Functional A Posteriori Error Control for Conforming Mixed Approximations of the Reaction-Convection-Diffusion Problem

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by  Immanuel Anjam

FUNCTIONAL A POSTERIORI ERROR CONTROL FOR CONFORMING MIXED
APPROXIMATIONS OF THE REACTION-CONVECTION-DIFFUSION PROBLEM
IMMANUELANJAM
Abstract. In this paper we show how to obtain the exact value of the global error of a conforming mixed
approximationofthe reaction-convection-diffusion problem. Weoperateinthe frameworkoffunctional type a
posteriorierrorcontrol. Theerrorismeasuredinacombinednormwhichtakesintoaccountboththeprimaland
dual variables. Our main results state that the exact global error value of a conformingmixed approximation
isgivenbyafunctionalwhichincludesonlyknownquantities. Thepresentederrorequalitiesholdwithcertain
6
restrictionsonthereactioncoefficientandtheconvectionvector,namely,undertheconditionswhenthesolution
1
operators of the corresponding problems are isometries. For the case where these restrictions are not satisfied
0
wederiveatwo-sidederrorestimate.
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u
A
Contents

2
1. Introduction 2
2
2. Notation and the Model Problems 2

] 3. Error Equalities 5
A
3.1. Error Equality for the Problem (2.2) 5
N
3.2. Error Equality for the Problem (2.10) 6
. 4. Two-Sided Error Estimates for the Case c=divb=0 7
h
t 5. Further Remarks 9
a
References 10
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Appendix A. Proof of Identity (2.5) 11
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Appendix B. Norm Equivalences 11

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Date:August23,2016.
1991 MathematicsSubject Classification. 65N15.
Key words and phrases. functional aposteriorierrorestimates,errorequalities,mixedformulations,combinednorm,reaction-
convection-diffusion.
Theauthor wouldliketo thank DirkPaulyandSergey Repinfordiscussions andsuggestions relatedto the resultsexposed in
thispaper. ThisresearchwassupportedbytheEmilAaltonenFoundation.
1
2 IMMANUELANJAM
1. Introduction
The results presented in this paper are based on the framework of functional type a posteriorierror control.
These type estimates and equalities are valid for any conforming approximation and contain only global con-
stants. For a detailed expositionof the theory see the books [10] by Repin and [7] by Repin and Neittaanm¨aki.
For a more computational point of view see [6] by Mali, Repin, and Neittaanm¨aki.
Inthispaperwewillconsideronlyconformingapproximations,andwewillmeasuretheerrorofourapproxi-
mationsinacombinednorm,whichincludestheerrorofboththeprimalandthedualvariable. Thisisespecially
useful for mixed methods where one calculates an approximation for both the primal and dual variables, see
e.g. the book [2] by Brezzi and Fortin. We call this approximation pair a mixed approximation. We note here
that we consider more regular mixed approximations than in [2]. However, for less regular approximations the
results of this paper canstill be used by the help ofpost-processingtechniques (this is briefly commented upon
in Section 5).
Functional a posteriori error estimates for mixed approximations in combined norms were first exposed in
the paper [11], where the authors studied real valued problems of the type
∗
A Au=f,
∗
where A is a differential operator, and A its adjoint. Here the primal variable is u, and the dual variable is
p := Au. The simplest partial differential equation contained in this class of problems is the Poisson equation
∗
with A= and A = div. In [11] the authors present two-sided estimates bounding the combined error by
∇ −
the same quantity from below and from above aside from multiplicative constants.
Complex (and real) valued problems with lower order terms, i.e., of the following two types
∗ ∗
A Au+u=f and A Au+iνu=f
were studied in [1]. Here i is the imaginary unit and ν R 0 . For example, the reaction-diffusion equation
∈ \{ }
belongs to the first class of problems, and the eddy-current problem to the second class of problems. In [1] the
authors derive an error equality for the first class of problems, meaning that the exact global error value of a
mixed approximation can be calculated by having the problem data and the approximation at hand. In the
real valued case the error equality is a special case of the very general result [7, (7.2.14)], and was also found
in [3, Remark 6.12] in the case of the real valued reaction-diffusion problem. For the second type of problems
the authors derive in [1] a two-sided estimate bounding the combined error by the same quantity from below
and from above aside from multiplicative constants.
Most work on functional type a posteriori error control for the reaction-convection-diffusion problem has
concentrated on error estimation for approximationsof the primal variable. The error estimates resulting from
this research can be found in [4,5,8,9], and some of these results are also exposed in the book [10]. In [8] the
authors also derive two-sided estimates for mixed approximations in combined norms. The error measure is
different to the one used in this paper, and is commented upon in Remark 3.3.
This paper concentrates in deriving error equalities and two-sided error estimates for mixed approximations
of the reaction-convection-diffusionproblem. We work in the complex valued setting, but the results hold also
in the real case. The paper is organized as follows. In Section 2 we state our model problem in two equivalent
forms. Weshowthatthesolutionoperatorsoftheseproblemsareisometries,andcommentonthefactthaterror
control immediately follows if the primal approximation contains additional regularity. Section 3 is dedicated
to a posteriori error equalities for the reaction-convection-diffusion equations. The main results of this section
arestatedinTheorems3.1and3.4. InSection4wederiveatwo-sidederrorestimatewhichisapplicablealsoin
the case where both the reaction coefficient and convection vector divergence are zero simultaneously in some
part of the domain. The main result of this section is stated in Theorem 4.1. In Section 5 we briefly comment
on some aspects of the derived results.
2. Notation and the Model Problems
In this paper all spaces are defined over the complex field, and if not otherwise indicated, functions are
complex valued. All results hold unchanged in the real valued case as well.
Let Ω Rd, d 1, be an arbitrary (bounded or unbounded) domain. We denote by , and the
ω ω
⊂ ≥ h· ·i | · |
inner product and norm for scalar or vector valued functions in L2(ω) for ω Ω. We introduce the following
⊂
subindex notation: we define , := γ , , which induces , where γ belongs to the space of
ω,γ ω ω,γ
essentially bounded functions Lh∞·(ω·i). If γ ishse·lf-·aidjoint and uniform|ly·p|ositive definite, they become an inner
product and a norm in L2(ω), respectively. We denote by ω,∞ the norm in L∞(ω). If ω = Ω, we will not
|·|
indicate the dependence on domain in our notations, i.e., = for functions in L2 =L2(Ω).
Ω
| · | | · |
We define the usual Sobolev spaces
H1 := ϕ L2 ϕ L2 , D:= ψ L2 divψ L2 .
{ ∈ | ∇ ∈ } { ∈ | ∈ }
FUNCTIONAL A POSTERIORI ERROR EQUALITIES 3
These are Hilbert spaces equipped with the respective graph norms H1, D. The space of functions
belonging to H1 and vanishing on the boundary is defined as the closu|re·o|f sm|oo·t|h and compactly supported
test functions
H1
H1 :=C∞ .
0 0
We have
(2.1) ϕ H1 ψ D ϕ,ψ = ϕ,divψ .
∀ ∈ 0 ∀ ∈ h∇ i −h i
The reaction-convection-diffusion problem in the mixed form consists of finding a scalar potential u H1
and a diffusive flux p D, such that ∈
∈
p=A u in Ω,
∇
(2.2) divp+b u+cu=f in Ω,
− ·∇
u=0 on ∂Ω,
where the source f belongs to L2. The diffusion matrix A L∞ is self-adjoint and uniformly positive definite.
∈
In particular
(2.3) ϕ L2 αϕ2 Aϕ,ϕ β ϕ2,
∀ ∈ | | ≤h i≤ | |
where 0 < α β. The convection vector b is a real valued function from L∞ such that divb L∞. The
real scalar valu≤ed reaction coefficient c belongs to L∞. The variables u and p are often called prim∈al and dual
variables, respectively.
By multiplying (2.2) with a test function, and using (2.1) we obtain the weak formulation of this problem:
find u H1 such that
∈ 0
(2.4) w H1 A u, w + b u+cu,w = f,w .
∀ ∈ 0 h ∇ ∇ i h ·∇ i h i
For any v,w H1 we have the identity (see Appendix A)
∈ 0
(2.5) b v,w = bv, w (divb)v,w .
h ·∇ i −h ∇ i−h i
By setting v =w we see that
(2.6) w2 = ( divb)w,w =2 b w,w .
| |−divb h − i ℜh ·∇ i
By setting w = u and applying (2.6) to the real part of the form generated by the left hand side of (2.4), we
see that
1
A u, u + b u+cu,u = A u, u + c divb u,u
ℜ h ∇ ∇ i h ·∇ i h ∇ ∇ i h − 2 i
(cid:18) (cid:19)
(2.7) (cid:0) (cid:1)=h∇u,∇uiA+hu,uic−21divb
= u2 + u2 .
|∇ |A | |c−21divb
The form is then coercive provided that
1
c divb λ>0
− 2 ≥
holds,andthen(2.4)hasaunique solutionu H1 bythe Lax-MilgramTheorem. Ifthedomainisbounded(by
∈ 0
boundedwemeanboundedatleastinonedirection,i.e.,itliesbetweentwoparallelhyperplanes)thiscondition
can be weakened by using the Friedrichs inequality
(2.8) w H1 w C w,
∀ ∈ 0 | |≤ F|∇ |
where C is the Friedrichs constant. We note that generally the exact value of the Friedrichs constant is
F
unknown, but it is easy to estimate it from above. Using (2.3) and (2.8) we can write (2.7) as
1 1
A u, u + b u+cu,u = u2 + u2 + u2
ℜ h ∇ ∇ i h ·∇ i 2|∇ |A 2|∇ |A | |c−21divb
(cid:0) (cid:1) 1 α
(2.9) u2 + u2+ u2
≥ 2|∇ |A 2C2| | | |c−21divb
F
1
= u2 + u2 .
2|∇ |A | |c−21divb+2Cα2
F
The form is then coercive provided that
1 α
c divb+ λ>0
− 2 2C2 ≥
F
holds, and then (2.4) has a unique solution u H1 by the Lax-MilgramTheorem.
∈ 0
4 IMMANUELANJAM
Byapplying(2.5)to(2.4)weobtainanalternativeformulationforthereaction-convection-diffusionproblem:
w H1 A u bu, w + (c divb)u,w = f,w ,
∀ ∈ 0 h ∇ − ∇ i h − i h i
from which we can deduce the Euler equations in the mixed form:
q =A u bu in Ω,
∇ −
(2.10) divq+(c divb)u=f in Ω,
− −
u=0 on ∂Ω.
The dualvariableq is the so-calledtotalflux, the combinationofthe diffusive flux p=A u andthe convective
∇
flux bu.
Remark 2.1. With additional assumptions on the material coefficients and equipped with suitable norms the
solution operators of (2.2) and (2.10) are isometries.
(i) Assume c c >0 and c divb 0. Then the solution operator
0
≥ − ≥
L :L2 H1 D;f (u,p)
1 → 0× 7→
of (2.2) is an isometry, i.e., L =1: from (2.2) we see that
1
| |
f 2 = cu+b u divp2
| |c−1 | ·∇ − |c−1
=|cu|2c−1 +|b·∇u−divp|2c−1 +2ℜhcu,b·∇u−divpic−1
= u2+ b u divp2 +2 u,b u divp .
| |c | ·∇ − |c−1 ℜh ·∇ − i
Together with u, divp = u,p = u,A u = u2 = p2 and (2.6) this becomes
h − i h∇ i h∇ ∇ i |∇ |A | |A−1
f 2 = u2+ b u divp2 + u2 + p2 + u2
| |c−1 | |c | ·∇ − |c−1 |∇ |A | |A−1 | |−divb
= u2 + u2 + p2 + b u divp2
| |c−divb |∇ |A | |A−1 | ·∇ − |c−1
=: (u,p) 2.
k k
(ii) Assume c 0 and c divb λ>0. Then the solution operator
≥ − ≥
L :L2 H1 D;f (u,q)
2 → 0× 7→
of (2.10) is an isometry, i.e., L =1: from (2.10) we see that
2
| |
f 2 = (c divb)u divq 2
| |(c−divb)−1 | − − |(c−divb)−1
=|(c−divb)u|2(c−divb)−1 +|divq|2(c−divb)−1 +2ℜh(c−divb)u,−divqi(c−divb)−1
= u2 + divq 2 +2 u, divq .
| |c−divb | |(c−divb)−1 ℜh − i
Together with u, divq = u,q = u,A u bu = u2 u,bu and (2.6) this becomes
h − i h∇ i h∇ ∇ − i |∇ |A−h∇ i
f 2 = u2 + divq 2 +2 u2 u2
| |(c−divb)−1 | |c−divb | |(c−divb)−1 |∇ |A−| |−divb
= u2+ u2 + q+bu2 + divq 2
| |c |∇ |A | |A−1 | |(c−divb)−1
=: (u,q) 2,
||| |||
where we have also used |∇u|A =|A−1(q+bu)|A =|q+bu|A−1.
Itiseasytoseethatthenorms and areequivalenttoweighedH1 D-norms(seeAppendixB),giving
k·k |||·||| ×
some justification on their use as errormeasures. In fact, Remark 2.1 immediately furnishes errorequalities for
sufficiently regular approximations: Let the primal approximation u˜ H1 be such that p˜= A u˜ D. Then,
∈ 0 ∇ ∈
the pair (u˜,p˜) can be seen as the exact solution of the problem
p˜ = A u˜ in Ω,
∇
divp˜+b u˜+cu˜=:f˜ in Ω,
− ·∇
u˜ = 0 on ∂Ω,
i.e., we have L (f˜)=(u˜,p˜). Then by Remark 2.1(i) we directly have
1
(u,p) (u˜,p˜) 2 = L (f f˜) 2 = f f˜2 = f cu˜ b u˜+divp˜2
k − k k 1 − k | − |c−1 | − − ·∇ |c−1
and this can obviously be normalized to obtain
(u,p) (u˜,p˜) 2 f cu˜ b u˜+divp˜2
k − k = | − − ·∇ |c−1.
(u,p) 2 f 2
k k | |c−1
A similar error equality for the formulation (2.10) follows directly from Remark 2.1(ii). The assumption of the
high regularity of u˜ is not very practical, and we will remove it in subsequent sections. However,we emphasize
FUNCTIONAL A POSTERIORI ERROR EQUALITIES 5
that for linear problems error control follows directly from estimates of the norm of the solution operator,
provided that the primal approximation is sufficiently regular.
3. Error Equalities
Inthissectionwederiveerrorequalitiesforthetwoformulationsofthereaction-convection-diffusionproblem
discussed in the previous section.
In the following,we willunderstandthe pairs (u˜,p˜),(u˜,q˜) H1 D without further requirements as(mixed)
approximations of the exact solutions (u,p),(u,q) H1 ∈D, r0e×spectively. In other words, the results are
∈ 0×
applicable to any conforming approximation, i.e., approximations which belong to the appropriate Hilbert
spaces and satisfy the boundary conditions exactly. To emphasize further, we do not attract any specific
properties of a numerical method.
In the previous section we listed the requirements for the material coefficients A,b, and c which guarantee
existence and uniqueness of a solution. For clarity we will not repeat these requirements in this section, but
simply list those requirements which are needed for the error equalities to hold. These requirements will be
same as the ones in Remark 2.1, i.e., the conditions under which the solution operators are isometries.
3.1. Error Equality for the Problem (2.2). We present the first main result of the paper.
Theorem 3.1. Let (u,p) H1 D be the exact solution of (2.2), and assume c c > 0 and c divb 0.
∈ 0× ≥ 0 − ≥
Let (u˜,p˜) H1 D be arbitrary. Then
∈ 0×
(3.1) (u,p) (u˜,p˜) 2 = (u˜,p˜)
1
k − k M
and the normalized counterpart
(u,p) (u˜,p˜) 2 (u˜,p˜)
1
(3.2) k − k = M
(u,p) 2 f 2
k k | |c−1
hold, where
(u,p) (u˜,p˜) 2 = u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2 ,
k − k | − |c−divb |∇ − |A | − |A−1 | ·∇ − − − |c−1
(u˜,p˜):= f cu˜ b u˜+divp˜2 + p˜ A u˜2 .
M1 | − − ·∇ |c−1 | − ∇ |A−1
Proof. We begin by using (2.2) and p=A u:
∇
(u˜,p˜)= f cu˜ b u˜+divp˜2 + p˜ A u˜2
M1 | − − ·∇ |c−1 | − ∇ |A−1
= c(u u˜)+b (u u˜)+div(p˜ p)2 + p˜ p+A (u u˜)2
| − ·∇ − − |c−1 | − ∇ − |A−1
=|c(u−u˜)|2c−1 +|b·∇(u−u˜)+div(p˜−p)|2c−1 +2ℜhc(u−u˜),b·∇(u−u˜)+div(p˜−p)ic−1
+|p˜−p|2A−1 +|A∇(u−u˜)|2A−1 +2ℜhp˜−p,A∇(u−u˜)iA−1
= u u˜2+ (u u˜)2 + p˜ p2 + b (u u˜)+div(p˜ p)2
| − |c |∇ − |A | − |A−1 | ·∇ − − |c−1
+2 p˜ p, (u u˜) +2 u u˜,div(p˜ p) +2 u u˜,b (u u˜) .
ℜh − ∇ − i ℜh − − i ℜh − ·∇ − i
The first two terms in the last line cancel each other due to (2.1), since u u˜ H1 and p˜ p D. By using
− ∈ 0 − ∈
(2.6) we see that the third term in the last line can be written as
2 u u˜,b (u u˜) = u u˜2 .
ℜh − ·∇ − i | − |−divb
We arrive at (3.1) by combining this term to the other weighed L2 norm of the difference u u˜. The equality
(3.2) follows by the isometry property in Remark 2.1(i). − (cid:3)
Corollary 3.2. If b = 0, the problem (2.2) becomes the reaction-diffusion equation. For the error equality to
hold we need to assume c c >0. The equality of Theorem 3.1 then becomes
0
≥
u u˜2+ (u u˜)2 + p p˜2 + div(p p˜)2 = f cu˜+divp˜2 + p˜ A u˜2 ,
| − |c |∇ − |A | − |A−1 | − |c−1 | − |c−1 | − ∇ |A−1
which is the result from [1,3,7]. Naturally, this error equality has a normalized counterpart.
Remark 3.3. We note the following:
(i) If c divb=0 boundedness of the domain is needed for proper error control (see Appendix B).
−
(ii) One of the terms in the error measure can be written as
b (u u˜) div(p p˜)2 = f cu b u˜+divp˜2 .
| ·∇ − − − |c−1 | − − ·∇ |c−1
(iii) By applying the first norm equivalence in Appendix B to the equality of Theorem 3.1 one obtains a
two-sided estimate where the error is measured in a weighed H1 D -norm. This two-sided estimate is
×
similar to the one derived in [8]. However, these bounds can be quite coarse, which is apparent from the
constants appearing in the norm equivalence.
6 IMMANUELANJAM
We emphasize here that the error equality is possible because of the presence of the reactive term cu.
Indeed, c cannot be zero. Unfortunately this means that the result of Theorem 3.1 cannot be used for the
convection-diffusion problem. However,the result of the next section allows also for this case.
3.2. Error Equality for the Problem (2.10). We present the second main result of the paper.
Theorem 3.4. Let (u,q) H1 D be the exact solution of (2.10), and assume c 0 and c divb λ > 0.
Let (u˜,q˜) H1 D be arbit∈rary0.×Then ≥ − ≥
∈ 0×
(3.3) (u,q) (u˜,q˜) 2 = (u˜,q˜)
2
||| − ||| M
and the normalized counterpart
(u,q) (u˜,q˜) 2 (u˜,q˜)
2
(3.4) ||| − ||| = M
(u,q) 2 f 2
||| ||| | |(c−divb)−1
hold, where
(u,q) (u˜,q˜) 2 = u u˜2+ (u u˜)2 + q q˜+b(u u˜)2 + div(q q˜)2 ,
||| − ||| | − |c |∇ − |A | − − |A−1 | − |(c−divb)−1
(u˜,q˜):= f (c divb)u˜+divq˜2 + q˜ A u˜+bu˜2 .
M2 | − − |(c−divb)−1 | − ∇ |A−1
Proof. We begin by using (2.10) and q =A u bu:
∇ −
(u˜,q˜)= f (c divb)u˜+divq˜2 + q˜ A u˜+bu˜2
M2 | − − |(c−divb)−1 | − ∇ |A−1
= (c divb)(u u˜)+div(q˜ q)2 + q˜ q+A (u u˜)+b(u˜ u)2
| − − − |(c−divb)−1 | − ∇ − − |A−1
= (c divb)(u u˜)2 + div(q˜ q)2
| − − |(c−divb)−1 | − |(c−divb)−1
+2 (c divb)(u u˜),div(q˜ q) (c−divb)−1
ℜh − − − i
+|q˜−q+b(u˜−u)|2A−1 +|A∇(u−u˜)|2A−1 +2ℜhq˜−q+b(u˜−u),A∇(u−u˜)iA−1
= u u˜2 + (u u˜)2 + q˜ q+b(u˜ u)2 + div(q˜ q)2
| − |c−divb |∇ − |A | − − |A−1 | − |(c−divb)−1
+2 q˜ q, (u u˜) +2 u u˜,div(q˜ q) 2 u u˜,b (u u˜) .
ℜh − ∇ − i ℜh − − i− ℜh − ·∇ − i
The first two terms in the last line cancel each other due to (2.1), since u u˜ H1 and q˜ q D. By using
− ∈ 0 − ∈
(2.6) we see that the third term in the last line can be written as
2 u u˜,b (u u˜) = u u˜2 .
− ℜh − ·∇ − i | − |divb
We arrive at (3.3) by combining this term to the other weighed L2 norm of the difference u u˜. The equality
(3.4) follows by the isometry property in Remark 2.1(ii). − (cid:3)
Note that we can set c = 0 in the result of Theorem 3.4, because there is ‘additional reaction’ provided by
the term (divb)u. This corresponds to the case of the convection-diffusion problem. However, if c = 0, then
−
the result is not valid if divb = 0. This means that b cannot be zero in this case, or even a constant vector.
The case c=divb=0 will be considered in the next section.
Corollary 3.5. If c = 0, the problem (2.10) becomes the convection-diffusion equation. For the error equality
to hold we need to assume divb b >0. The equality of Theorem 3.4 then becomes
0
− ≥
(u u˜)2 + q q˜+b(u u˜)2 + div(q q˜)2
|∇ − |A | − − |A−1 | − |(−divb)−1
= f +(divb)u˜+divq˜2 + q˜ A u˜+bu˜2 .
| |(−divb)−1 | − ∇ |A−1
Note that for proper error control boundedness of the domain is now required (see Appendix B). Naturally, this
error equality has a normalized counterpart.
Remark 3.6. We note the following:
(i) One of the terms in the error measure can be written as
q q˜+b(u u˜)2 = A u q˜ bu˜2 .
| − − |A−1 | ∇ − − |A−1
(ii) By applying the second norm equivalence in Appendix B to the result of Theorem 3.4 one obtains a
two-sided estimate in a weighed H1 D -norm. However, these bounds can be quite coarse, which is
×
apparent from the constants appearing in the norm equivalence.
FUNCTIONAL A POSTERIORI ERROR EQUALITIES 7
Ω
c=0 ΩΩcc c≥c0>0
χΩ0 =1 ΩΩ00 χΩ0 =0
Figure 1. Division of a bounded three dimensionaldomain Ω into two subdomains according
to the values of c.
4. Two-Sided Error Estimates for the Case c=divb=0
If c = 0 and divb = 0 simultaneously in some part of the domain, neither of the error equalities of the
previous section can be used. For this case we derive a two-sided error bound by using the error equality of
Theorem 3.1.
In this section, we consider the problem formulation (2.2) for a bounded domain Ω such that the Friedrichs
inequality (2.8) holds. Then we see from (2.9) that even in the case c = divb = 0 the form generated by the
left hand side of (2.4) is coercive so we have a unique solution u H1 by the Lax-Milgram Theorem.
Like before, the real scalar valued reaction coefficient c 0 b∈elon0gs to L∞. We denote by Ω the part of
0
≥
the domain where c = 0. In the other part Ω := Ω Ω we assume that c is uniformly positive definite, i.e.,
c 0
\
c c > 0. We define cˆ:= c+χ , where χ (defined on the whole domain Ω) is the characteristic function
of≥Ω 0. See Figure 1 for an exampΩle0. Note thaΩt0cˆbelongs to L∞ and is uniformly positive definite.
0
We present the third main result of the paper.
Theorem4.1. Let(u,p) H1 Dbetheexactsolution of (2.2), andassumec c >0inΩ andc divb 0.
∈ 0× ≥ 0 c − ≥
Let (u˜,p˜) H1 D be arbitrary. Then
∈ 0×
1
(4.1) f b u˜+divp˜2 + p˜ A u˜2 [(u,p) (u˜,p˜)]2 +(u˜,p˜)
| − ·∇ |Ω0 2| − ∇ |A−1 ≤| − | ≤M3
holds. If in addition divb 0 in Ω , we have the improved lower bound
c
− ≥
(4.2) −(u˜,p˜) [(u,p) (u˜,p˜)]2.
M3 ≤| − |
Here
[(u,p) (u˜,p˜)]2 := u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2 ,
| − | | − |c−divb |∇ − |A | − |A−1 | ·∇ − − − |cˆ−1
C2
+(u˜,p˜):=2 f cu˜ b u˜+divp˜2 + p˜ A u˜2 + 1+4 F f b u˜+divp˜2 ,
M3 | − − ·∇ |Ωc,c−1 | − ∇ |A−1 α | − ·∇ |Ω0
(cid:16) (cid:17) (cid:18) (cid:19)
1
−(u˜,p˜):= f cu˜ b u˜+divp˜2 + p˜ A u˜2 + f b u˜+divp˜2 .
M3 2 | − − ·∇ |Ωc,c−1 | − ∇ |A−1 | − ·∇ |Ω0
(cid:16) (cid:17)
Proof. By adding χ u to both sides of the problem (2.2) we obtain
Ω0
p=A u in Ω,
∇
divp+b u+cˆu=f +χ u in Ω,
− ·∇ Ω0
u=0 on ∂Ω,
for which we can apply the result of Theorem 3.1 with reaction coefficient cˆand right hand side f +χ u:
Ω0
(4.3) u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2
| − |cˆ−divb |∇ − |A | − |A−1 | ·∇ − − − |cˆ−1
= f +χ u cˆu˜ b u˜+divp˜2 + p˜ A u˜2 .
| Ω0 − − ·∇ |cˆ−1 | − ∇ |A−1
The rest of the proof concentrates on removing the unknown exact solution from the right hand side. The first
term on the right hand side can be written as
(4.4) f +χ u cˆu˜ b u˜+divp˜2
| Ω0 − − ·∇ |cˆ−1
= f +χ (u u˜) cu˜ b u˜+divp˜2
| Ω0 − − − ·∇ |cˆ−1
= f cu˜ b u˜+divp˜2 + f +u u˜ b u˜+divp˜2
| − − ·∇ |Ωc,c−1 | − − ·∇ |Ω0
= f cu˜ b u˜+divp˜2 + f b u˜+divp˜2 + u u˜2
| − − ·∇ |Ωc,c−1 | − ·∇ |Ω0 | − |Ω0
+2 f b u˜+divp˜,u u˜ .
ℜh − ·∇ − iΩ0
8 IMMANUELANJAM
The last term can be estimated from above by
(4.5) 2 f b u˜+divp˜,u u˜ 2f b u˜+divp˜ u u˜
ℜh − ·∇ − iΩ0 ≤ | − ·∇ |Ω0| − |Ω0
2f b u˜+divp˜ u u˜
≤ | − ·∇ |Ω0| − |
C
F
2 f b u˜+divp˜ (u u˜)
≤ √α| − ·∇ |Ω0|∇ − |A
C2 1
γ F f b u˜+divp˜2 + (u u˜)2,
≤ α | − ·∇ |Ω0 γ|∇ − |A
which holds for all γ >0. By combining (4.3)–(4.5) and choosing γ =2 we obtain
u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2
| − |cˆ−divb |∇ − |A | − |A−1 | ·∇ − − − |cˆ−1
C2
f cu˜ b u˜+divp˜2 + 1+2 F f b u˜+divp˜2 + p˜ A u˜2
≤| − − ·∇ |Ωc,c−1 α | − ·∇ |Ω0 | − ∇ |A−1
(cid:18) (cid:19)
1
+ u u˜2 + (u u˜)2.
| − |Ω0 2|∇ − |A
By moving the last two terms to the left hand side we obtain
1
(4.6) u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2
| − |c−divb 2|∇ − |A | − |A−1 | ·∇ − − − |cˆ−1
C2
f cu˜ b u˜+divp˜2 + 1+2 F f b u˜+divp˜2 + p˜ A u˜2 .
≤| − − ·∇ |Ωc,c−1 α | − ·∇ |Ω0 | − ∇ |A−1
(cid:18) (cid:19)
Since in Ω we have
0
(4.7) b (u u˜) div(p p˜)2 = b (u u˜) div(p p˜)2 = f b u˜+divp˜2
| ·∇ − − − |Ω0,cˆ−1 | ·∇ − − − |Ω0 | − ·∇ |Ω0
we can multiply (4.6) with any constant without affecting this term in the estimate. We have then arrived at
the upper boundin (4.1). The lowerbound in (4.1) is obtainedsimply from(4.7) andinserting p A u=0 in
− ∇
(4.8) p˜ A u˜2 = p˜ p+A (u u˜)2 2 p˜ p2 + (u u˜)2 .
| − ∇ |A−1 | − ∇ − |A−1 ≤ | − |A−1 |∇ − |A
If divb 0inΩ ,wecanalsowrite(notethatwehaveassume(cid:0)dthatc divb 0holdsi(cid:1)nthewholedomain)
c
− ≥ − ≥
(4.9) f cu˜ b u˜+divp˜2 = c(u u˜)+b (u u˜)+div(p˜ p)2
| − − ·∇ |Ωc,c−1 | − ·∇ − − |Ωc,c−1
2 u u˜2 + b (u u˜)+div(p˜ p)2
≤ | − |Ωc,c | ·∇ − − |Ωc,c−1
(cid:16) (cid:17)
2 u u˜2 + b (u u˜)+div(p˜ p)2
≤ | − |Ωc,c−divb | ·∇ − − |Ωc,c−1
(cid:16) (cid:17)
2 u u˜2 + b (u u˜)+div(p˜ p)2 .
≤ | − |c−divb | ·∇ − − |Ωc,c−1
(cid:16) (cid:17)
The lower bound (4.2) results then by combining (4.7)–(4.9). (cid:3)
The above result is more general than the error equalities of the previous section in the sense that there is
more freedom for the values of the reaction coefficient c and the convection vector b. However, to achieve this
we needed to assume boundedness of the domain, and naturally the Friedrichs constant entered the estimates.
We also note that the error equalities of the previous section cannot be recoveredfrom Theorem 4.1.
Corollary 4.2. Depending on the reaction coefficient and the convection vector the result of Theorem 4.1 takes
the following forms:
(i) If b=0 the problem (2.2) becomes the reaction-diffusion problem with reaction limited to Ω . The result
c
holds when c c >0 in Ω and it reads as
0 c
≥
1
f cu˜+divp˜2 + p˜ A u˜2 + f +divp˜2
2 | − |Ωc,c−1 | − ∇ |A−1 | |Ω0
(cid:16) u u˜2 + (u u˜)2 + p p˜(cid:17)2 + div(p p˜)2
≤| − |Ωc,c |∇ − |A | − |A−1 | − |cˆ−1
C2
2 f cu˜+divp˜2 + p˜ A u˜2 + 1+4 F f +divp˜2 .
≤ | − |Ωc,c−1 | − ∇ |A−1 α | |Ω0
(cid:16) (cid:17) (cid:18) (cid:19)
If Ω =Ω (then Ω = ) the error equality of Corollary 3.2 holds (and the domain may be unbounded).
c 0
∅
FUNCTIONAL A POSTERIORI ERROR EQUALITIES 9
(ii) If c=0 (Ω = ,Ω =Ω) the problem (2.2) becomes the convection-diffusion problem. The result holds
c 0
∅
when divb 0, and it reads as
− ≥
1
p˜ A u˜2 + f b u˜+divp˜2
2| − ∇ |A−1 | − ·∇ |
u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2
≤| − |−divb |∇ − |A | − |A−1 | ·∇ − − − |
C2
2p˜ A u˜2 + 1+4 F f b u˜+divp˜2.
≤ | − ∇ |A−1 α | − ·∇ |
(cid:18) (cid:19)
If divb b >0 the error equality of Corollary 3.5 holds.
0
− ≥
(iii) Ifc=b=0 (Ω = ,Ω =Ω) theproblem (2.2)becomes thediffusion equation andwehave theestimate
c 0
∅
1
p˜ A u˜2 + f +divp˜2
2| − ∇ |A−1 | |
(u u˜)2 + p p˜2 + div(p p˜)2
≤|∇ − |A | − |A−1 | − |
C2
2p˜ A u˜2 + 1+4 F f +divp˜2,
≤ | − ∇ |A−1 α | |
(cid:18) (cid:19)
which is similar to the two-sided estimate derived (in the real case) in [11].
Remark 4.3.
(i) It is easy to verify (in the case divb 0 in Ω ) that
c
− ≥
+(u˜,p˜) C2
M3 max 2, 1+4 F .
sM−3(u˜,p˜) ≤ ( r α )
(ii) By calculating estimates for the solution operator norm the two-sided bounds exposed in this section can
be normalized in the same way as in the previous section (one way to obtain estimates for the solution
operator norm is setting u˜=p˜=0 in Theorem 4.1).
(iii) The upper bound in Theorem 4.1 can be improved if one leaves the Young constant γ in the estimate (in
(4.5) this constant was chosen to be 2). Then the upper bound would contain this constant which could
be chosen such that it minimizes the value of the upper bound.
5. Further Remarks
In this section we briefly comment on deriving error estimates for the primal variable and error indication
properties formixed approximations. For brevitywe limit ourselvesto the problem(2.2) andthe errorequality
of Theorem 3.1. In the following we will utilize just the following inequalities:
(5.1) p˜ A u˜2 = p˜ p+A (u u˜)2
| − ∇ |A−1 | − ∇ − |A−1
2 p˜ p2 + (u u˜)2 ,
≤ | − |A−1 |∇ − |A
(5.2) |f −cu˜−b·∇u˜+divp˜|2c−1 =|c(cid:0)(u−u˜)+b·∇(u−u˜)−d(cid:1)iv(p−p˜)|2c−1
2 u u˜2+ b (u u˜) div(p p˜)2 .
≤ | − |c | ·∇ − − − |c−1
Theorem 3.1 reads as (cid:0) (cid:1)
(5.3) u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2
| − |c−divb |∇ − |A | − |A−1 | ·∇ − − − |c−1
= f cu˜ b u˜+divp˜2 + p˜ A u˜2 .
| − − ·∇ |c−1 | − ∇ |A−1
The first obvious estimate results from simply omitting the last two terms on the left hand side. Then the left
hand side becomes independent of p˜, and so it becomes arbitrary on the right hand side, and we have
φ D u u˜2 + (u u˜)2 f cu˜ b u˜+divφ2 + φ A u˜2 ,
∀ ∈ | − |c−divb |∇ − |A ≤| − − ·∇ |c−1 | − ∇ |A−1
wherewehavechangedp˜toφtoemphasizethatitisnowanarbitraryfunctionfromD. Asisusualforproblems
with a convective term, such a form of a functional upper bound may unfortunately not be sharp: to get an
idea ofthe over-estimationwe set φ=p; then the r.h.s. becomes u u˜2 + b (u u˜)2 + (u u˜)2.
| − |c−divb | ·∇ − |c−1 |∇ − |A
In[5]theauthorsdemonstratedthatsharpupperboundscanbederivedaswell. Theestimatefrom[5]contains
two free functions (in contrast to the one free function in the above estimate) with respect to which the upper
bound must be minimized in order to obtain good estimates of the global primal error.
Next we derive estimates for the L2-part of the primal error: by estimating the left hand side of (5.3) from
below by (omitting the last term and) using (5.1) we obtain
1
(5.4) φ D u u˜2 f cu˜ b u˜+divφ2 + φ A u˜2 .
∀ ∈ | − |c−divb ≤| − − ·∇ |c−1 2| − ∇ |A−1
10 IMMANUELANJAM
If divb b >0 we can apply both (5.1) and (5.2) to the error equality (5.3) and obtain
0
− ≥
1
(5.5) φ D u u˜2 f cu˜ b u˜+divφ2 + φ A u˜2 .
∀ ∈ | − |−divb ≤ 2 | − − ·∇ |c−1 | − ∇ |A−1
As mentioned in the introduction, mixed me(cid:0)thods usually produce approximations with l(cid:1)ess regularity than
assumed in this paper. A first simple idea might be to post-process the approximationto obtain upper bounds
for errors. For example, assume that u˜ L2 and that we have a post-processing operator G : L2 H1 at our
∈ → 0
disposal(forexample,forpiecewiseconstantu˜onecouldusestandardnodalaveraging). Byusing(5.4)wethen
have the estimate
u u˜c−divb u G(u˜)c−divb+ G(u˜) u˜c−divb
| − | ≤| − | | − |
1
≤ |f −cG(u˜)−b·∇G(u˜)+divφ|2c−1 + 2|φ−A∇G(u˜)|2A−1 +|G(u˜)−u˜|c−divb
r
for any φ D. If divb b >0 we can use (5.5) in an similar fashion to obtain
0
∈ − ≥
u u˜−divb u G(u˜)−divb+ G(u˜) u˜−divb
| − | ≤| − | | − |
1
≤ √2 |f −cG(u˜)−b·∇G(u˜)+divφ|2c−1 +|φ−A∇G(u˜)|2A−1 +|G(u˜)−u˜|−divb
q
for any φ D.
∈
Up until now only globalvalues of the errorhave been discussed. However,estimating the errordistribution
of an approximation is an important task as well since this is essential for adaptive mesh refinement. Let
T
denote a discretization of the domain Ω into a mesh of non-overlapping elements T. Note that we assume
T = Ω, i.e., in particular that the boundary of Ω is exactly represented by the mesh. This is necessary
T∈T
in order to have conforming approximations in the first place: they must satisfy exactly the imposed boundary
S
conditions. The upper bound from Theorem 3.1 generates the following error indicator:
1
M
η (u˜,p˜):= f cu˜ b u˜+divp˜2 + p˜ A u˜2 ,
T | − − ·∇ |T,c−1 | − ∇ |T,A−1
q
whose purpose is to indicate the exact error distribution
e (u˜,p˜):= u u˜2 + (u u˜)2 + p p˜2 + b (u u˜) div(p p˜)2 .
T | − |T,c−divb |∇ − |T,A | − |T,A−1 | ·∇ − − − |T,c−1
q
In the following
η := η2 and e:= e2.
T T
sT∈T sT∈T
X X
The often cited global requirement for error indicators reads as Cη e Cη, where C,C > 0 are called the
≤ ≤
globalefficiency constantand globalreliability constant, respectively. Obviously in this case C =C =1, which
is the best case possible. What remains to be checked are the values of the local efficiency constants C > 0
T
in the local requirement C η e . In the case divb 0 this is particularly simple: since the inequalities
T T T
≤ − ≥
(5.1) and (5.2) result simply from applying the triangle inequality and the Young inequality, they hold also in
any subdomain, and directly give η2 2e2. This shows that C =1/√2>0.7 for any element T regardlessof
T ≤ T T
shape or size.
Finallywenotethatthe(non-normalized)errorequalitiesandboundsexposedinthispaperholdalsofornon-
homogenousDirichlet boundary conditions providedthat the approximationu˜ satisfies the boundary condition
exactly: in the proofs we require that u u˜ vanishes on the boundary.
−
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