A-posteriori error analysis of linear finite elements

OUTLINE FOR THE LECTURES ON ADVANCED COMPUTATIONAL TECHNIQUES
A-posteriori error analysis of linear finite elements for a two-point value problem for a
complete 2nd order ordinary differential equation
by
ANTONIO ORLANDO
1
Strong formulation
We consider a two point value problem for a complete 2nd order linear differential equation as model
problem. Let Ω =]0, L[, consider then the following problem.
Data: k ∈ C1 (Ω); β, c and f ∈ C 0 ([0, L])
Find: u ∈ C 2 ([0, L]) such that
ODE: − (k(x)u′ (x))′ + β(x)u′ (x) + c(x)u(x) = f (x)
BCs: u(0) = 0
u(L) = 0 ,
for all x ∈ [0, L]
(BVP)
where ′ denotes the classical derivative. For sake of simplicity only homogeneous essential boundary
conditions are considered. The classical setting for the formulation of the above mathematical model
requires strong assumptions on the data if the boundary value problem BVP is expected to have solutions.
Moreover such assumptions restrict class of physical problems which can be described by BVP. In order
to further enlarge the application areas the mathematical model requires some generalizations.
2
Weak formulation
One possible generalisation of BVP is obtained by interpreting its derivatives in the weak sense and
assuming the appropriate smoothness of the problem data. Assume then f ∈ L2 (Ω), k, c ∈ L∞ (Ω) and
β ∈ H 1 (Ω) with
κ1 ≥ k(x) ≥ κ0 > 0 and c(x) −
β ′ (x)
≥0
2
a.e. x ∈ Ω̄ .
(2.1)
In (2.1), κ0 , κ1 are constants, that is, they do not depend on x. Condition (2.1)1 is referred to as uniform
ellipticity wheraes (2.1)2 is called Garding inequality.
By using test functions v meeting homogeneous boundary conditions on the part of ∂Ω where essential
boundary conditions are assigned (in our case this is all ∂Ω), the weak formulation corresponding to BVP
reads as follows:
Find: u ∈ H01 (Ω) such that
Z L
a(u, v) =
[k(x)u′ (x)v ′ (x) + β(x)u′ (x)v(x) + c(x)u(x)v(x)] dx =
0
=
Z
L
0
f (x)v(x) dx = (f, v)L2 (Ω) for all v ∈ H01 (Ω)
(WBVP)
Remark 2.1. If c = 0 then a = a(u, v) is symmetric.
The Sobolev space V = H01 (Ω) used in stating the WBVP is a Hilbert space. The assumptions (2.1) allow
one to conclude that the bilinear form a(u, v) and the linear functional ℓ = ℓ(v) = (f, v)L2 (Ω) satisfy the
hypotheses of the Lax-Milgram theorem. Hence, (WBVP) has solution and is unique. One obtains also
the following stability result. There exists a constant C > 0 such that there holds
kukH 1 (Ω) ≤ Ckf kL2 (Ω)
The value of the constant is important from the numerical point of view as it reflects the sensitivity of the
solution to data changes.
For a given u ∈ V (= H01 (Ω)),
ru (v) := (f, v)L2 (Ω) − a(u, v)
1
for all v ∈ V
March 15, 2013 7:22
(2.2)
denotes a linear functional defined over V which is continuous. Hence ru is an element of the dual of V 1 .
The functional ru is referred to as the residual functional associated with u. As result, WBVP may be also
expressed as the problem of finding u ∈ V such that its corresponding residual ru (v) is the zero element
of V ′ .
Remark 2.2. In mechanics, the residual functional expresses the difference between the internal and
external virtual power with the space of the test functions as space of the virtual velocities.
3
Discrete Problem
The difficulty in trying to solve WBVP arises in the infinite dimension of the space V over which the
functional ru (v) is defined (which is referred to as the space of the test or weighted functions), and in the
infinite dimension of the space V in which the solution u is sought for (which is referred to as the space
of the trial functions). The replacement of the space V with one of its finite dimensional subspaces Vh is
the underlying idea of the Galerkin methods. If Vh denotes a finite dimensional subspace of V we define
the discrete problem as follows
Find: uh ∈ Vh ⊂ V such that
a(uh , v) = (f, v)L2 (Ω) for all v ∈ Vh
3.1
(GWBVP)
Galerkin orthogonality
The comparison of the exact solution u of WBVP with the solution uh of GWBVP gives, for any v ∈ Vh ⊂ V
a(u − uh , v) = a(u, v) − a(uh , v) = (f, v) − a(uh , v) = 0
(3.1)
that is, the Galerkin approximation uh is such that the corresponding residual ruh (v) defined over V is
equal to zero only on Vh ⊂ V , and consequently uh is, in general, different from u, the exact solution of
WBVP. The element of V given by e = u − uh is referred to as the discretization error.
3.2
Error representation equation
The discretization error e is such that, for any v ∈ V
a(e, v) = a(u − uh , v) = a(u, v) − a(uh , v) = (f, v) − a(uh , v)
and recalling the definition (2.2), it follows
a(e, v) = (f, v) − a(uh , v) = ruh (v) for all v ∈ V
(3.2)
that is, the discretization error e is a solution of the following weak formulation
Find: ψ ∈ V such that
a(ψ, v) = ruh (v) for all v ∈ V
(ERWP)
Since this problem has solution and is unique, then the discretization error is the unique solution of the
problem ERWP. This property, along with (3.1) will be the point of departure for the error analysis of
the finite element method.
4
Considerations on the convergence of Galerkin approximations
The concept of closeness of the approximate solution uh to the exact solution u may be made precise if it
is related to the one of convergence. In the Galerkin method, for each choice of the discrete space Vh ⊂ V ,
an element uh of the space V is obtained. Therefore, any convergence consideration requires first a family
of discrete solutions to be determined (i.e. the criterion of choice of the space Vh to be given), and then,
1 Given a vector normed space V , the dual of V , denoted by V ′ , is the space of all linear continuous functionals ℓ defined
over V . For a linear functional ℓ : V → R defined over a normed space V , continuity is equivalent to boundness, i.e.
∃M > 0 : |ℓ(x)| ≤ M kxkV for all x ∈ V
.
2
March 15, 2013 7:22
the topological structure (necessary to analyse the convergence) to be specified. For this purpose, the
Cea’s lemma plays a fundamental role. It refers to a basic property of the Galerkin approximation uh
expressed by the following inequality
ku − uh kV ≤ C inf ku − vh kV
vh ∈Vh
where
(4.1)
inf ku − vh kV denotes the distance in V of the exact solution u to Vh and C is a constant
vh ∈Vh
independent of Vh given by the ratio of the continuity constant to the coercivity constant of the bilinear
form a = a(u, v). When the bilinear form a = a(u, v) is symmetric, i.e. when β = 0 in WBVP, since
a(u − uh , v) = 0 for all v ∈ Vh , it follows that uh represents the projection onto Vh of the exact solution
u ∈ V with respect to the inner product defined by a(u, v).
Equation (4.1) shows that the natural setting to study the convergence of a family of Galerkin approximations is given by the same topological structure as the space V in which WBVP is formulated. Furthermore,
the concept of convergence of a family of Galerkin approximations may be made precise as well. In fact,
if there exists a family (Vh )h∈]0,1[ with Vh ⊂ V such that the distance of u to the subspace Vh approaches
to zero, that is,
lim inf ku − vh kV = 0
(4.2)
h→0 vh ∈Vh
the family of the corresponding solutions (uh )h∈]0,1[ is convergent to the exact solution u, that is,
lim ku − uh kV = 0
h→0
⇐⇒
lim uh = u in V .
h→0
(4.3)
Proof. Just pass to the limit in (4.1) and account for (4.2).
Given a familty of spaces (Vh )h∈]0, 1[ with the property (4.2), the problem of estimating the distance of u
to the subspace Vh is object of the approximation theory in Sobolev spaces.
The previous considerations regard the convergence of a family of Galerkin approximations (uh )h∈]0,1[ in
the natural norm of V . However, the same family of Galerkin approximations (uh )h∈]0,1[ can be convergent
to the exact solution u even by assuming other metrics to measure the distance of uh to p
u. For instance, if
the bilinear form a = a(u, v) is continuous, elliptic and symmetric, the function k|u|k = a(u, u), referred
to as energy norm, may be used to define a norm on V which is equivalent to its natural norm (see below
next section). Since the natural norm of V , which is the H 1 -norm, is stronger than the L2 -norm, then
there holds
lim uh = u in V ⇒ lim uh = u in L2
h→0
h→0
but not viceversa.
The previous considerations refer to generic finite dimensional spaces Vh ⊂ V . By specifying Vh is then
possible to address, for instance, problems such as the study of the order of convergence of a family of
Galerkin approximations and the evaluation of quantitative bounds for the discretization error. These
bounds express estimates of the error and are generally referred to as error estimates, which will be object
of study of the following sections.
We conclude this section by introducing some notations which will be needed for building and analysing
the error estimates in the subsequent sections.
The finite element space Vh which we will refer to is defined as follows. Let T be a finite partition of
n
Ω =]0, L[ such that Ω = ∪ Ωi where Ωi =]xi−1 , xi [ and for every (i, j) with i, j ∈ (1, 2, . . . , n), Ωi ∩Ωj = ∅.
i=1
Assume x0 = 0 and xn = L and define the mesh size function h = h(x) as h(x) = hi = xi − xi−1 if x ∈ Ωi .
Once the function h(x) has been given, the set Vh of the continuous functions defined over Ω such that v
is a linear polynomial on each subinterval Ωi and v(0) = 0 and v(L) = 0 is uniquely determined2 . It is
trivial to show that Vh is a closed finite dimensional linear subspace of H01 and is therefore a Hilbert space
with the norm of H1 . Because Vh ⊂ V , the finite element is referred to as conforming finite element.
2 It is worth noting that for a given n, specifying the number of elements which the domain Ω is subdivided into, there
is an infinite number of partitions of Ω consisting of n elements, and consequently there is an infinite number of different
spaces Vh corresponding to the same number n of elements
3
March 15, 2013 7:22
5
A simple a-posteriori error estimate for the solution of a linear
system
Let A ∈ Rn×n be a positive definite matrix and f ∈ Rn . Consider then the following linear system
(5.1)
Ax = f
Given any x̃ ∈ Rn , in order to check whether this is solution of (5.1) one should evaluate Ax̃ and check
whether it is equal to f . If that is not the case, then we define the error e as
e = x − x̃
where x is the solution of (5.1). Now, we ask whether it is possible to give an estimate of some norm of
the error. From the definition of e, we soon check that e ∈ Rn is solution of the following linear system
Ae = r
(5.2)
r = f − Ax̃ .
(5.3)
where r is called residual and is defined as
Equation (5.2) shows that if r = 0, then e = 0 (since A is positive definite). Can we then assume a measure
of the size of r as a measure of the size of e?
Well this depends clearly on the matrix as it appears from the following proposition. Let k · kRn denote
the Euclidean norm in Rn induced by the Euclidean inner product in Rn , that is, for x, y ∈ Rn with
x = (xi )i=1,...,n and y = (yi )i=1,...,n we have
q
(5.4)
kxkRn = x21 + . . . + x2n and x · y = x1 y1 + . . . + xn yn .
Then we can state the following.
Proposition 5.1. Let e ∈ Rn be solution of (5.2) and r ∈ Rn be defined by (5.3). Then there holds
kekRn ≤
1
krkRn
λ1
where λ1 is the smallest eigenvalue of A.
Proof. Since A is positive definite, from the property of the Rayleigh quotient we have that
λ1
Ay · y
for all y ∈ Rn ,
kyk2Rn
(5.5)
with λ1 being the smallest eigenvalue of A. Since A is positive definite, then λ1 > 0. From (5.5) for y = e
and using Cauchy-Scwarz inequality it follows
kek2 ≤
1
1
1
Ae · e =
r·e≤
krkkek ,
λ1
λ1
λ1
which proves the statement.
As result of this proposition, we conclude that if λ1 is very small, a measure of the size of only the residual
cannot be an indication on the size of the error.
6
A-posteriori error estimates
Error estimates are usually distinguished as a-priori and a-posteriori. They are called a-priori when
they are based on an a-priori knowledge of the regularity properties of the solution such as its degree
of smoothness. Their main use is to establish convergence properties along with optimal global rate of
convergence. A-posteriori error estimates, on the other hand, express estimates of the error akin to the
current approximation and therefore are much more appealing to finite element practitioners.
As result of the Cea’s lemma, the error analysis of the Galerkin methods for second order elliptic problems
is most easily described in the H 1 -norm. The inequality (4.1) is the starting point in establishing a-priori
4
March 15, 2013 7:22
error estimates and in studying asymptotic rates of convergence of the finite element method. In these
analysis, a-priori error estimates are evaluated as bounds for the interpolation/approximation error of
polynomials in Sobolev spaces.
If the bilinear form a = a(u, v) is symmetric and elliptic,
a(u, v) may be used to define p
an inner product in
p
V with the corresponding norm given by k|u|k = a(u, u) called energy norm. Since a(u, u) is a norm,
one can investigate the convergence in V with the energy norm, that is, looking at the error measured
with the energy norm. If the bilinear form a(u, v) is also continuous, the energy norm is then equivalent
to the H 1 -norm that is, there holds
αkuk2H 1 ≤ a(u, u) ≤ M kuk2H 1
(6.1)
where α and M are the coercivity and continuity constants of the bilinear form, respectively. Hence, any
considerations in H 1 -norm applies to energy norm and viceversa.
If the bilinear form
(6.1) still
p a(u, v) is not symmetric, while it is coercive and continuous, the inequality
3
a(u,
u)
is
no
longer
a
norm
as
the
triangular
inequality
is
not
satisfied
.
As
a result,
holds.
However,
p
a(u, u) should be considered only simply as an estimate of the error in H1 -norm.
Finally, note that since from (6.1) it is
kukH 1 ≤
1p
a(u, u)
α
p
it follows that a(u, u) cannot be considered as a robust error estimator because it fails to be meaningful
for those problems having very small coercivity costant α, i.e. when kmin in (2.1) is very small. The case
kmin very small should deserve special attention as it leads in general to singularly pertubed boundary
value problems where a boundary or interior layer phenomenon emerges. An error estimate is said to be
robust if it performs well in relatively general circumstances. More precise and quantitative definitions
can be found in literature.
6.1
Explicit residual type error estimates in energy norm
Explicit a posteriori residual type error estimates are obtained by providing an explicit estimate of the
norm of the residual associated with the Galerkin approximation uh in the dual space. This residual
functional is constitued by the pointwise residual defined over the interior of each element and the jump
of the finite element solution across the element boundaries. Let
k|e|k2 = a(e, e)
(6.2)
with the convention that (6.2) is a norm if a(u, v) is symmetric. Using (3.2) and noting that the discretization error e ∈ V (for u ∈ V and uh ∈ Vh ⊂ V ), we have then
k|e|k2 = a(e, e) = ruh (e)
(6.3)
where
ruh (v) = (f, v) − B(uh , v) =
Z L
f (x)v(x) dx+
=
−
Z
0
L
0
[k(x)u′h (x)v ′ (x) + β(x)u′h (x)v(x) + c(x)uh (x)v(x)] dx
(6.4)
Since uh is smooth over each subinterval Ωi , the integral which gives a(uh , v) must be then evaluated as
follows
n Z xi
X
[k(x)u′h (x)v ′ (x) + β(x)u′h (x)v(x) + c(x)uh (x)v(x)] dx
a(uh , v) =
i=1
3 The
xi−1
triangular inequality of the norm, that is,
kx + ykV ≤ kxkV + kykV for all x, y ∈ V
is invoked to prove that given the set S of all open sheres, there is a unique topology having S as one of its bases. This
topology is referred to as topology generated by the metric.
5
March 15, 2013 7:22
In order to have the residual corresponding to uh with respect to the strong form of the differential
equation, a formal application of the integration by parts yields
a(uh , v) =
n Z
X
xi
xi−1
i=1
− (k(x)u′h (x))′ v(x) + β(x)u′h (x)v(x) + c(x)uh (x)v(x)] dx+
h
i xi + k(x)u′h (x)v(x)
(6.5)
xi−1
Replacing (6.5) into (6.4) gives
ruh (v) =
n Z
X
xi
xi−1
i=1
{f (x) − [(k(x)u′h (x))′ + β(x)u′h (x)+
h
i xi =
+ c(x)uh (x)]}v(x) dx + k(x)u′h (x)v(x)
xi−1
=
n Z
X
i=1
xi
Ruh (x)v(x)dx +
xi−1
n h
X
i=1
where we have set
Ruh : x ∈
[
i=1,...,n
i xi
k(x)u′h (x)v(x)
xi−1
(6.6)
Ωi → Ruh,i (x)
with
Ruh,i (x) = f (x) − [ − (k(x)u′h (x))′ + β(x)u′h (x) + c(x)uh (x)]
i = 1, . . . , n
(6.7)
the pointwise residual corresponding to uh . Notice that Ruh,i is well defined since uh is smooth over
Ωi =]xi−1 , xi [.
Let uh,j = uh,j (x) denote the restriction of the finite element solution uh over the element Ωj =]xj−1 , xj [,
rearranging (6.6) as follows:
ruh =
n Z
X
i=1
xi
Ruh,i (x)v(x)dx +
xi−1
n−1
X
j=1
d(xj )[u′h,j (x) − u′h,j−1 (x)]v(xj )
(6.8)
shows that ruh is made up of two terms: (i) the pointwise residual Ruh , which is referred to as the regular
part of the residual and is defined over each subdomain Ωi ; and (ii) the term k(xj )[u′h,j (x) − u′h,j−1 (x)]
which is referred to as the singular part of the residual and is defined by the jump of the finite element
solution uh across the element boundaries.
Let Πh e denote the Vh -nodal linear interpolant of e, that is Πh e ∈ C 0 (Ω), Πh e|Ωi ∈ P1 (Ωi ) and Πh e(xi ) =
e(xi ). Since Πh e ∈ Vh ,
ruh (Πh e) = 0 .
Due to the linearity of ruh , it follows
ruh (e) = ruh (e − Πh e)
which combined with (6.3) yields
k|e|k2 = ruh (e − Πh e)
(6.9)
where
ruh (e − Πh e) =
n Z
X
i=1
xi
Ruh (x)(e(x) − Πh e(x)) dx+
xi−1
h
i xi ′
+ k(x)u (x)[e(x) − Πh e(x)]
xi−1
(6.10)
Since in 1-D problems the boundary of each subdomain is formed by only nodes, the term involving the
boundary in ruh vanishes as Πh e(xi ) − e(xi ) = 0. Then, it follows
k|e|k2 = ruh (e − Πh e) =
6
n Z
X
i=1
xi
xi−1
Ruh (x)(e(x) − Πh e(x)) dx
March 15, 2013 7:22
(6.11)
Remark 6.1. Notice that this is a typical 1D result. For Ω ⊂ Rn for n > 1, the singular part of the
residual does not vanish and in some cases this is the dominant term.
Let kmin,i = min {k(x), x ∈ Ωi =]xi−1 , xi [ } and consider the following piecewise constant function
p(x) = pi if x ∈ Ωi with p2i =
h2i
kmin,i
.
We give now the main result of this section.
Proposition 6.1. Denote by uh = uh (x) the continuous piecewise linear finite element solution of
(GWBVP) and by u the exact solution of the (WBVP). Then the following bound on the energy norm of
the error e holds
1/2
n
1 X 2
,
(6.12)
η
k|e|k ≤
π i=1 i
where the error indicators ηi are defined for each element as follows:
Z xi
o2
h2i n
ηi2 =
f (x) − − (k(x)u′h (x))′ + β(x)u′h (x) + c(x)uh (x)
dx .
xi−1 kmin,i
(6.13)
Proof. Rewrite (6.11) as
ruh (e − Πh e) =
=
n Z
X
xi
pi Ruh (x)
i=1 xi−1
L
Z
p(x)Ruh (x)
0
1
[e(x) − Πh e(x)] dx =
pi
1
[e(x) − Πh e(x)] dx .
p(x)
The use of Schwartz’s inequality gives
1
|ruh | ≤ kpRuh kL2 (Ω) k (e − Πh e)kL2 (Ω)
p
where
kpRuh kL2 (Ω) =
=
X
n Z
i=1
xi
xi−1
X
n Z
i=1
X
n Z
xi
xi−1
1/2
p2i Ru2 h (x) dx
=
h2i
kmin,i
1/2
Ru2 h (x) dx
1/2
1
2
[e(x)
−
Π
e(x)]
dx
h
p2
i=1 xi−1 i
1/2
X
Z
n
kmin,i xi
2
[e(x) − Πh e(x)] dx
.
=
h2i
xi−1
i=1
1
k (e − Πh e)kL2 (Ω) =
p
(6.14)
xi
(6.15)
Now in 1D, it is possible to show that for any v ∈ H 1 (Ωi )
kv − Πh vk2L2 (Ωi ) ≤
where
ai (v, v) =
Z
xi
2
2
1
h
h
kv ′ k2L2 (Ωi ) ≤
ai (v, v)
π
π
kmin,i
[k(x)v ′2 (x) + β(x)v ′ (x)v(x) + c(x)v 2 (x)] dx .
xi−1
As a result, using (6.16) with v = e one obtains
Z
1
kmin,i xi
[e(x) − Πh e(x)]2 dx ≤ 2 ai (e, e)
2
hi
π
xi−1
7
March 15, 2013 7:22
(6.16)
which combined with (6.15) yields
1
k (e − Πh e)kL2 (Ω) =
p
≤
X
n
i=1
X
n
i=1
kmin,i
h2i
Z
xi
xi−1
1/2
≤
[e(x) − Πh e(x)] dx
2
1/2
1
a
(e,
e)
.
i
π2
In addition
n
X
ai (e, e) =
n Z
X
i=1
L
i=1
=
Z
xi
[k(x)e′2 (x) + β(x)e′ (x)e(x) + c(x)e2 (x)] dx =
xi−1
[k(x)e′2 (x) + β(x)e′ (x)e(x) + c(x)e2 (x)] dx = a(e, e)
0
hence,
1
1
k (e − Πh e)kL2 (Ω) ≤ [a(e, e)]1/2 .
p
π
In summary, we obtain the following error estimate
1
k|e|k2 = a(e, e) = ruh (e − Πh e) ≤ kpRuh kL2 (Ω) k (e − Πh e)kL2 (Ω)
p
1
≤ kpRuh kL2 (Ω) [a(e, e)]1/2
π
that is,
1
k|e|k ≤
π
X
n Z
xi
h2i n
f (x) − − (k(x)u′h (x))′ + β(x)u′h (x)+
kmin,i
1/2
o2
+ c(x)uh (x)
dx
,
i=1
xi−1
(6.17)
which is (6.12) by accounting for the definition of ηi given by (6.13).
The term on the right hand side of (6.12) is referred to as error estimate and gives an estimate to the global
discretization error e = u − uh . It is composed of locally computable quantities, ηi , named error indicators
of the approximate solution uh , which reflect the contribution of each element to the global error estimate.
Due to the structure of the error estimator obtained, it is suitable to adaptive mesh refinement strategies.
It must, however, be said that in the current literature the term of error indicator is adopted with a wider
meaning to denote simply a mesh quality indicator, i.e. it is meant to give rough information about the
distribution of the error and is sometimes not suitable to evaluate an error estimator. It is also worth
noting the dependence of the estimate on the first power of h which is consistent with the a-priori error
analysis in the enrgy norm. However, one must observe that in deducing this a-posteriori error estimate
we have not required u ∈ H 2 which was invoked for the a-priori error analysis.
We conclude this section by reporting the proof of (6.16). First we need to recall the following result
which can be easily checked.
Lemma 6.1. Let ϕ ∈ H01 (Ωi ), the smallest value for λ such that the following homogeneous two-point
value problem
− ϕ′′ + λϕ = 0
ϕ(xi−1 ) = ϕ(xi ) = 0
has no trivial solution is
λ1 =
and there holds
λ1 ≤
8
(6.18)
π2
h2i
(ϕ′ , ϕ′ )
for all ϕ ∈ H01 (Ωi ) .
(ϕ, ϕ)
March 15, 2013 7:22
(6.19)
Lemma 6.2. Let v ∈ H 1 (Ωi ) and Πh v be the nodal linear interpolant of v with Πh v(xi−1 ) = v(xi−1 ) and
Πh v(xi ) = v(xi ). Then there holds
k(v − Πh v)′ kL2 (Ωi ) ≤ kv ′ kL2 (Ωi ) .
(6.20)
Proof. For any v ∈ H 1 (Ωi ), v − Πh v ∈ H01 (Ωi ). Then we note that
k(v − Πh v)′ k2L2 (Ωi ) = ((v − Πh v)′ , (v − Πh v)′ )L2 = ((v − Πh v)′ , v ′ )L2 − ((v − Πh v)′ , (Πh v)′ )L2
and since
((v−Πh v)′ , (Πh v)′ )L2 =
Z
xi
xi−1
(v−Πh v)′ (Πh v)′ dx = −
Z
xi
xi−1
x i
(v−Πh v)(Πh v)′′ dx+ (v−Πh v)(Πh v)′
=0
xi−1
then we obtain after applying Schwartz inequality
k(v − Πh v)′ k2L2 (Ωi ) = ((v − Πh v)′ , v ′ )L2 ≤ k(v − Πh v)′ kL2 (Ω) kv ′ kL2 (Ω) ,
which yields (6.20).
If we combine (6.19) with (6.20) then we conclude that
hi
1
kv − Πh vkL2 (Ωi ) ≤ √ kv ′ kL2 (Ωi ) = kv ′ kL2 (Ωi ) for any v ∈ H 1 (Ωi ) .
π
λ1
Lemma 6.3. Given the conditions (2.1), there holds
2
2
h
1
h
′ 2
kv kL2 (Ωi ) ≤
ai (v, v)
π
π
kmin,i
where
ai (v, v) =
Z
xi
k(x)v ′2 (x) + β(x)v ′ (x)v(x) + c(x)v 2 (x) dx .
xi−1
Proof. Just observe that
2
2
h
kmin,i ′ ′
h
′ 2
kv kL2 (Ωi ) =
(v , v )L2 (Ωi )
π
π
kmin,i
2
2
h
h
1
1
′
′
(kmin,i v , v )L2 (Ωi ) ≤
ai (v, v) .
=
π
kmin,i
π
kmin,i
6.2
Dual variational formulation for the error
In the previous section we have seen that the error e = u − uh is solution of the weak formulation (ERWP)
which we rewrite for the reader convenience
Find: e ∈ V such that
a(e, v) = ruh (v) = ℓ(v) − a(uh , v) for all v ∈ V .
(6.21)
This formulation is referred to as the primal formulation for the error. If a(u, v) is symmetric, which will
be assumed hereafter, then (6.21) is equivalent to the minimization problem
Find: e ∈ V such that
J (e) ≤ J (v) for all v ∈ V .
(6.22)
with
1
a(v, v) − ℓ(v) + a(uh , v) for all v ∈ V .
2
It then follows that the energy norm of the error e given by (6.3) meets the following bound
J (v) =
k|e|k2 ≥ −2J (v) for all v ∈ V .
9
March 15, 2013 7:22
(6.23)
Proof. From the definition of k|e|k and of J (v) we have that
1
1
k|e|k2 = a(e, e) = J (e) + ℓ(e) − a(uh , e)
2
2
(6.24)
and using (6.21) with v = e,
a(e, e) = ℓ(e) − a(uh , e),
we have
1
1
k|e|k2 = a(e, e) = J (e) + a(e, e)
2
2
which gives
1
− a(e, e) = J (e)
2
and therefore
1
1
k|e|k2 = a(e, e) = −J (e) .
2
2
(6.25)
Recalling then (6.22), (6.23) follows.
Equation (6.23) states that in order to obtain a lower bound to k|e|k2 one just needs to evaluate J (v) at
any element of V . Notice that this bound can be, however, very poor unless v is chosen suitably.
Adaptive processes rely, on the other hand, on upper bounds to the discretization error. Upper bounds
can be obtained by means of the so-called dual variational formulation of (6.22).
We next detail how to obtain such dual formulation for the error e, using the following two-point value
problem as model problem. At variance of (BVP) here we consider the case that a natural boundary
condition is given at x = L).
− (k(x)u′ (x))′ + c(x)u(x) = f (x) for all x ∈ [0, L]
u(0) = u0
u′ (L) = u′L .
(6.26)
To put things in perspective, in order to get the dual formulation, we have to find a functional G such that
the ‘stress variable’
qe = (k(x)e′ , c(x)e)
represents a critical point for G. It is the objective of this note to show that such functional exists and
there holds
G(q − qh ) ≤ G(qe ) for all q ∈ Q
with Q the set of equilibrated stress fields, qh = (k(x)uh , c(x)uh ) where uh denotes any compatible function, that is, we mean a function that is continuous and meets the essential boundary conditions. For
instance, conforming finite element method will yield such uh .
By introducing the following variables
σ(x) = k(x)u′ (x)
T (x) = c(x)u(x) ,
(6.27)
(6.26) is transformed into the following system of first order differential equations in the unknowns σ, T
10
March 15, 2013 7:22
and u
Find σ, T, u such that:
a)
−σ ′ (x) + T (x) = f (x) for all x ∈ [0, L]
b) σ(x) − k(x)u′ (x) = 0 for all x ∈ [0, L]
c)
T (x) − c(x)u(x) = 0 for all x ∈ [0, L]
(6.28)
d) σ(L) = k(L)u′L
e)
T (0) = c(0)u0
f)
u(0) = u0
g)
u′ (L) = u′L .
Equation (6.28)a is called the equilibrium equation whereas (6.28)d represents a natural boundary condition. We introduce then the following notation: σex and Tex denote the exact solution of (6.28) whereas
for any given compatible function uh = uh (x), we set
σh (x) = k(x)u′h (x)
(6.29)
Th (x) = c(x)uh (x) .
Moreover, we denote by Q the set of elements q = (σ, T ) solutions of the following problem stated over
the whole domain Ω =]0, L[
− σ ′ (x) + T (x) = f (x) for all x ∈]0, L[
(6.30)
σ(L) = σex (L) = k(L)u′L ,
obtained from (6.28) considering only the equilibrium equation along with the natural boundary condition.
Given ri = (σi , Ti ) ∈ L2 (Ω) × L2 (Ω) i = 1, 2, we consider the linear space L2 (Ω) × L2 (Ω) equipped with
the following inner product
(r1 , r2 ) := hσ1 , σ2 i + hT1 , T2 i :=
Z
and denote by k|q|k the associated norm, that is,
s
Z
p
k|q|k = (q, q) =
0
L
0
L
σ1 (x)
σ2 (x) dx +
k(x)
σ 2 (x)
dx +
k(x)
Z
0
L
Z
0
L
T1 (x)
T2 (x) dx ,
c(x)
(6.31)
T 2 (x)
dx .
c(x)
We have then the following fundamental result.
Proposition 6.2. Let qex = (σex , Tex ), qh = (σh , Th ) then for any q ∈ Q there holds
(q − qex , qex − qh ) = hσ − σex , σex − σh i + hT − Tex , Tex − Th i = 0 .
(6.32)
Remark 6.2. Equation (6.32) expresses the orthogonality between q − qex and qe = qex − qh . Such
condition is known as Prager-Synge relationship.
Proof. Next the proof of (6.32) is presneted for the case that uh is a conforming finite elment solution. By
11
March 15, 2013 7:22
accounting for (6.31) and applying integration by parts over each subdomain where uh is smooth yields
Z
L
0
=
=
−
(a)
(b)
=
−
(σ − σex )
Z
Z
L
u′h )dx
+
σex − σh
dx +
k(x)
L
(σ −
0
n h
X
i=1
Z L
σex )(u′ex
−
0
0
Tex − Th
dx =
c(x)
L
Z
0
(T − Tex )(uex − uh )dx =
i xi
(σi (x) − σex,i (x))(uex (x) − uh (x)
xi−1
(σ − σex )′ (uex − uh )dx +
0
n h
X
i=1
Z L
(T − Tex )
Z
0
(6.33)
L
(T − Tex )(uex − uh )dx =
i xi
(σi (x) − σex,i (x))(uex (x) − uh (x))
+
xi−1
+
(σ − σex )′ − (T − Tex ) (uex − uh )dx .
By rewriting the term (6.33)a as follows
n h
i xi
X
(σi (x) − σex,i (x))(uex (x) − uh (x))
xi−1
i=1
(a′ )
(b′ )
(6.34)
=
= σ1 (x0 ) − σex,1 (x0 ) uex (x0 ) − uh (x0 ) + σn (xn ) − σex,n (xn ) uex (xn ) − uh (xn ) +
n−1
X n
o
σi (xi ) − σi+1 (xi ) + σex,i+1 (xi ) − σex,i (xi )
uex (xi ) − uh (xi )
+
i=1
we observe that
• The term (6.34)a′ vanishes either for the compatibility of the finite element solution uh = uh (x) or
because σ meets the natural boundary conditions;
• The term (6.34)b′ vanishes for the continuity of the functions σ and σex .
Moreover, also the term (6.33)b is equal to zero as q = (σ, T ) does satisfy the equilibrium equation in
Ω =]0, L[. As result, we have
Z
0
since
L
(σ − σex )′ − (T − Tex ) (uex − uh ) dx =
′
−σex
(x) + Tex (x) = f (x)
−σ ′ (x) + T (x) = f (x)
Z
0
L
′
(−σex
+ Tex ) − (−σ ′ + T ) (uex − uh ) dx = 0
for x ∈ Ωi =]xi−1 , xi [ i = 1, 2, ..., n
(6.35)
for x ∈ Ωi =]xi−1 , xi [ i = 1, 2, ..., n .
(6.36)
Combining these results all together yields finally (6.32).
Once we have established the orthogonality condition (6.32), we are now in the position to formulate the
dual variational principle for (6.22). For any q ∈ L2 (Ω) × L2 (Ω) define the following functional
1
1
G(q) = − (q, q) = − k|q|k2
2
2
(6.37)
with (·, ·) defined in (6.31). We can then state the following result.
Proposition 6.3. Let qex = (σex , Tex ) and qh = (σh , Th ) corresponding to a compatible function uh .
Then there holds
G(qex − qh ) = sup G(q − qh ) .
(6.38)
q∈Q
12
March 15, 2013 7:22
Proof. For any q ∈ Q we can write
(q − qh , q − qh ) = (q − qex + qex − qh , q − qex + qex − qh )
(6.39)
= (q − qex , q − qex ) + 2(qex − qh , q − qex ) + (qex − qh , qex − qh )
and accounting of (6.32) yields
(q − qh , q − qh ) = (q − qex , q − qex ) + (qex − qh , qex − qh ) ,
(6.40)
k|q − qh |k2 = k|q − qex |k2 + k|qex − qh |k2 ≥ k|qex − qh |k2 .
(6.41)
that is,
Recalling then the definition (6.37) of G it follows
(6.42)
−2G(q − qh ) ≥ −2G(qex − qh )
which finally yields (6.38) after multiplying by −1/2.
Exercise 6.1. Show that (6.40) can also be written as
k|q − qh |k = 2k|qex −
qh + q
|k for all q ∈ Q
2
The stationariety condition expressed by (6.38) represents the dual variational formulation of (6.22) once
we note after (6.31) that
− 2G(qex − qh ) = (qex − qh , qex − qh ) = hσex − σh , σex − σh i + hTex − Th , Tex − Th i
=
Z
(6.43)
L
0
k(x)e,2 (x) + c(x)e2 (x) dx = a(e, e) = −2J (uex − uh ) .
In summary, we have therefore
For any v ∈ V
− 2J (v) ≤ −2inf J (v) = −2J (e) = k|e|k2 = a(e, e)
v∈V
= −2G(qex − qh ) = −2sup G(q − qh ) ≤ −2G(q − qh )
q∈Q
for any q ∈ Q.
(6.44)
Examining (6.44) reveals that upper bounds on the discretisation error can be obtained by constructing
elements of the set Q and evaluating therein the value of the functional G.
6.3
Implicit residual type error estimates in energy norm
From the considerations of the previous section, it follows that we can compute upper bounds to the
energy norm of the error as long as we can build an element of the set Q. We recall that Q is the set of
q = (σ, T ) solutions of the equilibrium equations (6.30) stated all over Ω. In this section we are concerned
with finding ways to avoid this global construction and still retain the bound (6.38). The key observation
is that (6.38) holds even if q(x) = (σ(x), T (x)) is in equilibrium over each element Ωi ∈ T with σ ∈ q
continuous on Ω, meeting the natural boundary conditions of (6.30) and the following condition
n h
i xi
X
σi (x)v(x)
xi−1
i=1
h
i
= σn (xn )v(xn ) − σ1 (x0 )v(x0 )
(6.45)
for any compatible function v, that is, for any v ∈ H1 (Ω) that meets the homogeneous essential boundary
conditions. This follows because with such σ, the orthogonality condition (6.32) still holds. In proving
(6.32) in 1D we have invoked the continuity of σ across the nodes whereas for higher dimensions, one must
ensure the continuity of the normal traction σν across the interelement boundaries, with ν a fixed normal
to the element boundary. Now, a way to ensure that the functions σi solutions of the elemental equilibrium
equations can be patched together can be for instance the following. Denote by ∂T the set of all the edges
of the elements Ωi ∈ T the continuity conditions across γ ∈ ∂T can be enforced by associating a smooth
13
March 15, 2013 7:22
scalar field
gγ : γ ∈ ∂T → R
with each γ ∈ ∂T . In 1D, assigning the function gγ means to assign a family of values {gγ (xi ) i = 1, 2, ..., n}
for the value of σ at the nodes. The choice of gγ is arbitrary subject to the constraints to meet the natural boundary conditions on the part of ∂Ω where they have been prescribed and to ensure existence of
solutions for the following problem. Given a scalar field gγ , consider on each element Ωi ∈ T the following
problem
′
(x) + Ti,gγ (x) = f (x) for all x ∈ Ωi =]xi−1 , xi [
− σi,g
γ
σi,gγ (xi−1 ) = −gi (xi−1 ) = −ξi (xi−1 )gγ (xi−1 )
(6.46)
σi,gγ (xi ) = gi (xi ) = ξi (xi )gγ (xi ),
with gi (xi ) = ξgγ (xi ) where


+1 at the node xi−1
ξi = −1 at the node xi


+1 at the nodes x0 and xn , i.e. on the boundary of the domain
which ensures continuity across the nodes of the function σ built element by element by means
Q of (6.46).
QΩi ,gγ the
For the given gγ , let QΩi ,gγ denote the set of solutions qi,gγ = (σi,gγ , Ti,gγ ) of (6.46) and
Ωi ∈T
set of functions q = (σ, T ) such that the restriction over each Ωi is qi,gγ , that is, for x ∈ Ωi q(x)|Ωi =
qi,gγ (x) = (σi,gγ (x), Ti,gγ (x)). We have then the following result.
Q
QΩi ,gγ . If σ ∈ q then σ satisfies condition (6.45).
Proposition 6.4. Let q ∈
Ωi ∈T
Proof. It follows at once by observing that for any compatible v it is
n h
i xi
X
σi (x)vi (x)
i=1
xi−1
n h
i xi
X
gi (x)vi (x)
=
xi−1
i=1
= gγ (x0 )v0 (x0 ) +
n−1
X
i=1
gγ (xi )[vi+1 (xi ) − vi (xi )] + gγ (xn )vn (xn )
= σn (xn )v(xn ) − σ1 (x0 )v(x0 ) .
As result of Proposition 6.4, we can then conclude that (6.38) will hold also with elements of
Q
Ωi ∈T
that is,
G(q − qh ) ≤ G(qex − qh )
∀q ∈
Y
Ωi ∈P
QΩi ,gγ ,
(6.47)
QΩi ,gγ .
For the additivity property of the integral, from (6.44) and recalling the definition (6.37), we obtain
2
k|e|k ≤ −2
+
n Z
X
i=1
xi
xi−1
n
X
i=1
Gi (qi − qh,i ) =
n Z
X
i=1
xi
xi−1
2
1 Ti (x) − Th,i (x) dx
c(x)
2
1 σi (x) − σh,i (x) dx
k(x)
for any q ∈
n
Y
i=1
(6.48)
QΩi ,gγ .
where Gi is defined as (6.37) with the integration limits equal to xi−1 and xi .
Qn
By choosing the function gγ and solving (6.46) in each Ωi we can build in this way elements q ∈ i=1 QΩi ,gγ
with q|Ωi = (σi , Ti ). By evaluating G at this q, according to (6.48) we obtain an upper bound to the energy
norm of error. It is the different
choice of the function gγ and of the criterion to pick the ‘best’ element q
Q
Qi,gγ to deliver different error estimates and to determine its sharpness.
within the resulting set
Ωi ∈P
Remark 6.3. In (6.48) the localization of the upper bound has been obtained by simply using the additivity
of the integral. Consequently, no link is established between global error estimate and solution of element
by element complementary problems. However, the primal hybrid variational formulation of the primal
14
March 15, 2013 7:22
problem (6.22), where one relaxes the interelement continuity of the flux, leads to the following inequality
(for details see [1]).
X
k|e|k2 = −2J (e) ≤ −2
(6.49)
inf J Ωi ,gγ (v) for all Ωi ∈ T
1
Ωi ∈T v∈H (Ωi )
with
1
J Ωi ,gγ (v) = ai (v, v) − ℓi (v) + ai (uh , v) +
2
I
gi vds.
∂Ωi
Applying the complementary principle to the minimization of the functional J Ωi ,gγ (v) we, therefore, find
inf
v∈H 1 (Ωi )
J Ωi ,gγ (v) =
sup
qi,gγ ∈QΩi ,gγ
Gi (qi,gγ − qh ),
thus,
k|e|k2 = −2G(qex − qh ) ≤ −2
≤ −2
X
Ωi ∈T
X
sup
Ωi ∈T qi,gγ ∈QΩi ,gγ
Gi (qi,gγ − qh ) for any q ∈
Y
Ωi ∈T
Gi (qi,gγ − qh )
QΩi ,gγ , for any Ωi ∈ T .
(6.50)
Equation (6.49) links the estimate of the energy norm of the error to the solution of local primal problems
wheras (6.50) links the estimate of the energy norm of the error to the solution of local complementary
problems.
Remark 6.4. Recall that a differential problem stated over the whole domain Ω can be split into local
problems stated over each subdomain belonging to a decomposition T of Ω. Each of these local problems
defines the restriction of the exact solution to the subdomain, provided that suitable boundary conditions
are given on the interelement boundaries. It is the definition of the boundary conditions the most delicate
point as it involves the knowledge of the value of the exact solution (which is the unknown of the problem
we want to solve) or the value of its normal flux across the interior boundaries.
This observation may also be applied to the differential problem which defines the discretization error.
Thus, local problems can be stated over each subdomain such that their solution gives the restriction to
the subdomain of the error. For the generic element Ωi =]xi−1 , xi [ the local problem would read indeed as
follows
−(k(x)ψi′ (x))′ + c(x)ψi (x) = Ruh,i (x) for x ∈ Ωi =]xi−1 , xi [
ψi′ (xi−1 ) = e′i (xi−1 )
ψi′ (xi )
=
(6.51)
e′i (xi ),
where Ruh,i is the pointwise residual and ei (x) = uex (x) − uh,i (x) is the discretization error.
Due to the objective difficulty in stating the boundary conditions, the previous aim is difficult to pursue,
hence we can enquiry if it is possible to modify the local problems so that their solution can give still useful
information at least on the estimate of the error.
By looking at the estimate (6.49) we see that upper bounds on the discretization error have been linked to
the solution of element by element primal problems given by the solution of the infinimum of the functionals
J Ωi ,gγ (v) over H 1 (Ωi ). Hence, the strong form corresponding to each of the local minimizations gives how
the local problems have to be posed, which result in local Neumann boundary value problems. Thus, (6.49)
provides the tool to evaluate and state local problems so that their solution can give useful results for the
estimation of the error in energy norm.
Remark 6.5. From (6.50) follows that exact solution of local complementary problems would detect the
element of QΩi ,gγ which would give the sharpest estimate that one could get with any other element of
QΩi ,gγ . It is also clear that the estimate given by the solution of the local complementary problems could
not be sharp by itself unless the set QΩi ,gγ , and hence the function gγ has been appropriately chosen. In
fact, in (6.50) the sign equal holds if the function gγ is chosen equal to the normal flux of the exact solution
u across the interelement boundaries. With this choice
Q of the function gγ , qex is one of the solutions of
QΩi ,gγ . Furthermore, because of (6.50), nothing
the problem (6.46), hence it belongs to the resulting
Ωi ∈T
15
March 15, 2013 7:22
can be said in general on the relationship between solution of local problems and the restriction of the
error to the subdomain, that is, the solution of local problems does not give any type of information on the
restriction of the error to the subdomain. However, if gγ is chosen equal to the normal flux of the exact
solution across the boundaries, then the solution of the local problem is equal to the restriction of the error
to the subdomain Ωi .
Remark 6.6. The complementary principle appears as a tool by which we can obtain upper bounds on the
energy norm of the error without actually solving any complementary
Q problem. We just need to evaluate
QΩi ,gγ . Hence, the error estimates
the value of the functionals Gi (qi,gγ − qh ) at a suitable element q ∈
Ωi ∈T
presented in literature based on complementary
principle are obtained using the (6.48) and providing a
Q
QΩi ,gγ , given by the solution of (6.46). The price to pay for using
procedure to get an element q ∈
Ωi ∈T
(6.48) consists in the solution of the problem (6.46) and only estimate of the error in energy norm are
provided.
6.3.1
An application of an implicit residual error estimate
In this section we consider the error estimate developed in [2] with a = 0. Hence, the following equation
is analysed
− [k(x)u′ (x)]′ = f (x) for x ∈]0, L[
u(0) = u0 ,
u(L) = uL .
(6.52)
The inequality (6.48), which gives the upper bound on the energy norm of the error, becomes
k|e|k2 ≤ −2
n
X
i=1
Gi (σi − σh,i ) =
n Z
X
i=1
xi
xi−1
n
n
Y
X
2
1 QΩi ,gγ , (6.53)
ηi2 for all σ ∈
σi (x) − σh,i (x) dx =
k(x)
i=1
i=1
where we have let
ηi2 = −2Gi (σi − σh,i ) =
which are referred to as local error indicators.
Z
xi
xi−1
2
1 σi (x) − σh,i (x) dx,
k(x)
To use (6.53), we need, therefore,
n
Y
QΩi ,gγ and
(i) to define the set
i=1
(ii) to give the criterion to pick an element σ of the set
integrals −2Gi (σi − σh,i ).
Qn
i=1
(6.54)
QΩi ,gγ where then we will evaluate the
In the remaining part of this section, we show how for the problem (6.52) the above two questions are
tackeld with.
The set QΩi ,gγ is defined by (6.46) which for (6.52) reads as
′
(x) = f (x)
− σi,g
γ
x ∈ Ωi =]xi−1 , xi [
σi,gγ (xi−1 ) = −gi (xi−1 ) = −ξi (xi−1 )gγ (xi−1 )
(6.55)
σi,gγ (xi ) = gi (xi ) = ξi (xi )gγ (xi ) .
This problem is defined once a function gγ is assigned at the interelement boundaries upon the condition
that gγ meet the natural boundary conditions and (6.55) have solution. Since T = 0, the latter condition
does make sense. It expresses the data equilibrationfor apure Neumann problem and is obvious from the
structure of the differential problem (6.55). For being of the first order, only one initial value is necessary.
Thus, in order to problem (6.55) have solutions an additional condition involving the boundary data is
expected. It is immediate to find this condition, in fact, integrating (6.55)1 over Ωi yields
Z x
f (ξ) dξ for x ∈ Ωi ,
σi (x) = σi (xi−1 ) −
xi−1
16
March 15, 2013 7:22
hence, the following condition must hold
σi (xi ) = σi (xi−1 ) −
Z
xi
f (x)dx ,
(6.56)
xi−1
that is,
gγ (xi ) + gγ (xi−1 ) = −
Z
xi
f (x)dx .
(6.57)
xi−1
Thus, by giving the function gγ , which in 1D consists of the family of values {g(xi ); i = 1, . . . , n}, and
provided that one meets condition (6.57) and possible natural boundary conditions, problem (6.55) is
defined and it can be solved.
In 1D assigning gγ at the nodes is meaningful if essential boundary conditions are given at both the ends
of Ω =]0, L[. If at one of the ends a natural boundary condition is given, then the respect of this condition
along with the equilibration condition (6.57), defines unambiguously the values of gγ at all the nodes
recursively. In fact, in this case problem (6.52) can be split into a system of two indipendent differential
problems of the first order which can be solved separately. This circumstance describes a so-called statically
determined system,. It then follows that there exists only one equilibrated stress field which is the exact
one. For instance, if at x = L the natural boundary condition is given, say σL = k(L)u′L , it is gγ (L) = σL .
By accounting then for (6.57) consequent consistent values of gγ are obtained for the other nodes. The
equilibrated stress field, corresponding to the natural boundary condition σ(L) = σL prescribed at x = L
is then given as follows
Z x
f (ξ) dξ, for x ∈ Ωi =]xi−1 , xi [
σi (x) = σi (xi−1 ) −
xi−1
where
σi (xi−1 ) = σi (xi ) +
Z
xi
xi−1
with
f (x)dx i = {n, n − 1, . . . , 1}
(6.58)
σn (xn ) = σL
The indetermination in defining gγ arises if essential boundary conditions are given at both the ends of
Ω. In this case, which corresponds to a statically indetermined system, more equilibrated stress fields can
be defined and a choice must be done for defining a family of values of gγ . The function gγ is required to
meet only condition (6.57). In 1D, once gγ is given, problem (6.55) can be solved and it admits a unique
solution, hence the set QΩi ,gγ reduces to only one element. This will not be the case in higher dimensions
when the analogous of problems (6.55) will result indetermined.
In summary, if a natural boundary condition is given at one of the ends of Ω, say for instance at x = L,
then the unique equilibrated stress field is given by (6.58) which is also the exact stress field, that is
σ = σex . In this case the error estimate coincides with the energy norm of the error.
In order to provide more generality, we refer next to the case where equilibrated stress fields are indetermined, that is, when essential boundary conditions are given at both the ends of the domain Ω. In this
case, a function gγ must be chosen. Let Vh (Ωi ) denote the finite element space generated by the shape
functions defined over Ωi , in [2] gγ is introduced so that it does meet the following condition
Z xi
Z xi
h
i xi
′
f (x)v(x)dx for any v ∈ Vh (Ωi ).
(6.59)
σh,i (x)v (x)dx −
σi (x)v(x)
=
xi−1
xi−1
xi−1
which is obtained by imposing the orthogonality of σ − σh to the local space Vh (Ωi ), that is,
Z xi
σi (x) − σh,i (x) v ′ (x)dx = 0 for all v ∈ Vh (Ωi ) .
(σi − σh,i , v ′ )L2 (Ωi ) =
(6.60)
xi−1
Equation (6.60) says that σi is the local L2 projection onto Vh (Ωi ).
Condition (6.59) defines the values of the function σ at the nodes xi as result of the finite dimensionality of
the space Vh and of the linearity of condition (6.59) with respect to v. In fact, equation (6.59) is equivalent
17
March 15, 2013 7:22
to the following two conditions
σi (xi−1 ) = −
σi (xi ) =
xi
Z
xi
Z
σh,i (x)ϕ′h,xi−1 (x)dx
xi−1
σh,i (x)ϕ′h,xi (x)dx −
xi−1
Z
+
xi
Z
f (x)ϕh,xi−1 (x)dx
(6.61)
xi−1
xi
f (x)ϕh,xi (x)dx
(6.62)
xi−1
with ϕh,xi−1 and ϕh,xi the element lagrangian shape functions.
Remark 6.7. It is interesting to compare condition (6.60) to the orthogonality condition of the discretization error σex − σh to Vh (Ω) expressed as
(σex − σh , v ′ )L2 (Ω) =
L
Z
0
σex (x) − σh (x) v ′ (x) dx = 0 for all v ∈ Vh (Ω).
(6.63)
This means that the equilibrated stress field σ satisfying condition (6.60) is such that σ − σh is solution
of a problem similar to problem (6.21) characterising the discretization error, in the sense that at r.h.s.
of (6.21) there will still appear a functional vanishing over Vh (Ω) which is, however, different from the
residual functional.
Condition (6.59), albeit local, is consistent with the continuity of σ across the nodes, that is, it is
σi−1 (xi−1 ) = σi (xi−1 ).
Proof. Since it is
σi−1 (xi−1 ) =
σi (xi−1 ) = −
Z
xi−1
xi−2
Z xi
xi−1
σh,i−1 (x)ϕ′h,xi−1 (x)dx −
σh,i (x)ϕ′h,xi−1 (x)dx +
it follows
σi−1 (xi−1 ) − σi (xi−1 ) =
Z
Z
Z
xi
f (x)ϕh,xi−1 (x)dx
xi−1
f (x)ϕh,xi−1 (x)dx
xi−1
xi
xi−2
(6.64)
xi
σh (x)ϕ′h,xi−1 (x)dx −
Z
xi
f (x)ϕh,xi−1 (x)dx
(6.65)
xi−2
which is equal to zero, because the residual functional vanishes over Vh (Ω).
Replacing (6.61) and (6.62) in (6.56) shows that the choice suggested in [2] satisfies the equilibration
condition
Z xi
Z xi
′
f (x)ϕh,xi (x)dx +
σh,i (x)ϕh,xi (x)dx −
σi (xi ) − σi (xi−1 ) =
xi−1
xi
+
=
Z
xi−1
Z xi
xi−1
xi
−
Z
xi−1
and since,
xi−1
Z xi
σh,i (x)ϕ′h,xi−1 (x)dx −
f (x)ϕh,xi−1 (x)dx =
xi−1
′
σh,i (x) ϕh,xi−1 (x) + ϕh,xi (x) dx +
f (x) ϕh,xi−1 (x) + ϕh,xi (x) dx
ϕh,xi−1 (x) + ϕh,xi (x) = 1 for x ∈ Ωi =]xi−1 , xi [
(6.66)
(6.67)
the equilibration condition (6.56) is therefore identically satisfied.
In summary, the functions σi (x) to use in (6.53) are obtained as follows. If a natural boundary condition
is given at one of the ends of Ω, say at x = L, then the functions σi are given by equation (6.58); on the
other hand, if essential boundary conditions are given at both the ends of Ω, the functions σi are solution
18
March 15, 2013 7:22
of the following local problems,
− σi′ (x) = f (x) for x ∈ Ωi =]xi−1 , xi [
Z
Z xi
σh,i (x)ϕ′h,xi−1 (x)dx +
σi (xi−1 ) = −
σi (x)
Z x
σi′ (ξ)dξ =
σi (xi−1 ) +
xi−1
Z
Z xi
′
σh,i (x)ϕh,xi−1 (x)dx +
−
=
=
f (x)ϕh,xi−1 (x)dx.
xi−1
xi−1
that is,
(6.68)
xi
xi
xi−1
xi−1
f (x)ϕh,xi−1 (x)dx −
Z
(6.69)
x
f (ξ)dξ.
xi−1
Remark 6.8. The sharpness of the estimate depends on the choice of the equilibrated stress field σ. Now,
a condition which we would like to have, though no essential, can be that if it happens u′h = u′ex , so that the
error is zero, then also the estimate vanishes. This means to require that if σh happens to be equilibrated,
then the procedure that defines the equilibrated stress field σ should be such that it delivers σ = σh . This
condition is satisfied in 1D by the equilibrated stress field as defined in (6.69).
′
Proof. For f (x) = −σh,i
(x) over Ωi , equation (6.69) becomes
σi (x)
=
=
−
Z
xi
xi−1
σh,i (x)ϕ′h,xi−1 (x)dx −
xi
− σh,i (x)ϕh,xi−1 (x)
xi
xi−1
′
σh,i
(x)ϕh,xi−1 (x)dx +
x
+ σh,i (x)
xi−1
=
Z
Z
x
xi−1
′
σh,i
(ξ)dξ =
=
xi−1
σh,i (xi−1 ) + σh,i (x) − σh,i (xi−1 ) = σh,i (x)
that is the equilibrated stress coincides with the finite element solution if the latter is equilibrated.
If we let
pi = σi − σh,i for every Ωi ∈ T
(6.70)
with σi solution of (6.68), it might be more appropriate to transform (6.68) into another one which has
pi = pi (x) as solution. The differential problem which characterises the functions pi useful to deliver error
estimates is therefore given as follows.
′
′
−σi′ (x) + σh,i
(x) = f (x) + σh,i
= Ruh,i (x) for x ∈ Ωi = (xi−1 , xi ), for Ωi ∈ T
with boundary condition given by
pi (xi−1 ) = σi (xi−1 ) − σh,i (xi−1 ).
By accounting of (6.68)2 we obtain
Z
Z xi
σh,i (x)ϕ′h,xi−1 (x)dx +
pi (xi−1 ) = −
xi
xi−1
xi−1
f (x)ϕh,xi−1 (x)dx − σh,i (xi−1 ) ,
and applying integration by parts to the first integral yields
Z xi
i xi
h
σh,i (x)ϕ′h,xi−1 (x)dx = σh,i (x)ϕh,xi−1 (x)
xi−1
xi−1
that is,
pi (xi−1 ) =
Z
−
Z
xi
xi−1
′
σh,i
(x)ϕh,xi−1 (x)dx
xi
Ruh,i (x)ϕh,xi−1 (x)dx.
xi−1
Hence, in order to obtain an error estimate p must be solution of the following differential problem
− p′i (x) = Ruh,i (x) ∀x ∈ Ωi = (xi−1 , xi )
Z xi
Ruh,i (x)ϕh,xi−1 (x)dx.
pi (xi−1 ) =
xi−1
19
March 15, 2013 7:22
which gives,
pi (x) = pi (xi−1 ) −
that is,
pi (x) =
Z
ηi2 =
Z
xi
xi−1
1 h
k(x)
x
Ruh,i (ξ)dξ.
xi−1
xi
xi−1
so that,
Z
Z
Ruh,i (ξ)ϕh,xi−1 (ξ)dξ −
x
Z
Ruh,i (ξ)dξ.
xi−1
xi
xi−1
Ruh,i (ξ)ϕh,xi−1 (ξ)dξ −
Z
x
Ruh,i (ξ)dξ
xi−1
i2
dx .
Remark 6.9. The differential problem (6.71) applies also for i = 1, though the shape function ϕh,x0 =
ϕh,x0 (x) does not belong to the finite element space.
In fact, for the element Ω1 =]x0 , x1 [ the problem to solve should be the following
− p′1 (x) = Ruh,1 (x) ∀x ∈ Ω1 = (x0 , x1 )
Z x1
Ruh,1 (x)ϕh,x1 (x)dx .
p1 (x1 ) = −
(6.71)
x0
However, it results
Z x1
Ruh,1 (x)ϕh,x0 (x)dx +
Z
x1
Ruh,1 (x)ϕh,x1 (x)dx =
x1
Ruh,1 (x)dx =
x0
x0
x0
Z
h
i x1
= p1 (x0 ) − p1 (x1 )
= − p1 (x)
(6.72)
x0
that is,
p1 (x0 ) = p1 (x1 ) +
Z
x1
Ruh,1 (x)dx =
x0
Z
x1
Ruh,1 (x)ϕh,x0 (x)dx
x0
thus, even for i = 1 the differential problem (6.71) may be expressed as
− p′1 (x) = Ruh,1 (x) ∀x ∈ Ω1 = (x0 , x1 )
Z x1
Ruh,1 (x)ϕh,x0 (x)dx.
p1 (x0 ) =
x0
which is obtained by (6.71) for i = 1.
Remark 6.10. In [2] no link is given with the solution of local complementary problems. The gists of
the procedure is to establish the complementary problem characterising the
Q global discretization error and
indicate a strategy, introduced by assigning gγ , to detect the element q ∈ ni=1 QΩi ,gγ which would deliver
the best estimate.
References
[1] Ainsworth M., Oden J.T., A posteriori error estimation in finite element analysis, Comput. Methods
Appl. Mech. Engrg. 142 (1997) 1-88
[2] Ladeveze P., Leguillon D., Error estimate procedure in the finite element method and applications,
SIAM J. Numer. Anal. 30 (1983) 485-509
20
March 15, 2013 7:22