thefull (i.e. nondecomposed) problem was also constructed and solved, and error estimates based onthe adjoint to the full problem are reported in the following examples. We set,

� R.W ,�/ is the error estimate on the basis of the exact linearization of the global adjointproblem, and� R.W , Q�/ is the error estimates on the basis of the approximate linearization of the global

adjoint problem.

The approximate solutions W ¹iº were computed by rounding to the sixth decimal place. All adjointsolutions were computed with full precision. In each case, we estimated the iteration error usingEquation (2.11).



Cancelation between iteration and numerical error. Let ˛ D 1, ˇ D 10, and i D 40. The iterationconverges after 40 iterations when the norm of the residual is less than 10�4. Results are providedin Table II.

Notice that the adjoint to the coupled problem gives an excellent estimate of the error, yieldingfour significant figures even when using the approximate linearization. What is obscured is the can-celation that occurs between the iteration and numerical error. The methods developed here enablethe iteration and numerical errors to be estimated separately, and this is important when constructingadaptive algorithms. However, the iteration error estimate is polluted by the numerical error in U ¹kº.The partial sums of the history terms are plotted in Figure 2 where expected decay of history errorcontributions can be observed.

The effect of linearization error. Let ˛ D 2, ˇ D 8, i D 100. In this example, although w¹kº andW ¹kº approach have the same fixed point, they both do so in a nonmonotonic fashion. This producessignificant differences between Lf .w¹k�j º,W ¹k�j º/> and J.W ¹k�j º/> (see Equations (2.16) and(2.18), respectively) and consequently significant differences in the corresponding adjoint solutions� and Q� (Table III). Results are provided in Table IV where the practical numerical error estimate isseen to be completely inaccurate.

To obtain an estimate of how rapidly the adjoint solutions � and Q� can diverge, we see fromEquations (2.16) and (2.18) that

�Œj � � Q�Œj � D A�>�Lf .W ¹k�j º,w¹k�j º/>�Œj�1� � J.W ¹k�j º/> Q�Œj�1�

�

� A�>�Lf .W ¹k�j º,w¹k�j º/� J.W ¹k�j º/

�>�Œj�1�I

Table II. Error components for example 2.2.2. Note the cancellation of errors between the iteration andnumerical error. (R.W ¹k�jC1º,�Œj �IW ¹k�jº/�R.W ¹k�jC1º,�Œj �/.)

Error Estimated error Practical error Iteration error

jw2 �W2j R.W ,�/ R.W , Q�/ w2 �w¹kº2

1.1102276� 10�6 1.1102276� 10�6 1.10983� 10�6 �2.55547� 10�6

Numerical error Est. numerical error Pract. numerical error Est. iteration error

w¹kº2 �W

¹kº2

Pj R.W ¹k�jC1º,�Œj �/

Pj R.W ¹k�jC1º, Q�Œj �/

1.44524� 10�6 1.44524� 10�6 1.44592� 10�6 �6.6554� 10�6

5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

1.5

2

2.5

N.I

. Err

or E

stim

ate

# of History Terms

x 10−6

Figure 2. Numerical error estimatePmjD1R.U ¹k�jC1º,�Œj �IU ¹k�jº/ including m ‘history’ terms for

example 2.2.2.



Table III. Error components for example 2.2.2, with R.W ¹k�jC1º,�Œj �/�R.W ¹k�jC1º,�Œj �IW ¹k�jº/.


w2 �W2 R.W ,�/ R�W , Q�

�w2 �w

¹kº2

�0.048166 0.048166 7.06634 �0.050168


w¹kº2 �W

¹kº2

Pj R.W ¹j�kº,�Œj �/

Pj R.W ¹j�kº, Q�Œj �/

0.002002 0.002002 3.58� 106 �0.04568

Table IV. Error components for example 2.2.2. (R.W ¹k�jC1º,�Œj �/�R.W ¹k�jC1º,�Œj �IW ¹k�jº/.


jw2 �W2j R.W ,�/ R�W , Q�

�w2 �w

¹kº2

0.0999965 0.0999965 0.076506 0.0999963


w¹kº2 �W

¹kº2

Pj R.W ¹j�kº,�Œj �/

Pj R

�W ¹j�kº, Q�Œj �

�1.8524� 10�6 1.8524� 10�6 1.8530� 10�6 0.18273

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

1.5

Iteration

u−Uspec(inv(A)(LF−DF))

Figure 3. Differences between w¹kº and W ¹kº and the spectral radius of A�>.Lf .w,W / � J.W //> forexample 2.2.2.

so, the spectral radius of A�>�Lf .W ¹k�j º,w¹k�j º/� J.W ¹k�j º/

�>can be considered as an ampli-

fication factor to explain the exponential accumulation of linearization error in the history errorestimate (Figure 3).

Divergent iteration. Let ˛ D 2, ˇ D 7, and k D 100. The iteration fails to converge, but all errorestimates in Table IV are well-behaved but exhibit significant linearization error. Not surprisingly,because the iteration error estimate assumes a convergent iteration, the estimate of the iteration erroris poor. However, despite the fact that the iteration is divergent, the estimate of the numerical erroris accurate, and the effectivity ratio (defined as the ratio of the error estimate to the true error) shownin Figure 4 improves as the number of history terms is increased, although the practical numericalerror estimate is affected by errors by to linearization.

2.2.3. Discussion. An interesting case is the situation in which the numerical error is much higherthan the stopping criterion (say round U to the third decimal place), but the corresponding theoreti-cal iteration converges quickly. The computation does not converge because of the numerical error,



0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

Eff

ectiv

ity I

ndex

# of History Terms

Figure 4. Effectivity ratio for the numerical error estimatePmjD1R.W ¹k�jC1º,�Œj �IW ¹k�jº/ including

m ‘history’ terms for example 2.2.2.

but the quick computation of a few history terms leads to an exact estimate of the numerical error.See Section 4.2.4 for an adaptive solution to a similar problem.

In all three examples, the global adjoints are not diagonally dominant or symmetric positive dom-inant. But even when the operator decomposition iteration solution for the global adjoint (e.g. a‘Jacobi’ iteration) does not converge, the a posteriori error analysis still provides meaningful errorestimates. See Section 4.2.2 for more discussion.

3. ANALYSIS FOR SEMILINEAR ELLIPTIC SYSTEMS

For ease of presentation, we focus the analysis on a two component fully coupled elliptic system ofthe form

�r � a1.x/ru1C b1.x/ � ru1C c1.x/u1 D f1.x,u1,u2,Du2/, x 2�,

�r � a2.x/ru2C b2.x/ � ru2C c2.x/u2 D f2.x,u1,Du1,u2/, x 2�, (3.1)

u1 D u2 D 0, x 2 @�,

where � is a convex polygonal domain in Ri , i D 1, 2, 3, with boundary @�, and we assume thatai , bi , ci ,fi , i D 1, 2 are sufficiently smooth to establish optimal order a priori convergence for thefinite element method computed without operator decomposition iteration.

The weak formulation of Equation (3.1) is the following: find um 2 QW 12 .�/ satisfying

A1.u1, v1/D .f1.x,u1,u2,Du2/, v1/, 8v1 2 QW12 .�/,

A2.u2, v2/D .f2.x,u1,Du1,u2/, v2/, 8v2 2 QW12 .�/,

(3.2)

where

A1.u1, v1/� .a1ru1,rv1/C .b1 � ru1, v1/C .c1u1, v1/,

A2.u2, v2/� .a2ru2,rv2/C .b2 � ru2, v2/C .c2u2, v2/,

are assumed to be coercive bilinear forms on� and QW np .�/ represents the subspace ofW n

p .�/ withzero trace on @�.

If we were numerically solving Equation (3.2) without operator decomposition iteration, thenwe would introduce conforming discretizations Sh,m.�/ and solve the discretized system: findUm 2 Sh,m.�/ satisfying

A1.U1,�1/D .f1.x,U1,U2,DU2/,�1/, 8�1 2 Sh,m.�/,

A2.U2,�2/D .f2.x,U1,DU1,U2/,�2/, 8�2 2 Sh,m.�/.(3.3)



3.1. Analysis of operator decomposition iteration solutions

We analyze three different operator decomposition iterations for producing numerical approxima-tions of Equation (3.1). We recall that we decompose the error as in Equation (1.2), that is,

u�U ¹kº D u� u¹kº„ ƒ‚ …analytic iteration error

Cu¹kº �U ¹kº„ ƒ‚ …numerical error

D E¹kºC e¹kº,

where u¹kº respectively U ¹kº are the analytic and discrete solutions obtained by the particular iter-ation approach under consideration. The a posteriori error analysis is for the numerical error e¹kº.We estimate the analytic iteration error E¹kº by a natural application of the estimates discussed inSection 2.1.3.

For simplicity of presentation, we assume the quantity of interest is given as a linear functional of

the second solution component, determined by D .0, 2/>, that is, we estimate�e¹kº2 , 2

�. For

notational convenience and to be consistent with [1], we abbreviate the weak residual of a solutioncomponent,

Rm.�,�I �/D .fm.�/,�/�Am.�,�/, mD 1, 2.

3.1.1. Block Jacobi iteration. We first consider a block Jacobi operator decomposition iterativeapproach to the solution of the semilinear elliptic system (3.1).

Theorem 3.1The representation formula for the numerical error e¹kº is

.e¹kº, /DX

j6k, j odd

R2

�U¹k�jC1º2 ,�Œj �2 IU

¹k�j º1

�C

Xj6k, j even

R1


¹k�j º2

�,

(3.4)where the corresponding adjoint problems are

A�2��,�Œj �2

�D��, Œj �2

�, 8� 2 QW 1

2 .�/, j 6 k, j odd,

A�1��,�Œj �1

�D��, Œj �1

�, 8� 2 QW 1

2 .�/, j 6 k, j even,(3.5)

with

A�1.v,w/D .rv, a1rw/� .v, div.b1w//C .v, c1w/� .v, .J1.U //1w/,

A�2.v,w/D .rv, a2rw/� .v, div.b2w//C .v, c2w/� .v, .J2.U //2w/,(3.6)

and the additional adjoint data is defined recursively as

Œ1�2 D 1,�

�, Œj �2

�D�.J1.U

¹k�jC1º//2�,�Œj�1�1

�, j 6 k, j odd, (3.7)�

�, Œj �1

�D�.J2.U

¹k�jC1º//1�,�Œj�1�2

�, j 6 k, j even,



where Jm.V / is the Jacobian of fm evaluated at V .

If we wish to highlight the final (kth) iteration, then we can write

.e¹kº, /DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�C

X26j6k, j even

R1


¹k�j º2

�

CX

36j6k, j odd

R2


¹k�j º1

�. (3.8)

ProofIn the following, we simplify notation in the functions f ; so, for example, we writef1.x,u1,u2,Du2/� f1.u2/, and so on.

.e¹kº, /D�e¹kº2 , Œ1�2

�

DA�2�e¹kº2 ,�Œ1�2

�

DA2�e¹kº2 ,�Œ1�2

�

DA2�u¹kº2 ,�Œ1�2

��A2

�U¹kº2 ,�Œ1�2

�

D�f2

�u¹k�1º1

�,�Œ1�2

��A2

�U¹kº2 ,�Œ1�2

�

D�f2

�U¹k�1º1

�,�Œ1�2

��A2

�U¹kº2 ,�Œ1�2

�C�f2

�u¹k�1º1

�,�Œ1�2

��f2

�U¹k�1º1

�,�Œ1�2

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�C�Lf2.u

¹k�1º1 ,U ¹k�1º1 /e

¹k�1º1 ,�Œ1�2

�

�R2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�C��J2

�U¹k�1º1

��1e¹k�1º1 ,�Œ1�2

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�C�e¹k�1º1 , Œ2�1

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�CA�1

�e¹k�1º1 ,�Œ2�1

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�CA1

�e¹k�1º1 ,�Œ2�1

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�CR1

�U¹k�1º1 ,�Œ2�1 IU

¹k�2º2

�

C�Lf1.u

¹k�2º2 ,U ¹k�2º2 /e

¹k�2º2 ,�Œ2�1

�

�R2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�CR2

�U¹k�1º1 ,�Œ2�1 IU

¹k�2º2

�C��J1

�U¹k�2º2

��2e¹k�2º2 ,�Œ2�1

�

DR2

�U¹kº2 ,�Œ1�2 IU

¹k�1º1

�CR1

�U¹k�1º1 ,�Œ2�1 IU

¹k�2º2

�CA�2

�e¹k�2º2 , Œ3�2

�...

DX

j6k, j odd

R2


¹k�j º1

�C

Xj6k, j even

R1


¹k�j º2

�.

(3.9)

�



The estimate in this case takes into account the numerical error arising from each component solu-tion and the inherited errors passed between iterations. Following the discussion in Section 2.2.1,we are using the adjoint naturally associated with the analytic iterative solution operator. We use

‘local’ linearization Lfi�u¹kºj ,U ¹kºj

�between u¹kºj and U ¹kºj to define the required adjoints for

each component equation in the iteration. The effects of this linearization can be controlled assum-ing sufficient regularity of the solution and a priori convergence of the discretization, and we expectthe estimates to be accurate for all sufficiently accurate numerical solutions. The global adjoint ofthe iterated solution operator is therefore obtained by a sequence of ‘single physics solves’. We notethat we still have to estimate the analytic iteration error E¹kº.

3.1.2. Block Gauss–Seidel iteration. Next, we consider a block Gauss–Seidel operator decomposi-tion iteration.

Theorem 3.2The representation formula for the numerical error e¹kº is

.e¹kº, /DkXjD1

R2


¹k�jC1º1

�„ ƒ‚ …

within iteration errors

C

kXjD1

R1


¹k�j º2

�„ ƒ‚ …

between iteration errors

, (3.10)

where the corresponding adjoint problems are

A�2��,�Œj �2

�D��, Œj �2

�, 8� 2 QW 1

2 .�/, j D 1, : : : , k,

A�1��,�Œj �1

�D��, Œj �1

�, 8� 2 QW 1

2 .�/, j D 1, : : : , k,(3.11)

and the adjoint data is defined recursively as

Œ1�2 D ,

��, Œj �2

�D�.J1.U

¹k�jC1º//2�,�Œj�1�1

�, j D 2, : : : , k, (3.12)

��, Œj �1

�D�.J2.U

¹k�jC1º//1�,�Œj �2

�, j D 1, : : : , k.



Proof

.e¹kº, /D�e¹kº2 , Œ1�2

�D�f2.u

¹kº1 /,�Œ1�2

��A2

�U¹kº2 ,�Œ1�2

�DR2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�C�Lf2.u

¹kº1 ,U ¹kº1 /e

¹kº1 ,�Œ1�2

��R2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�C�.J2.U

¹kº1 //1e

¹kº1 ,�Œ1�2

�DR2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�C�e¹kº1 , Œ1�1

�DR2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�C�f1.U

¹k�1º2 /,�Œ1�1

��A1

�U¹kº1 ,�Œ1�1

�C�f1.u

¹k�1º2 /,�Œ1�1

��f1.U

¹k�1º2 /,�Œ1�1

�DR2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�CR1

�U¹kº1 ,�Œ1�1 IU

¹k�1º2

�C�Lf1.u

¹k�1º2 ,U ¹k�1º2 /e

¹k�1º2 ,�Œ1�1

��R2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�CR2

�U¹kº1 ,�Œ1�1 IU

¹k�1º2

�C�.J1.U

¹k�1º2 //2e

¹k�1º2 ,�Œ1�1

�DR2

�U¹kº2 ,�Œ1�2 IU

¹kº1

�CR1

�U¹kº1 ,�Œ1�1 IU

¹k�1º2

�CA�2

�e¹k�1º2 , Œ2�2

�...

D

kXjD1

R2


¹k�jC1º1

�C

kXjD1

R1


¹k�j º2

�. (3.13)

�

In this case, in contrast with Equation (3.4), there are contributions to the error reflecting both‘within’ and ‘between iteration’ errors.

3.1.3. Relaxed block Jacobi iteration. Both the Jacobi and Gauss–Seidel operator decompositioniterations can be ‘relaxed’ by letting U ¹kº D ˛ QU ¹kº C .1 � ˛/U ¹k�1º where � denotes a quantitycomputed before relaxation. The approximation is obtained via

This is often performed in practice to aid convergence of the iteration but poses more challengesfor a posteriori error analysis, and we present a partial analysis to explain how the results can be



extended to handle relaxation. Letting QU ¹kº, Qu¹kº, and Qe¹kº represent the corresponding quantitiescomputed without relaxation, we have�

u�U ¹kº, �D�u� u¹kº,

�C ˛

�Qe¹kº,

�C .1� ˛/

�e¹k�1º,

�. (3.14)

Using the notation earlier, we have

.e¹kº, /D�e¹kº2 , Œ1�2

�D ˛

�Qe¹kº2 , Œ1�2

�C .1� ˛/

�e¹k�1º2 , Œ1�2

�D ˛

hR2

�QU¹kº2 ,�Œ1�2 IU

¹k�1º1

�C�f2

�u¹k�1º1

�,�Œ1�2

��f2

�U¹k�1º1

�,�Œ1�2

�iC .1� ˛/

�e¹k�1º2 , Œ1�2

�� ˛

hR2


¹k�1º1

�C��J2

�U¹k�1º1

��1e¹k�1º1 ,�Œ1�2

�i

C .1� ˛/�e¹k�1º2 , Œ1�2

�D ˛

hR2


¹k�1º1

�C�e¹k�1º1 , Œ2�1

�iC .1� ˛/

�e¹k�1º2 , Œ1�2

�D ˛

hR2


¹k�1º1

�C�e¹k�1º1 , Œ2�1

�iC ˛.1� ˛/

hR2

�QU¹k�1º2 ,�Œ1�2 IU

¹k�2º1

�C�e¹k�2º1 , Œ2�1

�i.1� ˛/2

�e¹k�2º2 , Œ1�2

�...

D ˛

kXiD1

.1� ˛/i�1hR2

�QU¹k�iC1º2 ,�Œ1�2 IU

¹k�iº1

�C�e¹k�iº1 , Œ2�1

�i.

To obtain a full a posteriori error estimate, we have to repeat this argument to estimate�e¹k�iº1 , Œ2�1

�, i D 1, : : : , k. We refrain from doing this. Clearly, repeated application of this

analysis approach results in a great number of adjoint problems to be solved. It is to be hopedthat introducing relaxation means that relatively few number history terms need to be included foraccuracy in the estimate. As expected, the decay rate of the history terms decreases as ˛! 0.

4. NUMERICAL EXAMPLES FOR FULLY COUPLED SEMILINEAR ELLIPTIC SYSTEMS

We present a number of examples to illustrate characteristics of the a posteriori error estimate. Wealso illustrate the use of the estimates as the basis to adaptively adjust discretization parameters onthe basis of relative sizes of contributions to the error estimate.

Some of the examples are used to explore the accuracy of the a posteriori error estimate insituations in which the operator decomposition iteration is converging well. To measure the accuracyof the error estimate, we report on the ratio of the estimate to the true error (or an approximationcomputed with a highly accurate reference solution). This type of a posteriori error estimate tends tobe robustly accurate for a wide range of spatial meshes, as has been well reported in the literature.We observe the same robust accuracy in the estimates when the iteration is converging well. Ratherthan reporting extensively on that quality, we concentrate the experiments on situations in which theestimates do not work as well as hoped.

For adaptive error control, we employ a natural generalization of the ‘mark and refine’ strategy onthe basis of the Principle of Equidistribution [1,5–8]. In this approach, the error estimate is replacedby a bound obtained by replacing the signed contributions to the error estimate with their abso-lute values. Starting with a coarse discretization—mesh with large element size and small number



of iterations—we adjust the various discretization parameters according to the relative size of thecontributions and an estimate of the computational costs associated with changes in the parameters.This adaptive approach is described fully in the earlier paper [1].

We make one simplification as follows. In [1], we allowed the different components to have dif-ferent meshes, which requires modification of the a posteriori error analysis to account for the effectof mesh changes on the numerical solution. As will be discussed, we use the same mesh for all com-ponents of the system to avoid introduction of the additional terms in the a posteriori error estimate.The results from the first paper can be combined with the results in this paper in a straightforwardway.

4.1. Fully coupled linear system

The first example is a coupled system of linear equations for which we can compute an exactsolution. The coupled problem is the following: Find u1.x,y/,u2.x,y/ satisfying,

�r � .˛ru1/C b1 � ru1 D f1.u1,u2/, x 2�,

�u2 D b2 � ru1, x 2�,

u1 D u2 D 0, x 2 @�,

(4.1)

where � WD ¹.x,y/ W .x,y/ 2 Œ0, 1� Œ0, 1º,

˛ D 10C �2�5

13cos 4�xC

1

2cos�y

�

b1 �1

�

�8

13sin 4�x

1

2sin�y

>, b2 D

1

�

�1

2sin 4�x 2 sin�y

>

f1.u1,u2/D 85�2.2 sin 4�x sin�y C u2/.

This system has exact solution

u1 D sin 4�x sin�y

u2 D ��2.sin 8�x sin�y=65C sin 4�x sin 2�y=20/.

We select the quantity of interest u2.x0/, where x0 D .0.15, 0.15/.As mentioned, we solve for u1 and u2 on a common mesh. The Gauss–Seidel iteration proceeds

until kU ¹kº � U ¹k�1ºk1 6 10�5. Quadratic finite elements are used to compute ˆŒj �1 and ˆŒj �2approximations to �Œj �1 and �Œj �2 . Starting with a uniform initial mesh, the subsequent meshes areadapted identically using the sum of just the first four terms in the error representation formula,namely, the sum of the following:

1. the ‘primary’ error, "1 DR2

�U¹kº2 ,ˆŒ1�2 IU

¹kº1

�,

2. the ‘transfer’ error, "2 DR1

�U¹kº1 ,ˆŒ1�1 IU

¹k�1º2

�,

3. the ‘inherited’ error, "3 DR2

�U¹k�1º2 ,ˆŒ2�2 IU

¹k�1º1

�, and

4. the ‘inherited transfer’ error, "4 DR1

�U¹k�1º1 ,ˆŒ2�1 IU

¹k�2º2

�.

Using an approximate delta function ıregx0 , the corresponding adjoint problems used to compute

the weighted residuals are

A�2��,�Œ1�2

�D��, ıreg

x0

�, 8� 2 Sh2 .�/,

A�1��,�Œ2�1

�D��,�div.b2�

Œ1�2 /�

, 8� 2 Sh2 .�/,

A�2��,�Œ2�2

�D��,�div.b1�

Œ2�1 /�

, 8� 2 Sh2 .�/,

A�1��,�Œ3�1

�D��,�div.b2�

Œ2�2 /�

, 8� 2 Sh2 .�/.

(4.2)



Table V. The first four error terms, that is, the primary error and transfererror at iteration k and the inherited error and inherited transfer error from

the iteration .k � 1/ for example 4.1.

Primary error ("1) Transfer error ("2)

R2�U¹kº2 ,�Œ1�2 IU

¹kº1

�R1

�U¹kº1 ,�Œ1�1 IU

¹k�1º2

�

0.5070� 10�4 0.0940� 10�4

Inherited error ("3) Inherited transfer error ("4)

R2�U¹k�1º2 ,�Œ2�2 IU

¹k�1º1

�R1

�U¹k�1º1 ,�Œ2�1 IU

¹k�2º2

��0.0010� 10�4 0.0409� 10�4

The initial mesh is a uniform partition with step size h D 0.1 for 200 elements. The iteration isrepeated until the total error representation estimate for the quantity of interest is less than 10�4.The adaptive algorithm ran for three iterations before meeting the tolerance. The final mesh has2378 elements. The number of iterations used in Gauss–Seidel on the finest (last) mesh is five. Theiteration error estimate reports 1.43� 10�6.

The effectivity ratio (estimate/error) is 0.9976.Table V shows the contributions to the error. We observe that the ‘transfer’ error "2 is about 1/5th

of the size of the ‘primary’ discretization error at iteration "1, and that the ‘inherited transfer error’"4 is 1/10th of the size of the primary discretization error. By contrast, "3, the error inherited fromU2 at the iteration .k � 1/, is 1=500th the size of "1 Figure 5 shows the first four adjoint solutions.

4.2. Effects of iteration on the accuracy of the error estimate

We next consider a fully coupled semilinear system that is solved by the Jacobi operator decom-position iteration employing continuous piecewise linear elements with initial solutions U ¹0º1 D x,

U¹0º2 D 0. The coupled system is the following: Find u1.x/,u2.x/ for x 2�D Œ0, 1 such that

�u001 D e�2u2 , x 2�,

�u002 D sin.ˇ�u1/, x 2�,

u1.0/D 0, u1.1/D 1,u2.0/D u2.1/D 0.

The quantity of interest is the average value of u2 over the whole domain. The sequence of adjointproblems are

��Œj �1

�00D

Œj �1 , x 2�,

��Œj �2

�00D

Œj �2 , x 2�,

�1.0/D �2.0/D 0, �1.1/D �2.1/D 0,

with

Œ1�2 D 1

Œj �2 D �

2e�2U¹k�jº2 �

Œj�1�1 , j odd

Œj �1 D ˇ� cos

�ˇ�U

¹k�j º1

��Œj�1�2 , j even.

Quadratic finite elements are used to computeˆŒj �1 andˆŒj �2 approximations to �Œj �1 and �Œj �2 , and theiteration is performed until the iteration error estimator (2.11) is less than 10�6 or until a maximumof 30 Gauss–Seidel iterations are performed.

We examine different sets of parameters that affect the convergence of the iteration and,consequently, accuracy of the estimate.



(a)

(c) (d)

(b)

Figure 5. The first four adjoint solutions, that is, the primary, transfer, and the first two inherited adjointsolutions for example 4.1. (a) Solution for ˆŒ1�

2; (b) solution for ˆŒ1�

1; (c) solution for ˆŒ2�

2; and (d) solution

for ˆŒ2�1

.

4.2.1. Slowly decaying history contributions. The first example demonstrates problems that canarise when the history contributions decay very slowly. We fix ˇ D 10 and compute approximationsfor � D 2, 4, 8, and 16. We use a uniform mesh with h D 0.05 for the approximate solutions, andwe use a fine mesh with h D 0.005 to compute a ‘reference’ solution. We run the iterations untilkU ¹iº �U ¹i�1ºk6 10�6 up to a maximum of 30 iterations.

The iteration converges for �D 2, 4, and 8, but for �D 16, the iteration cannot converge (withoutrelaxation) even using a finer space mesh. We report the error estimate effectivity (estimate/error)ratios in Table VI.

These results the typical accuracy of the estimate. The estimate is even reasonably accurate in thelast case where, as noted, the iteration does not converge. This estimate in the last case is affectedby significant linearization errors because U2 does not represent u2 very well.

Figure 6 presents the partial sums of the first j history terms for the different values of �. For�D 2, 4, and 8, the error contributions become relatively constant when more than five error termsare included; but in all cases, taking simply the first error term only (the ‘primary error contribution’)leads to a poor error estimate.

4.2.2. The effect of large numerical error on the history contributions. Let � D 10, ˇ D10, and hD 0.05. The fixed point iteration fails on a coarser mesh of hD 0.2; but for hD 0.05, theiteration converges slowly. In this computation, the error estimate is affected both by the inaccuraciesbecause of linearizing about U ¹iº for i relatively small as well as the poor numerical resolution ofthe adjoint solution ˆ (computed using quadratic finite elements on the mesh for U ). Both sources



Table VI. Effectivity ratios for example 4.2.1.

� ratio

2 0.9999425884 0.9977374718 0.97826093416 1.647440221

0 5 10 15 20 25 30−6.5

−6

−5.5

−5

−4.5

−4

−3.5

# of history terms

log(

sum

his

tory

term

s) λ=2λ=4λ=8λ=16

Figure 6. Numerical error estimatePmjD1R

�U ¹k�jC1º,�Œj �IU ¹k�jº

�including m history terms, for

example 4.2.1 and �D 2, 4, 8, and 16.

0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8

log(

part

ial e

rror

) su

ms

# of history terms

Figure 7. Numerical error estimatePmjD1R

�U ¹k�jC1º,�Œj �IU ¹k�jº

�including m history terms for

example 4.2.2, demonstrating that numerical error can lead to nonconvergence and poor numerical errorestimates.

of numerical error destroy the accuracy of the estimate. We see the effect in Figure 7, where weobserve that the contributions to the error estimator increase as ‘older’ history terms are included.This is the opposite of the expected behavior that error contributions should decay and is in an indi-cator that the error estimator is not performing well. Large adjoint solutions relative to the size ofthe solution can be an indicator of unreliable error estimates.

4.2.3. The effect of a poor initial iterate on the history contributions. Let �D 10, ˇ D 10, and hD0.05. The choice of initial data for the iteration can strongly influence the performance of the error



0 5 10 15

−8

−6

−4

−2

0

2

4

log(

hist

ory

erro

r co

ntri

butio

n)

History Term (j)

u2{0} = x(1−x)

u2{0} = 0

Figure 8. Individual ‘history’ terms R�U ¹k�jC1º,�Œj �IU ¹k�jº

�for example 4.2.3, illustrating the

sensitivity of history error contributions to initial data.

estimator. In Figure 8, we plot the error contributions for U ¹0º2 D 0 versus U ¹0º2 D x.1 � x/.In this case, the norm of several of the early solution iterates is greater than 1014, whereasthe norm of the converged U ¹kº is O.1/. The difficulty is that each history adjoint problem issolved using quadratic finite elements of size h D 0.05, and the resulting use of ˆŒj � insteadof �Œj � in the resulting error representation formula yields an error that may be bounded byChkU ¹k�jC1ºkW 1

2.Œ0,1�/k�

Œj � �ˆŒj �kW 12.Œ0,1�/ D Ch

3kU ¹k�jC1ºkk Œj �k (assuming smooth , �,

and u). The relative size of U ¹k�jC1º and U ¹kº means that the resulting error estimate is useless,despite the exponential decay of �Œj � as j increases. Incorporating additional history terms canreduce the quality of the estimator.

4.2.4. Adaptive selection of relaxation parameter and mesh resolution. In the three previous exam-ples, problems with accuracy of the error estimate arise from slow convergence of the iterationand subsequent slow decay of the history contributions to the error. In Section 3.1.3, we describedan extension of the a posteriori error estimate to a version of the Jacobi iteration that employsrelaxation. The relaxation parameter directly enters into the history contributions of the errorestimate.

This suggests a generalization of an adaptive strategy in which the decay of the history contri-butions is monitored, and the iteration is interrupted and restarted with a new relaxation parametervalue if the history contributions are contributing too much relative to the other error contribu-tions. The efficiency question is balancing the error contributions between those arising from thediscretization against those arising from the iteration.

Consider once again the problem in Section 4.2, with quantity of interest equal to the value of u2at x D 0.1 (adjoint data Œ1�2 D ı0.1). When ˇ D 10 and � D 20, this iteration cannot convergewithout relaxation, regardless of mesh density.

We present the adaptive algorithm in Algorithm 4. We begin the adaptive procedure using coarsecommon mesh for both U1 and U2 with h D 0.2. We perform a Jacobi iteration with no relaxationuntil the convergence criterion (kU i � U i�1k1 6 10�7) is satisfied or a maximum of eight itera-tions is reached. At this point, the (computable) iteration error estimate (2.11) is calculated. We thencompute the error representation formula (3.4) using a variable number of history terms. In general,the minimum number of history terms is set to be min.4, k/, but the number of history terms com-puted is halted if the j th history term is too small (< C1hEtot ) or too large .> C2h�1Etot /, whereC1 D 0.1 and C2 D 10. We adapt the common mesh for U1 and U2 using a standard ‘mark andrefine’ strategy applied to the a posteriori estimate using klocal errork <TOL/(# of elements) as amarking criterion. If the iteration does not ‘converge’ or if the iteration error estimate is greater than



0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

U

x 10−3

x

Figure 9. Final adapted solution for U2 for example 4.2.4 showing the spatial grid.

the numerical error estimate, then we increase the relaxation parameter ˛ and repeat the process.Various choices for updating ˛ can be employed at this step. The process continues until both theiteration and numerical error estimate are less than 10�5.

We show the final adaptive solution for U2 in Figure 9. Generally, a quantity of interest of thevalue at a point leads to a mesh that is highly refined in a local neighborhood of the point. But thefinal refined mesh is nearly uniform as we see. This is a consequence of the errors inherited fromprevious iterations coupled to the fact that the solutions have significantly different scales, that is,U1 is O.1/, whereas U2 is O.10�3/). Thus, the localized contributions that should result in therefinement of the mesh for U2 near x D 0.1 are masked by the much larger—and nearly uniformlydistributed—contributions from U1 in previous iterations.



5. CONCLUSIONS

We have presented an a posteriori error analysis for operator decomposition iterative solution of sys-tems of coupled semilinear elliptic systems that use a block iterative solution technique. The analysisprovides the means to compute accurate error estimates that account for discretization errors in thesolution of each component at a given iteration, the errors passed between components at a giveniteration, numerical errors inherited from previous iterations, and errors arising due to the iterativesolution procedure. This paper specifically addresses the propagation of error between iterates inthe operator decomposition iteration solution and the effects of finite iteration on the error estimate.We extend the adaptive discretization strategy in [1] to systematically reduce the error.

ACKNOWLEDGEMENTS

This work was supported by the Defense Threat Reduction Agency (HDTRA1-09-1-0036), Department ofEnergy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909, DE-SC0001724), LawrenceLivermore National Laboratory (B573139, B584647), National Aeronautics and Space Administration(NNG04GH63G), National Science Foundation (DMS-1065046, DMS-0107832, DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559), Idaho National Laboratory (00069249, 00115474),National Institutes of Health (R01GM096192), and Sandia Corporation (PO299784).

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A posteriori analysis and adaptive error control for ...tavener/manuscripts/carey/coupled-semi... · In this paper, we develop an a ... model for the Belousov–Zhabotinsky reaction

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