Part IV, Chapter 20
Galerkin approximation
In the previous chapter, we have presented conditions that guarantee the wellposedness of the following model problem:
Find u ∈ V such that
(20.1)
a(u, w) = ℓ(w), ∀w ∈ W,
where V and W are Banach spaces, a is a bounded bilinear (or sesquilinear)
form on V ×W , and ℓ is a bounded (anti)linear form on W . In this chapter,
we study the approximation of (20.1) by the Galerkin method. The starting point is to replace the infinite-dimensional spaces V and W in (20.1) by
finite-dimensional spaces Vh and Wh , and possibly the forms a and ℓ by some
approximate forms ah and ℓh defined on Vh ×Wh and Wh , respectively. The
subscript h refers to the characteristic size of the mesh that is used to build
the spaces Vh and Wh ; examples can be found in Chapters 13 and 15. We give
discrete versions of the Lax–Milgram Lemma and the BNB Theorem, which
are respectively, sufficient conditions, and necessary and sufficient conditions
for well-posedness of the approximate problem. Finally, we derive a bound
on the approximation error (u − uh ) in a simple setting; the thorough error
analysis is postponed to Chapter 21.
20.1 Setting
The discrete problem takes the following form:
Find uh ∈ Vh such that
ah (uh , wh ) = ℓh (wh ), ∀wh ∈ Wh ,
(20.2)
where ah is a bounded bilinear (or sesquilinear) form on Vh ×Wh and ℓh is
a bounded (anti)linear form on Wh . Since the spaces Vh and Wh are finitedimensional, (20.2) is called a discrete problem. The space Vh is called the
discrete solution space (or trial space), and Wh the discrete test space.
262
Chapter 20. Galerkin approximation
Definition 20.1 (Standard Galerkin, Petrov–Galerkin). The Galerkin
approximation is said to be standard when Wh = Vh in the discrete problem (20.2). Otherwise, (20.2) is called a Petrov–Galerkin approximation or a
nonstandard Galerkin approximation.
Definition 20.2 (Conformity). The approximation is said to be conforming
if Vh ⊂ V and Wh ⊂ W .
There are circumstances when considering nonconforming approximations
is useful. Two important examples are discontinuous Galerkin methods where
discrete functions are discontinuous across the mesh interfaces (see Chapters 30 and 45) and boundary penalty methods where boundary conditions
are enforced weakly (see Chapters 29 and 44). Nonconforming approximations
make it often necessary to work with discrete bilinear (and sometimes linear)
formsRthat differ from their continuous counterparts. For instance, the bilinear
form D ∇v·∇w dx does not make sense if its arguments are discontinuous.
Another important example leading to a modification of the bilinear and linear
forms at the discrete level is the use of quadratures (see Chapter 26).
20.2 Discrete well-posedness
Our goal in this section is to study the well-posedness of the discrete problem (20.2). We equip the discrete spaces Vh and Wh with norms denoted k·kVh
and k·kWh . These norms can differ from those of V and W ; one reason can be
that the approximation is nonconforming.
20.2.1 Discrete Lax–Milgram
Lemma 20.3 (Discrete Lax–Milgram). Let Vh be a finite-dimensional
space over R. Set Wh = Vh in problem (20.2). Let ah be a bounded bilinear form on Vh ×Vh and let ℓh be a bounded linear form on Vh . Assume that
the bilinear form ah is coercive on Vh , i.e., there is a real number αh > 0 and
ξ = ±1 such that
ξah (vh , vh ) ≥ αh kvh k2Vh ,
∀vh ∈ Vh .
(20.3)
Then, the problem (20.2) is well-posed and the a priori estimate kuh kVh ≤
1
′
αh kℓh kVh holds. In the complex case, the same conclusions hold if there is a
real number αh > 0 and a complex number ξ with |ξ| = 1 such that
ℜ (ξah (vh , vh )) ≥ αh kvh k2Vh ,
∀vh ∈ Vh .
(20.4)
Proof. We give the proof in the real case; the complex case is treated similarly. Let Ah : Vh → Vh′ be the linear operator such that hAh vh , wh iVh′ ,Vh =
Part IV. Galerkin Approximation
263
ah (vh , wh ) for all vh , wh ∈ Vh . Well-posedness of (20.2) is equivalent to Ah being an isomorphism, which owing to the finite-dimensionality of Vh , is equivalent to Ah being injective, i.e., ker(Ah ) = {0}, since dim(Vh ) = dim(Vh′ ) < ∞.
Let vh ∈ ker(Ah ), then 0 = ξhAh vh , vh iVh′ ,Vh = ξah (vh , vh ) ≥ αh kvh k2Vh , which
proves that vh = 0. Hence, ker(Ah ) = {0}.
Remark 20.4 (Variational formulation). As in the continuous setting,
see Proposition 19.8, if ah is coercive (with ξ = 1) and symmetric in the real
case, uh solves (20.2) with Wh = Vh if and only if uh minimizes the functional
Eh (vh ) = 21 ah (vh , vh ) − ℓh (vh ) over Vh . In the conforming case where Vh ⊂ V ,
ah = a, and ℓh = ℓ, we observe that Eh (uh ) = E(uh ) ≥ E(u).
⊓
⊔
20.2.2 Discrete BNB
Theorem 20.5 (Discrete BNB). Let Vh and Wh be finite-dimensional
spaces over R. Let ah be a bilinear form on Vh ×Wh and let ℓh be a linear
form on Wh . Then, problem (20.2) is well-posed if and only if
inf
sup
vh ∈Vh wh ∈Wh
ah (vh , wh )
=: αh > 0,
kvh kVh kwh kWh
dim(Vh ) = dim(Wh ).
(20.5a)
(20.5b)
Moreover, the a priori estimate kuh kVh ≤ α1h kℓh kWh′ holds. Condition (20.5a)
is equivalent to the following inf-sup condition:
∃αh > 0,
αh kvh kVh ≤ sup
wh ∈Wh
ah (vh , wh )
,
kwh kWh
∀vh ∈ Vh .
(20.6)
In the complex case, ah (vh , wh ) must be replaced by ℜ(ah (vh , wh )) in the numerator in (20.5a).
Proof. We give the proof in the real case; the complex case is treated similarly.
Let Ah : Vh → Wh′ be the linear operator such that
hAh vh , wh iWh′ ,Wh = ah (vh , wh ),
∀(vh , wh ) ∈ Vh × Wh .
(20.7)
The well-posedness of (20.2) is equivalent to Ah being an isomorphism, which
owing to the finite-dimensional setting and the Rank Theorem, is equivalent
to the following conditions:
ker(Ah ) = {0} (i.e., Ah is injective),
dim(Vh ) = dim(Wh′ ).
(20.8a)
(20.8b)
Since dim(Wh ) = dim(Wh′ ), (20.5b) is equivalent to (20.8b). Let us prove
that (20.8a) is equivalent to the inf-sup condition (20.5a). Assume first
that (20.5a) holds and let vh ∈ Vh be such that Ah vh = 0. Then,
264
Chapter 20. Galerkin approximation
αh kvh kVh ≤ sup
wh ∈Wh
hAh vh , wh iWh′ ,Wh
= 0,
kwh kWh
whence vh = 0. Conversely, assume ker(Ah ) = {0} and let us prove the inf-sup
condition (20.5a). An equivalent statement of (20.5a) is that there is n0 ∈ N∗
such that for all Vh ∈ Vh with kvh kVh = 1, supwh ∈Wh
1
n0 .
hAh vh ,wh iW ′ ,W
h
h
kwh kWh
>
Reasoning by contradiction, consider a sequence (vhn )n∈N in Vh with
hAh vhn ,wh iW ′ ,W
h
h
≤ n1 . Since Vh is finite-dimensional,
kvhn kVh = 1 and supwh ∈Wh
kwh kWh
its unit sphere is compact. Therefore, up to a subsequence, vhn converges to
a certain vh in Vh . The limit vh satisfies kvh kVh = 1 and ah (vh , wh ) = 0 for
all wh ∈ Wh . The latter property implies vh ∈ ker(Ah ) which contradicts
kvh kVh = 1 since ker(Ah ) = {0}. Finally, the a priori estimate is proved as in
the continuous case.
⊓
⊔
Remark 20.6 (Verifying (20.5)). Since (20.5b) is, in general, simple to
verify, the key property for discrete well-posedness is (20.5a).
⊓
⊔
Remark 20.7 (Discrete vs. continuous BNB Theorems). The inf-sup
condition (20.5a) is nothing but the first condition (bnb1) of the BNB Theorem 19.10. The second condition (bnb2) amounts to
∀wh ∈ Wh ,
[ ah (vh , wh ) = 0, ∀vh ∈ Vh ] =⇒ [ wh = 0 ].
(20.9)
Let us introduce the adjoint operator A∗h : Wh → Vh′ (note that Wh is reflexive,
being finite-dimensional) such that
hA∗h wh , vh iVh′ ,Vh = ah (vh , wh ),
∀(vh , wh ) ∈ Vh × Wh .
(20.10)
Then, (20.9) amounts to A∗h being injective, and this statement is equivalent
to (20.5b) provided (20.5a) holds; see Exercise 20.1.
⊓
⊔
Remark 20.8 (Well-posedness of discrete adjoint problem). Consider
the adjoint operator A∗h : Wh → Vh′ defined in (20.10). Then, Ah is an isomorphism if and only if A∗h is an isomorphism; see Exercise 20.2. In other
words, the well-posedness of the discrete problem (20.2) is equivalent to the
well-posedness of the following adjoint problem:
Find yh ∈ Wh such that
ah (vh , yh ) = gh (vh ), ∀vh ∈ Vh ,
for any gh ∈ Vh′ . Moreover, owing to Corollary A.61 (and since Vh is reflexive,
being finite-dimensional), Ah and A∗h satisfy the inf-sup condition (20.5a) with
the same constant αh , i.e.,
inf
sup
vh ∈Vh wh ∈Wh
hAh vh , wh iWh′ ,Wh
hAh vh , wh iWh′ ,Wh
= inf sup
.
wh ∈Wh vh ∈Vh kvh kVh kwh kWh
kvh kVh kwh kWh
(20.11)
(Note that the numerator of the right-hand side equals hA∗h wh , vh iVh′ ,Vh .)
⊓
⊔
Part IV. Galerkin Approximation
265
20.3 Fortin’s criterion(♦)
Consider a conforming approximation, Vh ⊂ V and Wh ⊂ W , and assume that
the discrete problem (20.2) is formulated using the same bilinear form as in
the continuous problem, i.e., ah = a. Then, if a is coercive on V with coercivity
parameter α, a also coercive on Vh with coercivity parameter αh = α, and
the discrete problem (20.2) with Wh = Vh is automatically well-posed. In the
more general case of the BNB Theorem 19.10, the situation is different since
neither condition (bnb1) nor condition (bnb2) imply its discrete counterpart.
h ,w)
For instance, since Vh ⊂ V , (bnb1) implies that supw∈W a(v
kwkW ≥ αkvh kV
for all vh ∈ Vh , but it is not clear that the bound still holds if the argument in
the supremum is taken in the subspace Wh ⊂ W . This means that the inf-sup
condition (20.5a) needs to be proved. A powerful tool to achieve this is due
to Fortin [235]; see also Boffi et al. [64, Prop. 5.4.3]. Here, we consider the
Banach space setting and we include a converse statement.
Lemma 20.9 (Fortin’s criterion). Let V and W be two real Banach spaces
and let a ∈ B(V, W ). Assume that a satisfies the inf-sup condition (bnb1)
with constant α > 0. Let Vh ⊂ V and let Wh ⊂ W be equipped with the norms
of V and W , respectively. Consider the following two statements:
(i) There exists a mapping Πh : W → Wh and a real number γΠ > 0 such
that a(vh , Πh w − w) = 0, for all (vh , w) ∈ Vh × W , and γΠ kΠh wkW ≤
kwkW for all w ∈ W .
(ii) a satisfies the inf-sup condition
inf
sup
vh ∈Vh wh ∈Wh
a(vh , wh )
≥ β > 0,
kvh kV kwh kW
(20.12)
In the complex case, a(vh , wh ) is replaced by ℜ(a(vh , wh )) in (20.12).
Then, (i) ⇒ (ii) with β = γΠ α and (ii) ⇒ (i) with γΠ =
β
kakB(V,W ) .
Proof. We give the proof in the real case; the complex case is treated similarly.
(1) Assuming (i), we have
sup
wh ∈Wh
a(vh , wh )
a(vh , Πh w)
a(vh , w)
≥ sup
= sup
kwh kW
w∈W kΠh wkW
w∈W kΠh wkW
≥ γΠ sup
w∈W
a(vh , w)
≥ γΠ α kvh kV ,
kwkW
since a satisfies (bnb1) and Vh ⊂ V . This proves (20.12) with β = γΠ α.
(2) Conversely, assume that a satisfies (20.12). Let Ah : Vh → Wh′ be the
operator defined in (20.7), i.e., hAh vh , wh iWh′ ,Wh = a(vh , wh ). We now apply
the converse of Lemma A.57 with Wh in lieu of V , Vh′ in lieu of W , and
A∗h in lieu of A; note that we identify Vh and Vh′′ , Wh and Wh′′ , since Vh
and Wh are finite-dimensional. According to Lemma A.57, (20.12) implies
266
Chapter 20. Galerkin approximation
that for all θh ∈ Vh′ , there exists Φh (θh ) ∈ Wh such that A∗h Φh (θh ) = θh
and βkΦh (θh )kW ≤ kθh kVh′ . Let w ∈ W and define θh (w) ∈ Vh′ by setting
hθh (w), vh iVh′ ,Vh := a(vh , w) for all vh ∈ Vh . Defining Πh (w) = Φh (θh (w))
yields
a(vh , Πh (w)) = hAh vh , Φh (θh (w))iWh′ ,Wh = hA∗h Φh (θh (w)), vh iVh′ ,Vh
= hθh (w), vh iVh′ ,Vh = a(vh , w),
which proves that a(vh , Πh (w) − w) = 0 for all w ∈ W . Moreover
βkΠh (w)kW = βkΦh (θh (w))kW ≤ kθh (w)kVh′ ≤ kakB(V,W ) kwkW ,
which proves that
β
kakB(V,W ) kΠh (w)kW
≤ kwkW .
⊓
⊔
Remark 20.10 (Equivalence, linearity). Only the forward implication
in Fortin’s criterion is usually found in the literature. Note that there is a
gap in the stability constant γΠ between the direct and converse statements,
kakB(V,W )
since the ratio of the two is equal to
(which is independent of the
α
discrete setting). Moreover, in the converse statement, the mapping Πh can be
nonlinear (since the mapping Φh can be nonlinear). The existence of a linear
mapping is granted in a Hilbertian setting, see Remark A.58.
⊓
⊔
20.4 Basic error estimate
In this section, we bound the approximation error (u − uh ). We consider the
relatively simple setting of a conforming approximation (Vh ⊂ V and Wh ⊂
W ) with ah = a and ℓh = ℓ. We assume that the inf-sup condition (20.5a)
holds, as well as dim(Vh ) = dim(Wh ), so that the discrete problem (20.2) is
well-posed. Simple sufficient conditions for (20.5a) to hold are Wh = Vh and
a coercive on Vh (for which a sufficient condition is that a be coercive on V ).
Lemma 20.11 (Galerkin’s orthogonality). Let u solve (20.1) and let uh
solve (20.2) with Vh ⊂ V , Wh ⊂ W , ah = a, and ℓh = ℓ. Then, the following
holds:
a(u − uh , wh ) = 0,
∀wh ∈ Wh .
(20.13)
Proof. For all wh ∈ Wh , we observe that a(u, wh ) = ℓ(wh ) since Wh ⊂ W and
ℓ(wh ) = a(uh , wh ) since ah = a and ℓh = ℓ.
⊓
⊔
The starting point of the error analysis is to make sure that (20.2) is
consistent with the original problem (20.1). Loosely speaking, consistency is
checked by inserting the exact solution into the discrete problem and verifying
that the result is small. In the present situation, we infer that a(u, wh ) = l(wh )
for all wh ∈ Wh (since ah = a, ℓh = ℓ, and Wh ⊂ W ), i.e., the approximate
Part IV. Galerkin Approximation
267
problem is exactly consistent with the continuous one. This exact consistency
property is just a reformulation of Galerkin’s orthogonality.
Let us now bound the approximation error. Since the approximation
setting is conforming, we equip the subspaces Vh ⊂ V and Wh ⊂ W
with the norms k·kV and k·kW , respectively. Recall that kakB(V,W ) :=
.
sup(v,w)∈V ×W kvka(v,w)
V kwkW
Lemma 20.12 (Basic error estimate). Assume that problems (20.1) and
(20.2) are well-posed with solutions u and uh . Assume that the approximation
setting is conforming with ah = a and ℓh = ℓ. Let αh > 0 be the constant in
the inf-sup condition (20.5a). Then, the following holds:
kakB(V,W )
inf ku − vh kV .
(20.14)
ku − uh kV ≤ 1 +
αh
vh ∈Vh
Consequently, if approximability holds in the sense that
lim inf kv − vh kV = 0,
h→0 vh ∈Vh
∀v ∈ V,
(20.15)
then limh→0 ku − uh kV = 0.
Proof. Let vh ∈ Vh . Using stability (i.e., the inf-sup condition (20.5a)), consistency (i.e., Galerkin’s orthogonality), and the boundedness of a, we infer
that
αh kuh − vh kV ≤ sup
wh ∈Wh
= sup
wh ∈Wh
a(uh − vh , wh )
kwh kW
a(u − vh , wh )
≤ kakB(V,W ) ku − vh kV .
kwh kW
⊓
⊔
We conclude using the triangle inequality.
Lemma 20.12 is derived in Babuška and Aziz [33]. Note that (20.14) implies
that if the exact solution turns out to be in Vh , then uh = u. Whenever V
and W are Hilbert spaces, the error estimate can be sharpened as follows:
kakB(V,W )
(20.16)
inf ku − vh kV ,
ku − uh kV ≤
αh
vh ∈Vh
see Xu and Zikatanov [495] and Exercise 20.7. Moreover, if the bilinear form
a is coercive (with parameter α), W = V , and Wh = Vh , the error estimate is
known as Céa’s Lemma [136] and takes the form (see Exercise 20.6)
kakB(V,V )
inf ku − vh kV .
(20.17)
ku − uh kV ≤
α
vh ∈Vh
268
Chapter 20. Galerkin approximation
20.5 Stability loss(♦)
The purpose of this section is to illustrate on a simple one-dimensional
transport (advection) equation the loss of stability when applying a naive
Galerkin approximation technique to a first-order PDE. We consider the onedimensional transport equation
u′ = f
in D,
u(0) = 0,
(20.18)
with D := (0, 1) and given f ∈ L2 (D). Following §18.2.3), we consider the L2 based weak formulation with solution space V := {v ∈ H 1 (D) | v(0) = 0} and
R1
test space W := L2 (D) (we drop the superscripts). Setting a(v, w) = 0 v ′ w dt
for all (v, w) ∈ V ×W , the weak problem consists of finding u ∈ V such that
R1
a(u, w) = 0 f w dt for all w ∈ W ; this problem is well-posed, see Exercise 19.8.
Consider H 1 -conforming P1 (Lagrange) finite elements. Let N be a positive
integer. Set h = N1 and xi = ih for all i ∈ {0: N }. Let us use the finite element
space Vh = {vh ∈ C 0 (D) | ∀i ∈ {0: N −1}, vh|[xi ,xi+1 ] ∈ P1 | vh (0) = 0} both
as solution space and as test space. Since Vh is a subspace of V and of W , the
approximation is conforming, and we can consider using the bilinear form a
in the discrete problem in a standard Galerkin setting.
R 1The discrete problem
consists of finding uh ∈ Vh such that a(uh , wh ) = 0 f wh dt for all wh ∈
Vh . (The resulting linear system is identical to that obtained with centered
finite differences.) The well-posedness of the discrete problem falls into the
framework of Theorem 20.5. Since the discrete solution and test spaces have
the same dimension, well-posedness amounts to
inf
sup
vh ∈Vh wh ∈Vh
a(vh , wh )
=: αh > 0.
kvh kH 1 (D) kwh kL2 (D)
Unfortunately, it turns out that αh is not bounded away from zero from below
uniformly with respect to h.
Proposition 20.13 (Stability loss). There are c1 > 0 and c2 > 0, uniform
with respect to h, such that c1 h ≤ αh ≤ c2 h.
Proof. Assume that N is even, the odd case being treated similarly. Denote
by {ϕ1 , . . . , ϕN } the nodal basis of Vh . Recall from §22.2 the operator Rϕ :
RN → Vh that reconstructs a function in Vh from its coordinate vector. A
direct computation using Simpson’s rule (see §??) shows that 61 hkY k2ℓ2 (RN ) ≤
kRϕ (Y )k2L2 (D) ≤ hkY k2ℓ2 (RN ) for all Y ∈ RN . These bounds imply that
1
h− 2 kGkℓ2 (RN ) ≤ sup
wh ∈Vh
√
1
a(vh , wh )
≤ 6h− 2 kGkℓ2 (RN ) ,
kwh kL2 (D)
(20.19)
for all vh ∈ Vh with G ∈ RN such that Gi = a(vh , ϕi ) for all i ∈ {1: N }.
(2) The bound from above. The idea consists of constructing an oscillating
Part IV. Galerkin Approximation
269
PN
function vh ∈ Vh ; namely, set vh = i=1 Vi ϕi with V2i = 2ih if 1 ≤ i ≤ N2
and V2i+1 = 1 if 0 ≤ i ≤ N2 − 1. Set V0 = 0 and denote by [·] the integer part
operator. Then,
hkvh′ k2L2 (D) =
N
−1
X
i=0
N
2
≥
−1
X
i=0
h
Z
xi+1
xi
2
(vh′ ) dt =
N
−1
X
i=0
[N
4 ]
(1 − 2ih)2 ≥
X
i=0
(Vi+1 − Vi )2
(1 − 2ih)2 ≥ 41 ([ N4 ] + 1),
1
since 1 − 2ih ≥ 12 if i ≤ [ N4 ]. Using the inequality [ N4 ] + 1 > N4 = 4h
, yields
1
′
2
kvh kL (D) ≥ 4h . Furthermore, Gi = 0 if i is even, and Gi = h otherwise.
Hence, kGkℓ2 (RN ) ≤ h1/2 . Using the rightmost bound in (20.19) yields the
√
bound from above in (20.13) with c2 = 4 6.
(3) The bound from below. Let vh be arbitrary in Vh . Set V0 = 0 and
VN +1 = VN . Since Gi = 12 (Vi+1 − Vi−1 ) for all i ∈ {1: N }, we infer that,
Pi−1
for even indices, |V2i | ≤ 2 k=0 |G2k+1 |, while for odd indices, |V2i−1 | ≤
PN
PN
P N2
|G2k | ≤ 2 k=1 |Gk |. Therefore, kV kℓ∞ (RN ) ≤ 2 k=1 |Gk | ≤
|VN +1 | + 2 k=i
2h−1/2 kGkℓ2 (RN ) owing to the Cauchy–Schwarz inequality. Furthermore, a di√
rect computation shows that kvh′ kL2 (D) ≤ 2h−1 kV kℓ∞ (RN ) . Using the left√
most bound in (20.19) and the Poincaré inequality 2kvh′ kL2 (D) ≥ kvh kH 1 (D)
yields the bound from below in (20.13) with c2 = 14 .
⊓
⊔
The main conclusion we can draw from Proposition 20.13 is that a naive
Galerkin approximation of first-order PDEs cannot produce optimal error estimates, even though it yields an invertible linear system (c1 6= 0). In practice,
this problem manifests itself through the presence of spurious wiggles in the
approximate solution.
Exercises
Exercise 20.1 (Condition (bnb2)). Prove that (20.9) is equivalent to (20.5b)
provided (20.5a) holds. (Hint: use that dim(Wh ) = rank(Ah ) + dim(ker(A∗h )),
with A∗h defined in (20.10), together with the Rank Theorem.)
Exercise 20.2 (Bijectivity of A∗h ). Prove that Ah is an isomorphism if and
only if A∗h is an isomorphism. (Hint: use dim(Vh ) = rank(A∗h ) + dim(ker(Ah ))
and dim(Wh ) = rank(Ah ) + dim(ker(A∗h )).)
Exercise 20.3 (Minimal residual, Petrov–Galerkin). Let V, W be two
Hilbert spaces, let A ∈ L(V ; W ′ ) be an isomorphism, and let ℓ ∈ W ′ . Consider
270
Chapter 20. Galerkin approximation
a finite-dimensional subspace Vh ⊂ V . We want to approximate the unique solution u ∈ V satisfying Au = ℓ by using a conforming Petrov–Galerkin approx−1
imation with Wh = JW
AVh ⊂ W where JW : W → W ′ is the Riesz–Fréchet
isomorphism. The discrete bilinear form is ah (vh , wh ) = hAvh , wh iW ′ ,W , and
the discrete linear form is ℓh (wh ) = ℓ(wh ).
(i) Prove that the discrete problem (20.2) is well-posed.
(ii) Show that its unique solution minimizes the residual functional R(v) =
kAv − ℓkW ′ over Vh .
Exercise 20.4 (Variant of Fortin’s criterion). Write a variant of Fortin’s
criterion assuming V, W reflexive and using this time an operator Πh : V → Vh
such that a(Πh v − v, wh ) = 0 for all (v, wh ) ∈ V × Wh and γΠ kΠh vkV ≤ kvkV
for all v ∈ V for some γΠ > 0. (Hint: use (20.11) and Corollary A.61.)
Exercise 20.5 (Fortin’s criterion). Let V and W be two real Banach
spaces and let a ∈ B(V, W ). Let Vh ⊂ V and let Wh ⊂ W be equipped
with the norms of V and W , respectively. Assume that there are two operators (not necessarily linear) Π1,h , Π2,h : W −→ Wh and two uniform constants
c1 , c2 > 0 such that kΠ1,h wkW ≤ c1 kwkW , kΠ2,h (I − Π1,h )wkW ≤ c2 kwkW
and a(vh , Π2,h w) = a(vh , Π2,h w) for all vh ∈ Vh , w ∈ W . Prove that
Πh = Π1,h + Π2,h (I − Π1,h ) satisfies Fortin’s criterion.
Exercise 20.6 (Céa’s Lemma). Assume that a is coercive on V with parameter α. Take W = V and Wh = Vh . Assume Vh ⊂ V and ah = a. Prove (20.17)
1
kakB(V,V ) 2
inf vh ∈Vh ku − vh kV if a is symmetric.
and ku − uh kV ≤
α
Exercise 20.7 (Xu–Zikatanov Lemma). Let V, W be two Hilbert spaces,
let Vh ⊂ V and Wh ⊂ W with dim(Vh ) = dim(Wh ), and let a be a bounded
bilinear form on V ×W satisfying the stability property (??). Let Ph : V → Vh
map u ∈ V to the unique solution of a(uh − u, wh ) = 0 for all wh ∈ Wh .
kak
B(V,W )
(i) Prove that Ph ∈ L(V ; V ) with kPh kL(V ;V ) ≤
.
αh
(ii) Prove that Ph ◦Ph = Ph and that (I − Ph )(vh ) = 0 for all vh ∈ Vh .
(iii) Prove (20.16). (Hint: see the proof of Theorem 4.14.)
Exercise 20.8 (One-dimensional transport). Consider the bilinear form
a and the discrete space Vh from §20.5. Set Wh = {wh ∈ L∞ (D) | ∀i ∈
{0: N −1}, wh|[xi ,xi+1 ] ∈ P0 }. Verify that dim(Vh ) = dim(Wh ) and prove that,
a(vh ,wh )
uniformly with respect to h, inf vh ∈Vh supwh ∈Wh kvh k 1,1
kwh kL∞ (D) > 0.
W
(Hint: see the proof of Proposition 19.16.)
(D)