Part IX, Chapter 52
Parabolic PDEs: theory and space
approximation
We introduce in this chapter some basic properties of parabolic equations and
investigate various approximation techniques. The reader is strongly encouraged to consult Thomée [478] for a thorough analysis of parabolic equations.
52.1 Mathematical analysis
52.1.1 Functional setting and Bochner integral
In this section we review some basic concepts of functional analysis which
are useful in dealing with time-dependent functions with values in a Banach space, see Kufner et al. [332, §2.19]. Let I be an nonempty bounded
open set in R. Let V be a Banach space (real or complex). We say that
f : I −→ V is a simple function if there exit a collection {vk }k∈{1: m}
of elements in V , m ∈PN, and disjoint measurable subsets {Ak }k∈{1: m}
all t ∈ I. The Bochner inin I such that f (t) =
k∈{1: m} vk 1Ak (t) for
R
P
tegral of a simple function is defined by I f (t) dt :=
k∈{1: m} vk |Ak |.
R
R
Clearly k I f (t) dtkV ≤ I kf (t)kV dt for any simple function f . We say say
f : I −→ V is strongly measurable if there is a countable sequence of simple
functions {fn }n∈N such that limk→∞ kf (t) − fk (t)kV = 0 for a.e. t in I.
Lemma 52.1. Let f : I −→ V and assume that f is strongly measurable,
then the mapping I ∋ t 7−→ kf (t)kV ∈ R is Lebesgue measurable.
Proof. See [332, Lem. 2.19.2].
⊓
⊔
Definition 52.2. We say that f : I −→ V is Bochner integrable if there exists a countable sequence of simple functions
R {fn }n∈N such that limk→∞ kf (t)−
fk (t)kV = 0 a.e. t in I, and limk→∞ I kf (t) − fk (t)kV dt = 0. Let f be
Bochner integrable function andRB ⊂ I be a measurable
R set; the Bochner integral of f over B is defined by B f (t) dt := limk→∞ I 1B (t)fk (t) dt.
726
Chapter 52. Parabolic PDEs: theory and space approximation
The coherence of this definition is verified in Exercise 52.1.
Theorem 52.3 (Bochner). A strongly
mesurable function f : I −→ V is
R
Bochner integrable if and only if I kf (t)kV dt < ∞.
Proof. See [332, Thm. 2.19.8]
⊓
⊔
In this book we are only going to manipulate strongly measurable functions.
Then the above theorem says that to verify that f is Bochner integrable it
suffices to verify that I ∋ t 7−→ kf (t)kV ∈ R is in L1 (I; R).
Definition 52.4. For 1 ≤ p ≤ +∞, we define Lp (I; V ) to be the space of
V -valued functions that are strongly measurable and such that the following
norm is finite:
( R
1
if 1 ≤ p < +∞,
ku(t)kpV dt p
I
(52.1)
kukLp(I;V ) =
ess supt∈I ku(t)kV if p = +∞.
It follows
definitions and results that Lp (I; V ) ֒→ L1 (I; V )
R
R from the above
and k I f (t) dtkV ≤ I kf (t)kV dt := kf kL1 (I;V ) for every f ∈ Lp (I; V ). We
henceforth denote by C 0 (I; V ) the space composed of the functions u : I −→ V
such that the mapping I ∋ t −→ ku(t)V kV is continuous. We also denote
C 0 (I; V ) = C 0 (I; V ) ∩ L∞ (I; V ). These two spaces are Banach spaces.
Theorem 52.5. The space Lp (I; V ) is a Banach space for all p ∈ [1, ∞].
Proof. See [332, Thm. 2.20.4]
⊓
⊔
Definition 52.6. Given q ∈ [1, ∞] and a second Banach space W with continuous embedding V ֒→ W , we say that u has a weak derivative in Lq ((0, T ); W )
if there is w ∈ Lq ((0, T ); W ) such that
Z T
Z T
φ(t)w(t) dt, ∀φ ∈ C0∞ ((0, T ); R),
(52.2)
φ′ (t)u(t) dt =
−
0
0
and we denote ∂t u := w.
Henceforth we set I := (0, T ) with T > 0. Since time-evolution problems
are initial-value problems, it is important to determine which type of functions
have a trace over the time interval (0, T ). The following Lemma gives an
answer to this question.
Lemma 52.7. Let 1 ≤ p, q ≤ +∞, let B0 ֒→ B1 be two Banach spaces with
continuous embeddings, and set
W(B0 , B1 ) := {v ∈ Lp ((0, T ); B0 ) | ∂t v ∈ Lq ((0, T ); B1 )}.
(i) W(B0 , B1 ) is a Banach space when equipped with the norm kukW(B0 ,B1 ) =
kukLp((0,T );B0 ) + k∂t ukLq ((0,T );B1 ) .
Part IX. Time-Dependent PDEs
727
(ii) W(B0 , B1 ) is continuously embedded in C 0 ([0, T ]; B1 ).
⊓
⊔
Proof. See Exercise 52.4 for the proof of item (ii).
We henceforth denote kukC 0 ([0,T ];B) = supt∈[0,T ] ku(t)kB and |u|C 1 ([0,T ];B) =
supt∈[0,T ] k∂t u(t)kB , etc.
Lemma 52.8 (Aubin-Lions-Simon). Let 1 ≤ p, q ≤ +∞ and let B0 ֒→
B ֒→ B1 be three Banach spaces with B0 compactly embedded in B and B
continuously embedded in B1 .
(i) The embedding W(B0 , B1 ) ֒→ Lp ([0, T ]; B) is compact.
(ii) The embedding W(B0 , B1 ) ֒→ C 0 ([0, T ]; B) is compact if p = ∞ and
q > 1.
Proof. See Aubin [23]; see also, e.g., Amann [9], Lions [353], Lions and Magenes [354], Simon [452].
⊓
⊔
Henceforth, we specialize the above setting by restricting ourselves to
Hilbert spaces and taking p = q = 2 in Lemma 52.7. Let V ֒→ L be two
Hilbert spaces with continuous embedding. The norm of the embedding operator is denoted by c−1
P , i.e.,
∀v ∈ V,
cP kvkL ≤ kvkV .
(52.3)
We assume that V is dense in L, and we identify L with L′ so that we are in
the situation where V ֒→ L ≡ L′ ֒→ V ′ , i.e., the duality paring h·, ·iV ′ ,V can be
viewed as an extension of the inner product in L. Note that cP kf kV ′ ≤ kf kL
and hf, viV ′ ,V = (f, v)L for all f ∈ L and all v ∈ V . In this setting, the
following result justifies integration by parts with respect to time (see Dautray
and Lions [190, Thm. 2, p. 477]):
Lemma 52.9 (Integration by parts). Under the above assumptions, for
all u, v ∈ W(V, V ′ ), the following identity holds:
Z
0
Z T
T
h∂t u(t), v(t)iV ′ ,V dt = (u(T ), v(T ))L − (u(0), v(0))L − h∂t v(t), u(t)iV ′ ,V dt.
0
52.1.2 Well-posedness
We now state a general result for time-dependent problems which plays a
similar role to that played by the Lax–Milgram Lemma for elliptic equations.
Let V ֒→ L ≡ L′ ֒→ V ′ be a Hilbertian setting as defined above. Consider
a mapping a : (0, T )×V ×V → R such that a(t, ·, ·) is bilinear for a.e. t in
(0, T ). Moreover, assume that a satisfies the following properties:
(p1) The function t 7→ a(t, u, v) is measurable ∀u, v ∈ V .
(p2) ∃M such that |a(t, u, v)| ≤ M kukV kvkV for a.e. t ∈ [0, T ], ∀u, v ∈ V .
728
Chapter 52. Parabolic PDEs: theory and space approximation
(p3) ∃α > 0 and γ > 0 such that a(t, u, u) ≥ αkuk2V − γkuk2L for a.e. t ∈ [0, T ]
and for all u ∈ V .
For f ∈ L2 ((0, T ); V ′ ) and u0 ∈ L, consider the following problem:

′

 Find u ∈ W(V, V ) such that
h∂t u, viV ′ ,V + a(t, u, v) = hf (t), viV ′ ,V , a.e. t ∈ (0, T ), ∀v ∈ V ,


u(0) = u0 .
(52.4)
The initial data u(0) = u0 is meaningful according to Lemma 52.8. Note that
the time evolution equation has to be understood in the distribution sense,
RT
RT
i.e., 0 (−φ′ (t)(u(t), v)L + φ(t)a(t, u(t), v)) dt = 0 φ(t)hf (t), viV ′ ,V dt, for
all φ ∈ C0∞ ((0, T ); R).
Definition 52.10 (Parabolic equation). Equation (52.4) is said to be
parabolic whenever the bilinear form a satisfies the conditions (p1), (p2),
and (p3).
Up to a change of variable, it is always possible to modify condition (p3)
so that γ = 0. Indeed, upon setting
ã(t, φ, v) = a(t, φ, v) + γ(φ, v)L ,
it is clear that ã satisfies conditions (p1) and (p2), and that ã is V -coercive
since ã(t, φ, φ) ≥ αkφk2V . Furthermore, setting φ = e−γt u and g = e−γt f ,
problem (52.4) is recast in the following equivalent form:

′

 Find φ ∈ W(V, V ) such that
h∂t φ, viV ′ ,V + ã(t, φ, v) = hg(t), viV ′ ,V , a.e. t ∈ (0, T ), ∀v ∈ V, (52.5)


φ(0) = u0 .
We shall henceforth assume that a is coercive on V , i.e., γ = 0.
Before stating the main existence and uniqueness result of this section,
we reformulate (52.4) in a setting which, at this point in the book, should be
more familiar to the reader, namely that of the BNB Theorem. For the sake
of simplicity, assume u0 = 0. Consider the Hilbert spaces
Y = L2 ((0, T ); V )
and
X = {v ∈ W(V, V ′ ) | v(0) = 0}.
RT
For g ∈ Y ′ and y ∈ Y , set hg, yiY ′ ,Y = 0 hg(t), y(t)iV ′ ,V dt and define the
bilinear form b : X×Y → R such that
Z T
[h∂t x, yiV ′ ,V + a(t, x, y)] dt.
∀(x, y) ∈ X×Y, b(x, y) =
0
It is clear that b is continuous. Then, consider the following problem:
Part IX. Time-Dependent PDEs
729
Find u ∈ X such that
b(u, y) = hf, yiY ′ ,Y , ∀y ∈ Y.
(52.6)
Using the distribution theory, it can be shown that (52.6) and (52.4) are
equivalent. As a result, proving that (52.4) is well-posed amounts to proving
that b satisfies the two conditions of the BNB Theorem. This is the purpose
of the following:
Theorem 52.11 (J.-L. Lions). Under the hypotheses (p1), (p2), and (p3),
problem (52.4) has a unique solution.
Proof. See Lions and Magenes [354, p. 253-258] or Dautray and Lions [190,
Thm. 2, p. 513]. Let us prove the theorem using the formulation (52.6) and
assuming u0 = 0.
(1) For a.e. t ∈ (0, T ), define A(t) ∈ L(V ; V ′ ) such that hA(t)u, viV ′ ,V =
a(t, u, v) for all u, v ∈ V . Clearly, kA(t)kL(V ;V ′ ) ≤ M . Owing to the coercivity
hypothesis (p3), A(t) is an isomorphism for a.e. t ∈ (0, T ). Hence, A(t)−1 is
continuous and kA(t)−1 kL(V ′ ;V ) ≤ α−1 . Moreover, A(t)−1 is coercive since,
for all x ∈ V ′ ,
hx, A(t)−1 xiV ′ ,V = hA(t)A(t)−1 x, A(t)−1 xiV ′ ,V ≥ αkA(t)−1 xk2V ≥
2
α
M 2 kxkV ′ .
Hence, the coercivity constant of A(t)−1 is bounded from below by αM −2 .
(2) Let us prove that condition (bnb1) holds. Let u ∈ X, µ > 0, and set
v = A(t)−1 ∂t u + µu. Step 1 shows that kvkY ≤ c kukX . Moreover,
RT
h∂t u + A(t)u, A(t)−1 ∂t u + µuiV ′ ,V dt
RT
≥ µ2 ku(T )k2L + 0 Mα2 k∂t uk2V ′ + µαkuk2V − kAkkA−1 kkukV k∂t ukV ′ dt
RT
RT
RT
′
≥ Mα2 0 k∂t uk2V ′ dt + µα 0 kuk2V dt − M
α 0 kukV k∂t ukV dt
RT
RT
M4
α
2
2
≥ 2M 2 0 k∂t ukV ′ dt + (µα − 2α3 ) 0 kukV dt.
b(u, v) =
0
Taking µ = M 4 α−3 yields
b(u, v) ≥ ckuk2W(V,V ′ ) ≥ c′ kukX kvkY .
The inf-sup condition (bnb1) follows easily.
(3) Let us prove that condition (bnb2) holds. Let v ∈ Y be such that b(u, v) =
0 for all u ∈ X. Let φ ∈ C0∞ ((0, T ); R) and w ∈ V , then φ(t)w ∈ X and
RT
0
hw,
φ′ (t)(w, v(t))L dt = −
RT
0
φ′ (t)v(t) dtiV ′ ,V
RT
φ(t)hA(t)w, viV ′ ,V dt
RT
= −hw, 0 φ(t)A(t)T v dtiV ′ ,V .
0
RT
RT
i.e., 0 φ′ (t)v(t) dt = − 0 φ(t)A(t)T v dt in V ′ . This proves that ∂t v =
−A(t)T v ∈ L2 ((0, T ); V ′ ). Let us now use u = tw, where w ∈ V . Then using
the integration by part formula from Lemma 52.9, we infer that
730
Chapter 52. Parabolic PDEs: theory and space approximation
RT
0 = b(tw, v) = 0 ((w, v(t))L + thw, A(t)T v(t)iV ′ ,V dt
RT
= 0 ((w, v(t))L + thw, ∂t v(t)iV ′ ,V dt = T (w, v(T ))L ,
∀w ∈ V.
This in turn implies that v(T ) = 0. We finally use u = tv (note that
RT
RT
u ∈ X), then upon observing that 0 (∂t (tv), v)L dt = − 0 t(v, ∂t v)L dt =
RT
RT 1
− 2 t∂t kvk2L dt = 0 21 kvk2L dt, since v(T ) = 0, we infer that
0
0 = b(tv, v) ≥
RT
0
1
2
2 kvkL
+ αtkvk2V ) dt,
⊓
⊔
which eventually yields v = 0. Hence, (bnb2) holds.
Theorem 52.12 (A priori estimates). For f ∈ L2 ((0, T ); V ′ ), the solution
to (52.4) satisfies the energy estimate

2
1
1

 kukC 0 ([0,T ];L) ≤ ku0 kL e− 2 αcP t + √ kf kL2 ((0,T );V ′ ) ,

α
(52.7)
1
1


 kukL2((0,T );V ) ≤ √ ku0 kL + kf kL2 ((0,T );V ′ ) .
α
α
Furthermore, if f ∈ L∞ ((0, +∞); V ′ ),
lim sup ku(t)kL ≤
t→+∞
1
αcP
kf kL∞((0,+∞);V ′ ) .
(52.8)
2
Proof. Let t ∈ (0, T ). Choose ueαcP τ as a test function. The coercivity of a
together with Lemma 52.9 implies that
R t αc2 τ 1 d
R t αc2 τ
2
2
P (
P kf k ′ kuk
V dτ
V
2 dτ kukL + αkukV ) dτ ≤ 0 e
0 e
R t αc2 τ α
1
2
≤ 0 e P ( 2 kukV + 2α
kf k2V ′ ) dτ.
This, together with (52.3), gives the following estimate:
Rt
d
αc2P τ
kuk2L ) dτ
0 dτ (e
=
≤
which in turn yields
Rt
αc2P τ d
( dτ kuk2L
0 e
R t αc2 τ d
e P ( dτ kuk2L
0
2
ku(t)k2L ≤ ku(0)k2L e−αcP t +
1
α
+ αc2P kuk2L ) dτ
+ αkuk2V ) dτ ≤
Rt
0
1
α
Rt
0
2
eαcP τ kf k2V ′ dτ,
2
e−αcP (t−τ ) kf (τ )k2V ′ dτ.
Hence (52.8) and the first inequality in (52.7). The second inequality in (52.7)
is obtained by testing with u.
⊓
⊔
Remark 52.13. Theorem 52.12 establishes the continuous dependence of u
with respect to the data. It is wise to retain, or at least mimic, this stability
result when devising an approximation with respect to time and/or space. ⊓
⊔
Part IX. Time-Dependent PDEs
731
52.1.3 Example: the heat equation
We illustrate the notions introduced above on


 ∂t u − ∇·(κ(x)∇u) = f
u(x, t) = 0


u(x, 0) = u0 (x)
the heat equation:
x ∈ D, t ≥ 0,
x ∈ ∂D, t > 0,
x ∈ D.
(52.9)
The above problem models heat transfers in D: the dependent variable u(x, t)
is the temperature at the point x ∈ D and time t; f is a source term; u0 is
the initial temperature; and κ is the thermal conductivity. For the sake of
simplicity, we assume that κ is scalar-valued and that homogeneous Dirichlet
conditions are enforced on ∂D, i.e., that the temperature is prescribed on ∂D.
To formulate (52.9) in a weak sense, assume that u0 ∈ L2 (D) and f ∈
L2 ((0, T ); H −1 (D)). Take a test function v ∈ H01 (D), multiply (52.9) by v,
and integrate over D. This yields
Z
κ(x)∇u(t)·∇v dx = hf (t), viH −1 ,H01 ,
h∂t u(t), viH −1 ,H01 +
(52.10)
D
where recall that (52.10) has to be understood in the distribution sense on
the time interval (0, T ). A possible weak formulation of problem (52.9) is:

1
H −1 (D)) such that, a.e. t, ∀v ∈ H01 (D),

 Find u ∈ W(H0 (D),
R
h∂t u, viH −1 ,H01 + D κ(x)∇u(t)·∇v = hf (t), viH −1 ,H01 ,
(52.11)


u(0) = u0 .
R
Setting a(t, u(t), v) = D κ(x)∇u(t)·∇v and V = H01 (D), Theorem 52.11 implies that problem (52.11) is well-posed if κ ∈ L∞ (D) and if there is κ0 > 0
such that κ(x) ≥ κ0 a.e. in D. Indeed, letting CP,D be the constant from
the Poincaré inequality (see Lemma 2.25), we infer that a(t, u, u) ≥ αkuk21,D
C2
P,D
. Note that we have assumed here that the space variwhere α = κ0 1+C
2
P,D
ℓ2 c 2
D D
ables have be non-dimensionalized, otherwise we would have α = κ0 1+ℓ
2 c2 ,
D D
where ℓD is the reference length scale. Theorem 52.12 yields

2
1
1

 kukC 0 ([0,T ];L2 (D)) ≤ ku0 kL2 (D) e− 2 αcD t + √ kf kL2 ((0,T );H −1 (D)) ,

α
1
1


 kukL2((0,T );H 1 (D)) ≤ √ ku0 kL2 (D) + kf kL2 ((0,T );H −1 (D)) .
α
α
Conversely, it can be shown using standard density arguments together with
the distribution theory, that if u solves (52.11) and u is smooth enough, then
u is a classical solution to (52.9).
Remark 52.14. Theorem 52.11 applies to problems that are more complicated than the heat equation. Actually, it applies to the time-dependent version of all the problems studied in Chapter 24. For instance, setting
732
Chapter 52. Parabolic PDEs: theory and space approximation
a(t, u, v) =
Z
D
∇v·d(x, t)·∇u + (β(x, t)·∇u)v + µ(x, t)u v dx,
(52.12)
one readily verifies that the hypotheses of Theorem 52.11 are satisfied provided
the fields d(x, t), β(x, t), and µ(x, t) are sufficiently smooth and are such that
the bilinear form a(t, ·, ·) is coercive for a.e. t ∈ (0, T ); see Proposition 24.5 for
sufficient conditions yielding coercivity. The theory generalizes to Neumann
and Robin boundary conditions as well.
⊓
⊔
We conclude this section by stating two remarkable properties of the solutions to (52.4) when using the bilinear form (52.12).
Proposition 52.15 (Positivity). Let u0 ∈ L2 (D) and f ∈ L2 ((0, T ); L2 (D)).
Let u ∈ W(H01 (D), H −1 (D)) solve (52.4) with a(t, ·, ·) defined in (52.12). Assume u0 (x) ≥ 0 a.e. in D and f (x, t) ≥ 0 a.e. in Q. Then, u(x, t) ≥ 0 a.e. in
Q.
Proof. Observe first that u− = 12 (|u| − u) ∈ W(H01 (D), H −1 (D)) is an admissible test function. Observe that a(t, u, u− ) = −a(t, u− , u− ) to obtain
1 d
− 2
2 dt ku k0,D
+ a(t, u− , u− ) = −(f, u− )0,D ≤ 0,
implying ku− (t)kL2 (D) ≤ ku−
0 kL2 (D) = 0.
⊓
⊔
Proposition 52.16 (Maximum principle). Let u0 ∈ L∞ (D) and assume
f = 0. Let u ∈ W(H01 (D), H −1 (D)) be the solution to (52.4) with a(t, ·, ·)
defined in (52.12). Assume µ ≥ 0. Then, kukL∞ (Q) ≤ ku0 kL∞ (D) .
Proof. Set M = ku0 kL∞ (D) and note that (u − M )+ = 21 (|u − M | + u − M ) ∈
W(H01 (D), H −1 (D)) is an admissible test function. The property
a(t, u, (u − M )+ ) = a(t, u − M, (u − M )+ ) + a(t, M, (u − M )+ )
Z
µM (u − M )+ ≥ 0,
= a(t, (u − M )+ , (u − M )+ ) +
D
implies
d
dt k(u
− M )+ k2L2 (D) ≤ 0. The desired result follows easily.
⊓
⊔
Remark 52.17. It is generally difficult to retain Maximum Principle properties in discrete settings. If Vh ֒→ H01 (D) is a finite element space and vh is an
arbitrary function in Vh , it is unlikely that vh− and (vh − M )+ are in Vh . ⊓
⊔
52.2 Space semidiscretization
52.2.1 The method of lines
In problem (52.4) the space and time variables play different roles. This observation advocates for the following approach: First, approximate the solution in
Part IX. Time-Dependent PDEs
733
space only so as to obtain a system of coupled ordinary differential equations
(ODEs), where the time is the only independent variable. Second, construct an
approximation in time by making use of the vast theory of solution techniques
for ODEs. This approach is often called the method of lines.
Let (Vh )0≤h≤1 be a sequence of finite-dimensional subspaces of V . For instance, one may think of L = L2 (D), V = H01 (D), and Vh is a H 1 -conforming
finite element space, see Chapter ??. Consider the approximate problem:

1

 Find uh ∈ C ([0, T ]; Vh ) such that
(∂t uh , vh )L + a(t, uh , vh ) = hf, vh iV ′ ,V , ∀t ∈ [0, T ], ∀vh ∈ Vh , (52.13)


uh (0) = u0h ,
where u0h ∈ Vh is an approximation of u0 , the precise nature of which will
be clarified later. Since (52.13) is a finite coupled system of linear ODEs;
the Cauchy–Lipschitz Theorem guarantees the existence and uniqueness of a
solution uh (t) in C 1 ([0, T ]; Vh ); see, e.g., Brezis [100, Thm. 7.3].
Let {ϕ1 , . . . , ϕN } be a basis of Vh (for instance, the global shape functions
of Vh ). The approximate solution uh (t) ∈ Vh can be expanded as follows for
all t ∈ [0, T ]:
N
X
Ui (t)ϕi .
uh (t) =
i=1
T
Set U (t) = (U1 (t), . . . , UN (t)) , F (t) = (hf (t), ϕ1 iV ′ ,V , . . .)T , and introduce
the stiffness matrix A(t) ∈ RN ×N and the mass matrix M ∈ RN ×N such that
Aij (t) = a(t, ϕj , ϕi ),
Mij = (ϕi , ϕj )L ,
1 ≤ i, j ≤ N.
The mass matrix is symmetric definite positive, see §15.4.2, and the stiffness
matrix A(t) is positive-definite for all t ∈ [0, T ], see §22.2. Using the above
notation, (52.13) is recast as follows:
(
M∂t U (t) = −A(t)U (t) + F (t), t ∈ [0, T ],
(52.14)
U (0) = U0 ,
where U0 ∈ RN is the coordinate vector of u0h relative to the basis {ϕ1 , . . . , ϕN }.
Remark 52.18 (Mass lumping). Note that (52.14) is a standard system
of ODEs if M is diagonal. It is sometimes possible to force M to be diagonal by appropriately choosing the quadratures (see §??). For instance, if
{ϕ1 , . . . , ϕN } is a Lagrange basis, one can pick a quadrate rule that uses the
Lagrange nodes as quadrature points. This process is called mass lumping. ⊓
⊔
52.2.2 Error analysis
We now investigate whether uh (t) is an accurate approximation of u(t). Let
us introduce the projection Pht ∈ L(V ; Vh ), t ∈ [0, T ], such that for all w ∈ V ,
Pht (w) is the solution to
734
Chapter 52. Parabolic PDEs: theory and space approximation
∀vh ∈ Vh ,
a(t, Pht (w), vh ) = a(t, w, vh ),
∀vh ∈ V.
(52.15)
We henceforth abuse the language by saying that Pht is the elliptic projector.
Theorem 52.19. Let u be the solution to (52.4). Assume that u ∈ C 1 ([0, T ]; L)∩
C 0 ([0, T ]; V ) and let η(t) := u(t) − Pht (u), then
T
2
ku − uh kC 0 ([0,T ];L) ≤ (ku0 − u0h kL + kηh (0)kL )e−cP α 2
√1
T
+ kηkC 0 ([0,T ];L) +
ku−uhkL2 ([0,T ];V ) ≤
1
αc2P
(52.16)
|η|C 1 ([0,T ];L) ,
√1 (ku0 − u0h kL + kηh (0)kL )
αT
+ √1T kηh kC 0 ([0,T ];V ) + αc1P |ηh |C 1 ([0,T ];L) .
(52.17)
(52.18)
(52.19)
Proof. (1) Set eh (t) = Pht (u) − uh (t) and η(t) = u(t) − Pht (u) so that u(t) −
uh (t) = eh (t) + η(t). Subtracting (52.13) from (52.4) leads to
∀vh ∈ Vh ,
(∂t eh , vh )L + a(t, eh , vh ) = −(∂t η, vh )L .
Note that the term a(t, η(t), vh ) is zero by definition of Pht (u). Choosing eh (t)
as test function in this identity yields
1d
2
2 dt keh kL
+ αkeh k2V ≤ k∂t ηkV ′ keh kV ≤
α
2
2 keh kV
+
1
2
2α k∂t ηkV ′ .
The relation kvkV ≥ cP kvkL , see (52.3), implies that
d
2
dt keh kL
+ αc2P keh k2L ≤
d
2
dt keh kL
+ αkeh k2V ≤
1
αc2P
k∂t ηk2L .
(52.20)
(2) Owing to Gronwall’s Lemma (see item (v) in Exercise 52.5),
Z t
2
2
2 −αc2P t
1
e−αcP (t−τ ) kdτ ηk2L dτ
keh (t)kL ≤ keh (0)kL e
+ αc2
P
0
2
2 −αc2P t
1
≤ keh (0)kL e
+ (αc2 )2 1 − e−αcP t k∂t ηk2C 0 ([0,T ];L).
P
Then use the triangle inequality ku − uh kL ≤ keh kL + kηkL to infer that
2
t
ku(t) − uh (t)kL ≤ keh (0)kL e−αcP 2 + kηkC 0 ([0,T ];L) +
1
αc2P
|η|C 1 ([0,T ];L) .
The first desired bound from above follows easily.
(3) Integrating the second inequality in (52.20) with respect to time yields
Z T
Z T
T
1
2
k∂t ηk2L + keh (0)k2L ≤ 2 k∂t ηk2C 0 ([0,T ];L) + keh (0)k2L .
α keh kV ≤ 2
αc
αc
0
P 0
P
Hence,
1
T
Z
0
T
keh k2V ≤
1
1
k∂t ηk2C 0 ([0,T ];L) +
keh (0)k2L .
α2 c2P
αT
The second bound from above follows from the triangle inequality.
(52.21)
⊓
⊔
Part IX. Time-Dependent PDEs
735
Example 52.20 (Finite elements). Let u be the solution to the heat equation or the solution to the PDE corresponding to the bilinear form (52.12)
(assuming appropriate boundary conditions). Assume that Vh is composed of
conforming finite elements of degree k, see §25.1.2 and assume that the adjoint of the differential operator associated with the elliptic projector has a
smoothing property in H 1+s (D), s ∈ [0, 1], (see e.g., Theorem ??), then it is
possible to show that there is a uniform constant c such that
|u − Pht (u)|C 1 ([0,T ];L2 (D)) ≤ chk+s |u|C 1 ([0,T ];H k+1 (D))
ku − Pht (u)kC 0 ([0,T ];H 1 (D)) ≤ chk kukC 0([0,T ];H k+1 (D)) .
Assuming that u0 is well approximated (take for instance u0h = Ph0 u
or u0h = Ih u where Ih is an interpolation or quasi-interpolation operator), if kukC 1 ([0,T ];H k+1 (D)) is bounded, Theorem (52.19) implies the following error estimates: ku − uh kC 0 ([0,T ];L2 (D)) ≤ c hk+1 kukC 1([0,T ];H k+1 (D)) and
ku − uh kC 0 ([0,T ];H 1 (D)) ≤ c hk kukC 1 ([0,T ];H k+1 (D)) .
Remark 52.21.
(i) The error induced by the approximation of u0 exponentially decreases
1
with T in the L-norm and decreases in the mean like T − 2 in the V -norm.
Insensitivity to initial data as time grows is a characteristic property of dissipative equations (i.e., parabolic).
(ii) Note that in the eror estimate in the L-norm, (52.16), all the terms on
the right-hand side are measured in the L-norm. This implies that the error
estimate is optimal. The use of the elliptic projector is essential to obtain
optimality. This trick, introduced by Wheeler [504], avoids having to use a
duality argument involving the adjoint problem, i.e., a retrograde equation
from T to 0.
(iii) Note that as a byproduct of the use elliptic projector, the estimate
(52.21) is superconvergent, i.e., the error on the left-hand side is measured
in the V -norm whereas the terms on the right-hand are measured in the Lnorm. This property is central to the so-called Postprocessing Galerkin and
Nonlinear Galerkin Methods.
⊓
⊔
Exercises
Exercise 52.1 (Bochner integral). . Let I be a nonempty bounded open
set of R. Let V be a Banach space. Let f : I −→ V be a Bochner integrable
function. Let {fn }n∈N be a countable sequence of simple functions satisfying
the assumptions
of Definition
52.2. Let B ⊂ I be a measurable set. (i) Show
R
R
thatRk I s(t) dtkV ≤ I ks(t)kV dt for any simple function s. (ii) Show that
the I 1B (t)fk (t) dt has a limit when k → ∞. (iii) Show that if {fn }n∈N and
{gn }n∈N are two sequences of Rsimples functions satisfying
R the assumptions of
Definition 52.2, then limk→∞ I 1B (t)fk (t) dt = limk→∞ I 1B (t)gk (t) dt.
736
Chapter 52. Parabolic PDEs: theory and space approximation
R
exists such that limk→∞ kf (t)−fk (t)kV = 0 a.e. t in I, and limk→∞ I kf (t)−
fk (t)k
integrable, the Bochner integral of f is defined
R
R V dt = 0. If f is Bochner
by I f (t) dt = limk→∞ I fk (t) dt.
Exercise 52.2. Taking u0 = 0 in (52.4) , prove that the solution sets of
P
(52.4) and (52.6) are identical. (Hint: Show that { M
m=1 φm (t)vm | φm ∈
C0∞ ((0, T ); R), vm ∈ V, m ∈ {1:M }, M ∈ N} is dense in L2 ((0, T ); V )).
Exercise 52.3 (Hilbertian setting). Let V be a Hilbert space. Using the
notation of §52.1.1, show that W(V, V ′ ) is a Hilbert space.
Exercise 52.4 (W(B0 , B1 ) ֒→ C 0 ([0, T ]; B1 )). The objective of this exercise
is to prove item (ii) in Lemma 52.7. Let u ∈ W(B0 , B1 ), set v(t) := ∂t u(t)
Rt
and w(t) = 0 v(τ ) dτ .
(i) Show that w ∈ C 0 ([0, T ]; B1 ).
−
1
(ii) Let ρ(τ ) = ηe 1−|τ |2 if |τ | ≤ 1 and ρ(τ ) = 0 otherwise. Let the constant
R +∞
η be such that −∞ ρ(τ ) dτ = 1. Let 0 < s < t < T and let N be the
smallest integer such that N ≥ max( 1s , T 1−t ). Define ρn (τ ) = nρ(nτ ) for
Rτ
n ≥ N . Consider the sequence of smooth functions φn (τ ) = 0 (ρn (s −
ξ) − ρ(t − ξ)) dξ. What is limn→∞ φn (τ )?
RT
R1
(iii) Show that δ(s, t) := −1 ρn (y)(u(s− ny )−u(t− ny )) dy = 0 u(τ )φ′n (τ ) dτ .
(iv) limn→∞ δ(s, t) (Hint: Pass to the limit in the above equatility)
(v) Proves that u ∈ C 0 ([0, T ]; B1 ).
Exercise 52.5 (Gronwall’s lemma). Let β ∈ L1 ((0, T ); R), let u and α be
two real-valued functions on (0, T ) such that βu ∈ L1 ((0, T ); R) and u(t) ≤
Rt
α(t) + 0 β(r)u(r) dr for all t ∈ [0, T ].
Rt
Rt
(i) Let v(t) := e− 0 β(r) dr 0 β(r)u(r) dr. Show that v ∈ W 1,1 ((0, T ); R).
Rr
Rt
(ii) Conclude that v(t) ≤ 0 α(r)β(r)e− 0 β(s) ds dr.
(iii) Show that
Z t
Rt
u(t) ≤ α(t) +
α(s)β(s)e s β(r) dr ds.
Rt
0
Rt
(Hint: use item (ii) and 0 β(r)u(r) dr = v(t)e 0 β(r) dr .)
(iv) Assume now that α is non-decreasing. Conclude that
u(t) ≤ α(t)e
Rt
0
β(r) dr
,
∀t ∈ [0, T ]
(v) Prove the following estimate when β is constant and α ∈ L1 (0, T ):
u(t) ≤ α(0)eβt +
Z
0
t
α′ (t)eβ(t−s) dt.