Part V, Chapter 25
H 1 -conforming approximation
The goal of this chapter is to analyze the approximation of scalar, secondorder, elliptic PDEs using H 1 -conforming finite element spaces. We focus the
presentation on Dirichlet boundary conditions for simplicity, but most of the
concepts presented herein extend naturally to Neumann and Robin conditions.
We start the analysis with homogeneous Dirichlet conditions and derive H 1 and L2 -error estimates by invoking the results of Chapter 21. We continue
by investigating how the discrete maximum principle can be achieved using
finite elements. Then we consider theoretically and practically the handling
of non-homogeneous Dirichlet conditions.
25.1 Homogeneous Dirichlet conditions
Let D be a Lipschitz domain in Rd and let f ∈ L2 (D). The model problem
we want to approximate is the homogeneous Dirichlet problem:
(
−∇·(d∇u) + β·∇u + µu = f in D,
(25.1)
u=0
on ∂D,
with d ∈ L∞ (D), β ∈ W 1,∞ (D), µ ∈ L∞ (D). We assume that d is a symmetric second-order tensor field and the smallest eigenvalue of d is uniformly
bounded from below by λ♭,D > 0 and that (µ − 12 ∇·β) is nonnegative. Then
the model problem
Find u ∈ V := H01 (D) such that
R
(25.2)
a(u, w) = D f w dx, ∀w ∈ V,
is well-posed
R owing to the Lax–Milgram Lemma since the bilinear form
a(v, w) := D (∇v·d·∇w + (β·∇v)w + µvw) dx is bounded and coercive on
V ; see Proposition 24.5.
Chapter 25. H 1 -conforming approximation
338
25.1.1 Discrete problem and well-posedness
We assume for simplicity that D is a polyhedron and we consider a shaperegular sequence (Th )h>0 of simplicial affine meshes of D to approximate (25.2)
with H 1 -conforming finite elements. Let Pkg (Th ) be the H 1 -conforming finite
element space of degree k ≥ 1 introduced in §15.2.2. In this chapter, we
want to enforce the homogeneous Dirichlet condition explicitly in the discrete
g
problem. This leads us to consider the subspace Pk,0
(Th ) = Pkg (Th ) ∩ H01 (D)
(see §15.4) and the discrete problem
g
Find uh ∈ Vh := Pk,0
(Th ) such that
R
(25.3)
a(uh , wh ) = D f wh dx, ∀wh ∈ Vh .
This problem is well-posed owing to the Lax–Milgram Lemma since the approximation setting is H 1 -conforming. An alternative technique to enforce the
Dirichlet condition weakly by means of a boundary penalty method is studied
in Chapter 29.
Remark 25.1 (Other boundary conditions). For Robin/Neumann conditions one uses the entire space Pkg (Th ), see §24.3. For Dirichlet–Neumann
conditions, see §24.3.3, one must be careful to construct meshes that are compatible with the boundary partition ∂D = ∂Dd ∪ ∂Dn , i.e., boundary faces
must belong either to ∂Dd or to ∂Dn .
⊓
⊔
25.1.2 Error analysis
Our goal is to estimate the approximation error in the H 1 -seminorm (which
is a legitimate norm on H01 (D)) and in the L2 -norm. Recall that DK =
int(∪K∈TK K) where TK is the union of all the mesh cells that share global
shape functions with the mesh cell K ∈ Th .
Theorem 25.2 (H 1 -error). Let u solve (25.2) and let uh solve (25.3). Then
limh→0 |u − uh |H 1 (D) = 0. Assume that u ∈ H 1+r (D). If r ∈ {1: k}, then
|u − uh |H 1 (D) ≤ c
X
K∈Th
2
h2r
K |u|H 1+r (DK )
! 21
≤ c hr |u|H 1+r (D) ,
(25.4)
and one can replace |u|H 1+r (DK ) by |u|H 1+r (K) if 1 + r > d2 . Estimate (25.4)
also holds for r ∈ ( 21 , 1) with c depending on |r − 21 |. Finally, if r ∈ (0, 12 ),
then
|u − uh |H 1 (D) ≤ c hr |u|H 1+r (D) .
(25.5)
Proof. (1) We first prove (25.4) and (25.5). We infer from Lemma 20.12 that
the approximation error is bounded by the best-approximation error:
Part V. Elliptic PDEs
339
|u − uh |H 1 (D) ≤ c inf |u − vh |H 1 (D) .
vh ∈Vh
(25.6)
g,av
g,av
We conclude by taking vh = Ih0
(u) to bound the infimum, with Ih0
defined
g,av
in §16.4, and invoke Theorem 16.12 to bound |u−Ih0 (u)|H 1 (K) for all K ∈ Th
g,av
if r > 12 or to bound |u−Ih0
(u)|H 1 (D) if r ∈ (0, 21 ); for 1+r > d2 , the canonical
g
interpolation operator Ih using point values (see §15.3) can be used.
(2) We now prove the convergence of uh to u as h → 0, assuming only that
u ∈ V = H01 (D). We use a density argument. Let ǫ > 0. Since H 2 (D) ∩ V is
dense in V , there is uǫ ∈ H 2 (D) ∩ V such that |u − uǫ |H 1 (D) ≤ ǫ. Let uǫ,h ∈ Vh
be the best approximation of uǫ in H01 (D). Using again Theorem 16.12 yields
|uǫ − uǫ,h |H 1 (D) ≤ ch|uǫ |H 2 (D) . Then inf vh ∈Vh |u − vh |H 1 (D) ≤ |u − uǫ |H 1 (D) +
|uǫ − uǫ,h |H 1 (D) , so that that lim suph→0 (inf vh ∈Vh |u − vh |H 1 (D) ) ≤ ǫ. Since ǫ
is arbitrary, the bound (25.6) implies that limh→0 |u − uh |H 1 (D) = 0 .
⊓
⊔
We now use the duality technique from §21.4 to derive an improved error estimate in the L2 -norm. The differential operator associated with the
PDE in (25.1) is A ∈ L(V ; V ′ ) with V = H01 (D), V ′ = H −1 (D), and
Av = −∇·(d∇v) + β·∇v + µv for all v ∈ H01 (D). The adjoint operator
A∗ ∈ L(V ; V ′ ) is such that A∗ v = −∇·(d∇v) − β·∇v + (µ − ∇·β)v; see Exercise 25.1. Let g ∈ L2 (D). The adjoint problem consists of seeking ζ ∈ H01 (D)
such that A∗ ζ = g in L2 (D); one easily verifies that this problem is well-posed
owing to the Lax–Milgram Lemma. We assume that the adjoint operator has
the following smoothing property (see Definition 21.13): There is s ∈ (0, 1] and
a uniform constant csmo such that the adjoint solution ζ satisfies the a priori
bound kζkH 1+s (D) ≤ csmo kgkL2 (D) . Sufficient conditions for the smoothing
property to hold follow from elliptic regularity, see §24.4. For instance, this
property holds if the domain D is Lipschitz and the data d, β, and µ are
smooth; the maximal value s = 1 is obtained on convex domains.
Theorem 25.3 (L2 -error). Assume that there is s ∈ (0, 1] such that the
adjoint operator has the smoothing property in H 1+s (D). Then,
ku − uh kL2 (D) ≤ chs |u − uh |H 1 (D) .
(25.7)
Proof. We apply the Aubin–Nitsche Lemma 21.16. The approximation setting
being conforming, we infer that V♭ = H01 (D), and we equip all the spaces
with the |·|H 1 (D) -seminorm. We observe that exact adjoint consistency holds
since a(v, ζ) = (v, A∗ ζ)L2 (D) for all v ∈ H01 (D) and all ζ ∈ H01 (D) such that
A∗ ζ ∈ L2 (D). The estimate (25.7) results from (21.21) and the approximation
properties of the finite element space Vh .
⊓
⊔
Remark 25.4 (Flux recovery). In Chapter 38 we show that it is possible to
post-process locally the H 1 -conforming finite element approximation so as to
built a discrete flux σh∗ ∈ H(div; D) approximating the exact flux σ = −∇u
⊓
⊔
in H(div; D).
Chapter 25. H 1 -conforming approximation
340
25.1.3 Elliptic projection
g
The elliptic (also called Riesz) projector Ihell : H01 (D) → Pk,0
(Th ) is the linear
operator such that
g
∀wh ∈ Pk,0
(Th ),
a(Ihell (v), wh ) = a(v, wh ),
(25.8)
for all v ∈ H01 (D). That (25.8) uniquely defines Ihell (v) follows from the Lax–
Milgram Lemma. In the context of the finite element approximation of the
model problem (25.2), one readily sees that uh = Ihell (u). The elliptic projection can be defined more generally for any function v ∈ H01 (D) without
assuming that v solves some PDE.
g
(Th ) is pointwise invariant by the elliptic projector Ihell ;
The space Pk,0
ell
1
moreover Ih is H -stable and delivers optimal approximation errors in H 1
with improved L2 -estimates provided the suitable smoothing property holds,
see Exercise 25.3. Stability and approximation properties in other norms
have been extensively explored in the literature. One important result is
quasi-optimal approximation in L∞ on quasi-uniform meshes, see Schatz and
Wahlbin [422]: There is c, uniform with respect to h, such that
ku − Ihell (u)kL∞ (D) ≤ c(ln(h))α
inf
g
vh ∈Pk,0
(Th )
ku − vh kL∞ (D) ,
(25.9)
with α = R1 for k = 1 and α = 0 if k ≥ 2. Moreover, for the bilinear form
a(v, w) = D ∇v·∇w dx in H01 (D) with a quasi-uniform mesh sequence on a
convex polygonal domain (d = 2), it is shown in Rannacher and Scott [401]
that kIhell (v)kW 1,p (D) ≤ ckvkW 1,p (D) , for all p ∈ [2, ∞] and all v ∈ W01,p (D).
Recent extensions to three dimensions can be found in Guzmán et al. [279]
and extensions to graded meshes can be found in Demlow et al. [191].
25.2 Non-homogeneous Dirichlet conditions
In this section, we consider the PDE (25.1) with the same assumptions as
in §25.1 for d, β, µ, and f , but with the non-homogeneous Dirichlet condition
u = g on ∂D. We rewrite the weak formulation of §24.2.2 in the following
equivalent form:
Find u ∈ V := H 1 (D) such that u|∂D = g and
R
(25.10)
a(u, w) = D f w dx, ∀w ∈ V0 := H01 (D).
25.2.1 Discrete problem and well-posedness
Let {ϕa }a∈Ah and {σa }a∈Ah be the global shape functions and degrees of
freedom in Pkg (Th ), respectively, see §15.2. Recall that the degrees of freedom
for Lagrange elements are point-values; they are point-values or integrals over
Part V. Elliptic PDEs
341
edges, faces, or cells for the canonical hybrid element. Let t > d2 and let
Ih : H t (D) → Pkg (Th ) denote the
P canonical or Lagrange interpolation operator
from §15.3 such that Ih (v) = a∈Ah σa (v)ϕa for all v ∈ H t (D). Recall from
Definition 15.16 that the set A∂h is the collection of the boundary degrees of
freedom; recall also that σa (v) only depends on v|∂D for all a ∈ A∂h , i.e., we
can write σa (v) = σa∂ (v|∂D ), where σa∂ can be a value at a boundary point
or an integral over a boundary edge or face. We assume for simplicity that
gP∈ H t (∂D) with t > d−1
so that g has point-values, and we set gh :=
2
∂
a∈A∂ σa (g)ϕa|∂D . The discrete problem is:
h
(
Find uh ∈ Vh := P g (Th ) such that uh|∂D = gh and
R
g
(Th ).
a(uh , wh ) = D f wh dx, ∀wh ∈ Vh0 := Pk,0
(25.11)
Note that the discrete trial space includes boundary degrees of freedom (which
are explicitly prescribed), but not the discrete test space.
Lemma 25.5 (Well-posedness). (25.11) is well-posed.
P
Proof. Setting ugh = a∈A∂ σa∂ (g)ϕa (notice that ugh ∈ Vh ), we observe that
h
the function uh0 := uh − ugh is in Vh0 (since uh0|∂D vanishesR identically)
and that a(uh0 , wh ) = ℓ(wh ) for all wh ∈ Vh0 with ℓ(wh ) = D f wh dx −
a(ugh , wh ). Since a is coercive on Vh0 , the Lax–Milgram Lemma implies that
uh0 is uniquely defined.
⊓
⊔
25.2.2 Error analysis(♦)
Theorem 25.6 (H 1 -estimate). Let u solve (25.10) and let uh solve (25.11).
Assume that k + 1 > d2 and u ∈ H 1+r (D) with r ∈ N and d2 <1 + r ≤ k + 1.
The following holds:
|u − uh |H 1 (D) ≤ c hr |u|H 1+r (D) .
(25.12)
Proof. The proof is similar to that of Céa’s Lemma, except that now we need
to account for the interpolation of the Dirichlet condition. Since 1 + r > d2 ,
Ih (u) is well-defined. The key observation is that Ih (u) − uh ∈ Vh0 since
X
σa∂ (u|∂D )ϕa|∂D − gh = 0.
(Ih (u)u − uh )|∂D = (Ih (u))|∂D − gh =
a∈A∂
h
Moreover, (25.10) and (25.11) imply that a(u − uh , vh ) = 0 for all vh ∈ Vh0 ⊂
H01 (D). The coercivity of a on V0 (with parameter α > 0) implies that
α|Ih (u) − uh |H 1 (D) ≤ sup
vh ∈Vh0
= sup
vh ∈Vh0
a(Ih (u) − uh , vh )
|vh |H 1 (D)
a(Ih (u) − u, vh )
≤ M ku − Ih (u)kH 1 (D) ,
|vh |H 1 (D)
342
Chapter 25. H 1 -conforming approximation
where M is the boundedness parameter of a on H 1 (D) × H01 (D). Combining
this bound with the triangle inequality, we obtain |u−uh |H 1 (D) ≤ (1+ M
α )ku−
Ih (u)kH 1 (D) , and we conclude using Corollary 15.12.
⊓
⊔
We now use duality techniques to derive an improved L2 -norm error estimate. The adjoint operator A∗ and the adjoint solution ζ ∈ H01 (D) are the
same as in §25.1.2; notice that ζ satisfies a homogeneous Dirichlet condition.
Theorem 25.7 (L2 -estimate). Assume that there is s ∈ ( 12 , 1] such that
the adjoint operator has a smoothing property in H 1+s (D). Assume that d is
Lipschitz. Assume that k + 1 > d2 and u ∈ H 1+r (D) with d2 ≤ 1 + r ≤ k + 1.
Then, the following holds:
ku − uh kL2 (D) ≤ c (hr+s |u|H 1+r (D) + kgh − gkL2 (∂D) ).
(25.13)
Proof. The proof is similar to that of the Aubin–Nitsche Lemma, except that
we need to account for the interpolation of the Dirichlet condition. Let ζ ∈
H01 (D) be the adjoint solution with data g = Ih (u) − uh . Since Ih (u) − uh ∈
Vh0 ⊂ H01 (D), we obtain that
kIh (u) − uh k2L2 (D) = a(Ih (u) − uh , ζ)
= a(Ih (u) − u, ζ) + a(u − uh , ζ) =: T1 + T2 .
The term T1 is bounded using Lemma 25.8 below, leading to
|T1 | ≤ c (ku − Ih (u)kH 1−s (D) + kg − gh kL2 (∂D) )kζkH 1+s (D) ,
where we have used that (Ih (u) − u)|∂D = gh − g. For the term T2 , we use
the fact that a(u − uh , vh ) = 0 for all vh ∈ Vh0 , the boundedness of a on
g,av
to obtain
H 1 (D)×H01 (D), and the approximation properties of Ih0
g,av
|T2 | ≤ M ku − uh kH 1 (D) |ζ − Ih0
(ζ)|H 1 (D) ≤ c hs ku − uh kH 1 (D) |ζ|H 1+s (D) .
Collecting the bounds for T1 and T2 , using that ku − uh kH 1 (D) ≤ c(|u −
uh |H 1 (D) +kg −gh kL2 (∂D) ) from (24.19), and using elliptic regularity to bound
kζkH 1+s (D) , we infer that
kIh (u)−uh kL2 (D) ≤ c (hs |u − uh |H 1 (D) +ku−Ih (u)kH 1−s (D) +kgh −gkL2 (∂D) ).
We conclude using Theorem 25.6, the approximation properties of Ih yielding
ku − Ih (u)kH 1−s (D) ≤ chr+s |u|H 1+r (D) (this bound is derived by invoking the
⊓
⊔
Riesz–Thorin Theorem), and the triangle inequality.
Lemma 25.8 (H 1+s -boundedness). Assume that
lowing holds:
d is Lipschitz. The fol-
a(v, ζ) ≤ c(kvkH 1−s (D) + kvkL2 (∂D) )kζkH 1+s (D) ,
for all v ∈ H 1 (D) and all ζ ∈ H01 (D) ∩ H 1+s (D) with s ∈ ( 12 , 1].
(25.14)
Part V. Elliptic PDEs
343
Proof. Since d is Lipschitz and ∇ζ ∈ H s (D), we infer that d·∇ζ ∈ H s (D).
1
This implies that d·∇ζ has a trace in H s− 2 (∂D) and hence n·d·∇ζ ∈ L2 (∂D)
1
−1+s
(D) with k∇·(d·∇ζ)kH −1+s (D) ≤
(since s > 2 ). Moreover, ∇·(d·∇ζ) ∈ H
1−s
(D) = H01−s (D) (since 1 − s < 21 ), the
c|ζ|H 1+s (D) . Observing that H
linear form ∇·(d·∇ζ) can act on any v ∈ H01−s (D) despite that v may have
a nonzero trace at the boundary. Denoting
by h·, ·i the corresponding
duality
R
R
product, we infer that h∇·(d·∇ζ), vi + D ∇v·d·∇ζ dx = ∂D (n·d·∇ζ)v ds. As
Ra result, the bilinearR form a can be rewritten as a(v, ζ) = −h∇·(d·∇ζ), vi +
(n·d·∇ζ)v ds + D (−β·∇ζ + (µ − ∇·β)ζ)v dx. Denote by T1 , T2 , T3 the
∂D
three terms on the right-hand side. These terms can be bounded as follows:
|T1 | ≤ k∇·(d·∇ζ)kH −1+s (D) kvkH 1−s (D) ≤ c |ζ|H 1+s (D) kvkH 1−s (D) ,
|T2 | ≤ kn·d·∇ζkL2 (∂D) kvkL2 (∂D) ≤ c |ζ|H 1+s (D) kvkL2 (∂D) ,
|T3 | ≤ ckζkH 1 (D) kvkL2 (D) ≤ c kζkH 1+s (D) kvkH 1−s (D) .
⊓
⊔
25.2.3 Algebraic viewpoint
The algebraic realization of the discrete problem (25.11) is the linear system
AU = B of size card(Ah ). To identify a block structure, we enumerate the
degrees of freedom in A◦h first and then those in A∂h . Set N ◦ := card(A◦h ),
N ∂ := card(A∂h ). This leads to the following block-decomposition of the linear
system (with obvious notation)
◦ ◦
U
B
A◦◦ A◦∂
(25.15)
∂ =
∂ ,
U
B
O
I
′
◦
◦∂
′
with A◦◦
aa′ = a(ϕa′ , ϕa ) for all a, a ∈ Ah , Aaa′ = a(ϕa′ , ϕa ) for all (a, a ) ∈
◦
∂
∂
◦
Ah × Ah , O is a zero rectangular matrix of order
R card(Ah ) × card(Ah ), I
is the identity matrix of order card(A∂h ), Ba◦ = D f ϕa dx for all a ∈ A◦h ,
and Ba∂ = σa∂ (g) for all a ∈ A∂h . The matrix A◦◦ is invertible owing to the
H01 (D)-coercivity of a.
A first option to solve (25.15) is to eliminate U ∂ , i.e., to solve the linear
system A◦◦ U ◦ = B ◦ − A◦∂ B ∂ . The advantage is that the final size of the
linear system is optimal since only the internal degrees of freedom are unknown. However, this technique requires assembling two matrices, A◦◦ and
A◦∂ , instead of one, and the two matrices have a different sparsity profile. An
alternative technique consists of assembling first the stiffness matrix for all
the degrees of freedom in Ah (this is the stiffness matrix for the Neumann
problem) and then correcting the rows for a ∈ A∂h by setting the entries to zero
except the diagonal ones which are set to 1. The right-hand side of (25.15) is
assembled similarly. Despite the slight increase in the number of unknowns,
this technique is computationally effective. It has the apparent drawback of
breaking the symmetry of the model problem (recall that A◦◦ is symmetric
if the advective velocity is zero) since the matrix in (25.15) is not symmetric.
Actually, when using an iterative solution method based on a Krylov space,
344
Chapter 25. H 1 -conforming approximation
if the initial residual is zero for the boundary degrees of freedom, it is always
zero during the iterations, and the iterative algorithm behaves exactly as if
the boundary degrees of freedom are eliminated; see Exercise 25.5.
Remark 25.9 (Penalty method). Another technique to enforce Dirichlet
conditions without elimination consists of using a penalty method. First, one
assembles the matrix and the right-hand side of the Neumann problem. Then,
for each row associated with a ∈ A∂h , one adds ǫ−1 to the diagonal entry
of the stiffness matrix and ǫ−1 σa∂ (g) to the right-hand side, see Lions [344],
Babuška [32]. If ǫ−1 is not large enough, the method suffers from a lack of
consistency. This problem can be avoided by adding extra boundary terms
ensuring consistency, see Nitsche [376] and Chapter 29.
⊓
⊔
25.3 Discrete maximum principle(♦)
The maximum principle is an important property of scalar, second-order, elliptic PDEs that sets them apart from higher-order PDEs and systems of
PDEs. We focus here on Dirichlet boundary conditions.
Theorem 25.10 (Maximum principle). Let D be a Lipschitz domain in
Rd . Let d, β, and µ satisfy the assumptions in §24.1.1. Assume
R µ ≥ 0 in D.
Let f ∈ L2 (D) and let u ∈ H 1 (D) be such that a(u, w) = D f w dx for all
w ∈ H01 (D). Then, the following holds:
[ u ≤ 0 on ∂D and f ≤ 0 in D ] =⇒ [ u ≤ 0 in D ].
(25.16)
Moreover, if µ ≡ 0 in D, then f ≤ 0 in D implies u ≤ sup∂D u in D, and
f = 0 in D implies inf ∂D u ≤ u ≤ sup∂D u in D.
Proof. See Gilbarg and Trudinger [249, Thm. 8.1]; see also Brezis [99, Prop. 9.29]
and Evans [229, §6.4].
⊓
⊔
Remark 25.11 (Sign change). Owing to the linearity of the PDE, the
statements of Theorem 25.10 remain valid by changing signs, e.g., u ≥ 0 on
∂D and f ≥ 0 in D imply u ≥ 0 in D, and f ≥ 0 in D implies u ≥ inf ∂D u in
D.
⊓
⊔
The discrete analogue of a maximum principle is called the Discrete Maximum Principle (DMP); see Ciarlet and Raviart [154] for one of the pioneering
works on this topic which was motivated by the derivation of uniform a priori
error estimates. As in [154], we focus for simplicity on linear finite elements on
simplicial meshes, and we consider homogeneous Dirichlet conditions (see Vejchodský and Šolı́n [473] for a 1D example of DMP with high-order elements).
g
Let P1,0
(Th ) be the H01 (D)-conforming finite element space using linear finite
g
elements. Let {ϕi }i∈{1: N } denote the global shape functions in P1,0
(Th ) (with
N the number of interior mesh vertices).
Part V. Elliptic PDEs
345
Definition 25.12 (DMP). We say thatR the DMP holds for the discrete probg
lem (25.3) with Vh = P1,0
(Th ) and Fi = D f ϕi dx, i ∈ {1: N }, if the following
holds:
[ F ≤ 0 in D ] =⇒ [ uh ≤ 0 in D ].
(25.17)
For a vector X ∈ RN , X ≤ 0 means that Xi ≤ 0 P
for all i ∈ {1: N }. Then,
uh ≤ 0 in D if and only if U ≤ 0 in RN with uh = i∈{1: N } Ui ϕi since the
linear shape functions are non-negative.
Proposition 25.13 (DMP and stiffness matrix). The DMP holds if and
only if the stiffness matrix A associated with (25.3) is such that (A−1 )ij ≥ 0
for all i, j ∈ {1: N }.
Proof. A−1 F ≤ 0, ∀F ≤ 0, if and only if (A)−1
ij ≥ 0, ∀i, j ∈ {1: N }.
⊓
⊔
Lemma 25.14 (Sufficient condition). Assume that the stiffness matrix is
a Z-matrix, i.e., assume that the following holds (see Definition 22.4):
a(ϕi , ϕj ) ≤ 0,
∀i, j ∈ {1: N }, i 6= j.
(25.18)
Then, the DMP holds; that is to say, A is a non-singular M -matrix.
Proof. We present a proof in the spirit of Stampacchia’s truncation method;
see also Jiang and Nochetto [306] as well as Exercise 25.7 for an alternative,
more direct proof. Let Π + : RN → RN be the nonlinear map defined by
its components Π + (V )i = max(0, Vi ) for all i ∈ {1: N }. Let us prove that
Π + (V )T AV ≥ Π + (V )T AΠ + (V ) for all V ∈ RN . Indeed,
X
(V − Π + (V ))T AΠ + (V ) =
min(0, Vi )Aij max(0, Vj )
i,j∈{1: N }
=
X
i6=j
min(0, Vi )Aij max(0, Vj ) ≥ 0,
since A is a Z-matrix. Let B ≤ 0 and let U ∈ RN solve AU = B. Then,
0 ≥ Π + (U )T B = Π + (U )T AU ≥ Π + (U )T AΠ + (U ),
whence we infer that Π + (U ) = 0 since A is positive-definite (owing to the
coercivity of the bilinear form a); in other words, U ≤ 0, i.e., the DMP
holds. The equivalence with A being a non-singular M -matrix results from
Proposition 25.13 and Definition 22.4. Note in passing that the above proof
implies that any positive-definite Z-matrix is an M -matrix.
⊓
⊔
Realizing (25.18) on general diffusion-advection-reaction PDEs is somewhat delicate, so that we continue the discussion by restricting ourselves to
the Poisson problem, i.e., d = Id and β = 0, µ = 0.
Chapter 25. H 1 -conforming approximation
346
Definition 25.15 (Weakly acute simplicial mesh). The simplicial mesh
Th is said to be weakly acute if (25.18) holds for the Poisson problem, i.e., if
Z
∇ϕi ·∇ϕj dx ≤ 0,
∀i, j ∈ {1: N }, i 6= j.
(25.19)
D
The condition (25.19) is always satisfied in dimension d = 1, and in higher
dimensions it boils down to some geometric restriction on the mesh. More
precisely, it is shown in Xu and Zikatanov [494] that (25.19) holds if and only
if the following inequality holds for any (interior) edge e in the mesh:
X
|FK,e,1 ∩ FK,e,2 | cot αK,e ≤ 0,
(25.20)
K∈Te
where Te is the set of mesh cells containing e, FK,e,1 and FK,e,2 are the two
faces of K not containing e (FK,e,1 ∩ FK,e,2 is a (d − 2)-dimensional simplex,
and for d = 2, its measure is conventionally equal to one), and αK,e is the
interior dihedral angle between these two faces. In dimension d = 2, the
inequality (25.20) means that the sum of the two angles opposite to any
(interior) edge is less than or equal to π, which in turn implies that the
triangulation is Delaunay. A more stringent condition is to require that the
simplicial mesh be nonobtuse, meaning that all the dihedral angles of a given
simplex in the mesh are less than or equal to π2 . Indeed, it is shown in Brandts
et al. [93] that the following identity holds for all d ≥ 2,
Z
|FK,i ||FK,j |
∇ϕi ·∇ϕj dx =
cos(αK,i,j ),
(25.21)
d2 |K|2
D
where FK,i (resp., FK,j ) is the face of K opposite to the vertex of K associated
with ϕi (resp., ϕj ), and αK,i,j is the interior dihedral angle between FK,i and
FK,j . Summing (25.21) over all the simplices in the support of ϕi and ϕj ,
we infer that a nonobtuse simplicial mesh is weakly acute. See also Brandts
et al. [94] for further insight on nonobtuse simplicial partitions. Incidentally,
it is noticed in Brandts et al. [95] that the condition (25.19) cannot hold on
Cartesian meshes in space dimension d ≥ 4, and the strict inequality cannot
hold in three dimensions.
Remark 25.16 (Nonlinear stabilization). An alternative approach to the
DMP that avoids geometric requirements on the mesh is based on adding
nonlinear viscosity to the discrete problem; see [112, 113].
⊓
⊔
Exercises
Exercise 25.1 (Adjoint operator). Let A ∈ L(H01 (D); H −1 (D)) be such
that Av = −∇·(d∇v) + β·∇v + µv for all v ∈ H01 (D). Show that the adjoint
operator is such that A∗ v = −∇·(d∇v)−β·∇v+(µ−∇·β)v. (Hint: use (24.4).)
Part V. Elliptic PDEs
347
Exercise 25.2 (Negative-norm estimate). Let t ≥ 1 and consider the
norm kvk−t,D = supz∈H t (D)∩H01 (D)
−t
(v,z)L2 (D)
kzkH t (D)
(except for t = 1, this is not the
norm of the dual space H (D)). Assume that, for all g ∈ H t (D), the adjoint
solution ζ ∈ H01 (D) such that A∗ ζ = g satisfies kζkH 1+t+s (D) ≤ csp kgkH t (D)
for some s ∈ ( 12 , 1]. Assume that the polynomial degree k of the finite elements
composing Vh satisfies k ≥ t + s. Prove that ku − uh k−t,D ≤ cht+s ku −
uh kH 1 (D) . (Hint: consider the dual problem A∗ ζ = z.)
Exercise 25.3 (Elliptic projector). Let Ihell be defined by (25.8).
g
(i) Prove that Ihell leaves Pk,0
(Th ) pointwise invariant and that Ihell is H 1 stable. (Hint: use coercivity and boundedness of a.)
(ii) Prove that kv − Ihell (v)kH 1 (D) ≤ chr |v|H 1+r (D) for all r ∈ (0, k] and all
g,av
g,av
v ∈ H 1+r (D). (Hint: observe that Ihell (Ih0
(v)) = Ih0
(v).)
∗
(iii) Assume that the adjoint operator A has a smoothing property in
H 1+s (D) for some real number s ∈ (0, 1]. Prove that kv −Ihell (v)kL2 (D) ≤
chr+s |v|H 1+r (D) . (Hint: consider the adjoint problem A∗ ζ = v − Ihell (v).)
Exercise 25.4 (Regularity assumption). Let uh solve (25.11). Assume
that u ∈ H 1+r (D) with r ∈ (0, k]. Prove that ku−uh kH 1 (D) ≤ c(hr |u|H 1+r (D) +
P
P
1
g,av
kg−gh k2L2 (F ) ) 2 ). (Hint: consider vh = Ih0
( F ∈F ∂ h−1
(u)+ a∈A∂ σa∂ (g)ϕa ;
F
h
h
to bound ku − vh kH 1 (D) , see the proof of Theorem 16.12.)
Exercise 25.5 (Non-homogeneous Dirichlet). Let A denote the system matrix in (25.15). Let R ∈ RN and let k ≥ 1. Define the Krylov
space to be K := span{R, AR, . . . , Ak−1 R}. For any vector X ∈ RN , write
X = (X ◦ , X ∂ )T . Assume that R∂ = 0.
(i) Prove that Y ∂ = 0 for all Y .
(ii) Prove that if A◦◦ is symmetric, the restriction of A to K is symmetric.
Exercise 25.6 (Stiffness matrix).
b1 , λ
b2 , λ
b3 } and {θb1 , θb2 , θb3 , θb4 } be
Let {λ
the shape functions of the P1 and Q1
b
Lagrange element, respectively, with K
shown on the right.
Evaluate the stiffR
ness matrix for Kb ∇v·∇w dx, numbering nodes from the lower right one.
Consider the meshes of D =
(0, 3)×(0, 2) shown on the left.
Evaluate
the stiffness matrix for
R
∇v·∇w
dx. (Hint: the geometric
D
maps are isometries.)
(0,1)
(0,0)
(1,0)
(0,1)
(1,1)
(0,0)
(0,1)
348
Chapter 25. H 1 -conforming approximation
Exercise 25.7 (DMP). Assume
Pthat the stiffness matrix is a Z-matrix. Assume the following: (i)PAii ≥ − j6=i Aij for all i ∈ {1: N }; (ii) ∃i∗ ∈ {1: N }
such that Ai∗ i∗ > − j6=i∗ Ai∗ j ; (iii) For all i ∈ {1: N }, i 6= i∗ , there exists a path [i = i1 , . . . , iI = i∗ ] such that Aij ij+1 < 0 for all j ∈ {1: I − 1}.
Prove that A is a non-singular M -matrix. (Hint: let B ≤ 0, let U = A−1 B;
proceeding by contradiction, assume that there is i ∈ {1: N } such that
Ui = maxj∈{1: N } Uj > 0.)
Exercise 25.8 (Obtuse mesh).
The mesh shown on the right contains three interior nodes with coordinates (1, 1), (3, 1), and
(2, 23 ). The sum of the two angles opposite the
edge linking the first two nodes is > π. Assemble the stiffness matrix A.
(Hint: for d = 2, the local stiffness matrix is translation- and scale-invariant;
there are four shapes of triangles in the mesh, and one can work on triangles
with vertices ((0, 0), (1, 0), (0, 1)), ((0, 0), (1, 0), (0, 12 )), ((−1, 0), (1, 0), (0, 12 )),
and ((−1, 1), (1, 1), (0, 1)).) Is A a Z-matrix? Compute A−1 .