Part II, Chapter 8
Local interpolation in affine meshes
We have seen in the previous chapter how to build finite elements and local
interpolation operators in each cell K of a mesh Th . In this chapter, we analyze
the local interpolation error for smooth Rq -valued functions, q ≥ 1. We restrict
the material to affine meshes and to transformations ψK such that
ψK (v) = AK (v ◦ TK ),
(8.1)
where AK is a matrix in Rq×q . Non-affine meshes are treated in Chapter 9.
We introduce the notion of shape-regular families of affine meshes, we study
the transformation of Sobolev norms using (8.1), and we present important
approximation results collectively known as the Bramble–Hilbert lemmas. We
finally prove the main result of this chapter, which is an upper bound on the
local interpolation error over each mesh cell for smooth functions.
8.1 Shape-regularity for affine meshes
Let Th be an affine mesh. Let K ∈ Th . Since the geometric map TK is affine,
there is a matrix JK ∈ Rd×d such that
TK (b
x) − TK (b
y ) = JK (b
x − yb),
b
∀b
x, yb ∈ K.
(8.2)
The matrix JK is invertible since the map TK is bijective. Moreover, the
b for
b = JK h
(Fréchet) derivative of the geometric map is such that DTK (b
x)(h)
d
d
b
all h ∈ R . We denote the Euclidean norm in R by k·kℓ2 (Rd ) , or k·kℓ2 when
the context is unambiguous. We abuse the notation by using the same symbol
for the induced matrix norm.
Lemma 8.1 (Bound on JK ). Let Th be an affine mesh and let K ∈ Th . Let
ρK be the diameter of the largest ball that can be inscribed in K and let hK
♥♥
90
Chapter 8. Local interpolation in affine meshes
hK
ρK
θK,z
z
Fig. 8.1. Triangular cell K with vertex z, angle θK,z , and largest inscribed ball.
hKb be defined similarly.
be the diameter of K (see Figure 8.1). Let ρbKb and b
Then, the following holds:
|det(JK )| =
|K|
,
b
|K|
kJK kℓ2 ≤
hK
,
ρKb
kJ−1
K kℓ2 ≤
hKb
.
ρK
(8.3)
Proof. The first equality results from the fact that
Z
Z
b
|K| =
|det(JK )| db
x = |det(JK )||K|.
dx =
b
K
K
Regarding the bound on kJK kℓ2 , we observe that
b ℓ2
kJK hk
1
=
b ℓ2
ρKb
b =0 khk
h6
kJK kℓ2 = sup
sup
b 2 =ρ c
khk
ℓ
K
b ℓ2 .
kJK hk
b = TK (b
b =x
b and use JK h
b1 −b
b1 , x
b2 ∈ K
x1 )−TK (b
x2 ) = x1 −x2
Write h
x2 with x
b
to infer that kJK hkℓ2 ≤ hK . This proves the bound on kJK kℓ2 . The bound
b
on kJ−1
⊓
⊔
K kℓ2 is obtained by exchanging the roles of K and K.
Since the analysis of the interpolation error (implicitly) invokes sequences
of successively refined meshes, we henceforth denote by (Th )h>0 a sequence of
meshes approximating a domain D ⊂ Rd , where the index h takes values in
a countable set having zero as the only accumulation point. A mesh sequence
is said to be affine if all the meshes in the sequence are affine.
Definition 8.2 (Shape-regularity). A sequence of affine meshes (Th )h>0 is
said to be shape-regular if there exists σ♯ such that
σK :=
hK
≤ σ♯ ,
ρK
∀K ∈ Th , ∀h > 0.
(8.4)
Owing to Lemma 8.1, a shape-regular sequence of affine meshes satisfies
kJK kℓ2 kJ−1
b,
K kℓ2 ≤ σ♯ σK
∀K ∈ Th , ∀h > 0.
(8.5)
Example 8.3 (Dimension 1). Every sequence of one-dimensional meshes
is shape-regular, since hK = ρK when d = 1.
⊓
⊔
Part II. Finite Element Interpolation
91
Example 8.4 (Triangulations). A shape-regular sequence of affine triangulations can be obtained from an initial triangulation by connecting all the
edge midpoints and repeating this procedure as many times as needed.
⊓
⊔
Remark 8.5 (Angles). Let (Th )h>0 be a shape-regular sequence of affine
simplicial meshes and let K ∈ Th . Assume that d = 2, let K be a triangle
in Th and let z be a vertex of K; then, the angle θK,z ∈ (0, 2π) formed
by the (two) edges of K sharing z is uniformly bounded away from zero.
Indeed, the angular sector centered at z of angle θK,z and radius hK covers
the ball of diameter ρK that is inscribed in K (see Figure 8.1); as a result,
−2
1
1
1 2
2
2 hK θK,z ≥ 4 πρK , which in turn implies θK,z ≥ 2 πσ♯ . Assume now that
d = 3, let K be a tetrahedron and let z be a vertex of K; then, the solid angle
ωK,z ∈ (0, 4π) formed by the (three) faces of K sharing z is uniformly bounded
away from zero. Reasoning as above, with volumes instead of surfaces, leads
to 13 h3K ωK,z ≥ 16 πρ3K , thereby implying that ωK,z ≥ 21 πσ♯−3 .
⊓
⊔
8.2 Transformation of Sobolev seminorms
The question we investigate now is the following: given a function v ∈
b Rq ) compare
W m,p (K; Rq ), how does the seminorm of ψK (v) in W m,p (K;
m,p
q
to that of v in W
(K; R ) when ψK is defined by (8.1)?
Lemma 8.6 (Norm scaling by ψK ). Let Th be a mesh. Let m ∈ N and
1
p ∈ [1, ∞] (with z ± p = 1, ∀z > 0 if p = ∞). There exists c depending
only on m and d such that the following bounds hold for all K ∈ Th and all
v ∈ W m,p (K; Rq ):
1
−p
m
|v|W m,p (K;Rq ) ,
|ψK (v)|W m,p (K;R
b q ) ≤ c kAK kℓ2 kJK kℓ2 |det(JK )|
(8.6a)
1
p
−1 m
|v|W m,p (K;Rq ) ≤ c kA−1
b q ).
K kℓ2 kJK kℓ2 |det(JK )| |ψK (v)|W m,p (K;R
(8.6b)
Proof. Let α be a multi-index with length |α| = m, i.e., α = (α1 , . . . , αd ) ∈
b Owing to (B.8), we infer that
b ∈ K.
{1: l}d with α1 + . . . + αd = m. Let x
∂ α (ψK (v))(b
x) = AK Dm (v ◦ TK )(b
x)(e1 , . . . , e1 , . . . , ed , . . . , ed ),
| {z }
| {z }
×α1
×αd
b and (e1 , . . . , ed )
with Dm (v◦TK )(b
x) the m-th Fréchet derivative of v◦TK at x
the canonical Cartesian basis of Rd . We now apply the chain rule (see
Lemma B.13) to v ◦ TK . Since TK is affine, the Fréchet derivative of TK
b and its higher-order Fréchet derivatives vanish. Hence,
is independent of x
92
Chapter 8. Local interpolation in affine meshes
Dm (v ◦ TK )(b
x)(h1 , . . . , hm ) =
X 1
(Dm v)(TK (b
x))(DTK (hσ(1) ), . . . , DTK (hσ(m) )),
m!
σ∈Sm
for all h1 , . . . , hm ∈ Rd , where Sm is the set of the permutations of {1: m}.
Since DTK (h) = JK h for all h ∈ Rd owing to (8.2), we infer that
m
|∂ α (v ◦ TK )(b
x)| ≤ kJK km
x))kMm (Rd ,...,Rd ;Rq ) ,
ℓ2 k(D v)(TK (b
where we have set kAkMm (Rd ,...,Rd ;Rq ) := sup(y1 ,...,ym )∈Rd ×...×Rd
kA(y1 ,...,ym )kℓ2
ky1 kℓ2 ...kym kℓ2
for any multilinear map A ∈ Mm (Rd , . . . , Rd ; Rq ). Owing to the multilinearity of Dm v and using again (B.8), we infer that (see Exercise 8.1)
X
k(Dm v)(TK (b
x))kMm (Rd ,...,Rd ;Rq ) ≤ c
k(∂ β v)(TK (b
x))kℓ2 ,
|β|=m
where c only depends on m and d. As a result,
X
k∂ α (ψK (v))(b
x)kℓ2 ≤ c kAK kℓ2 kJK km
k(∂ β v)(TK (b
x))kℓ2 ,
ℓ2
|β|=m
b
and (8.6a) follows by taking the Lp (K)-norm
of the functions on both sides
of the inequality. The proof of (8.6b) is similar.
⊓
⊔
Remark 8.7 (Seminorms). The upper bounds in (8.6a) and (8.6b) involve
only seminorms because the geometric maps are affine.
⊓
⊔
8.3 Bramble–Hilbert Lemmas
This section contains important results for the analysis of the approximation
properties of finite elements. Lemma 8.8 is the key result. The estimate (8.7)
is proved in Bramble and Hilbert [78, Thm. 1] and in Ciarlet and Raviart
[149, Lem. 7], see also Deny and Lions [190]. The more general estimate (8.8),
rarely mentioned in the literature, is useful to estimate the local interpolation
error for a larger class of functions and polynomial degrees; see Theorem 8.12
below. We state the results for scalar-valued functions; they readily extend to
the vector-valued case by working componentwise.
Lemma 8.8 (Pk -Bramble–Hilbert/Deny–Lions). Let S be a connected,
open, bounded Lipschitz set in Rd . Let p be a real number such that p ∈ [1, ∞].
Let k ∈ N. Then, there exists c > 0 (depending on k, p, S) such that
inf kv + qkW k+1,p (S) ≤ c |v|W k+1,p (S) ,
q∈Pk,d
∀v ∈ W k+1,p (S).
(8.7)
Part II. Finite Element Interpolation
93
More generally, for all l ∈ N with l ≥ k + 1, there exists c > 0 such that
inf kv + qkW l,p (S) ≤ c
q∈Pk,d
l
X
m=k+1
|v|W m,p (S) ,
∀v ∈ W l,p (S).
(8.8)
Proof. It suffices to prove (8.8), since (8.7) is just (8.8) with
l = k + 1.
(1) Let Nk,d = dim Pk,d , and introduce the Nk,d = k+d
continuous linear
d
forms
Z
l,p
∂ α v dx ∈ R,
∀α ∈ Nd , |α| ≤ k,
fα : W (S) ∋ v 7−→ fα (v) =
S
where α = (α1 , . . . , αd ) is a multi-index. The key property of these linear forms
is that the restriction to Pk,d of the map Φk : W l,p (S) ∋ q 7→ (fα (q))|α|≤k ∈
RNk,d is an isomorphism (it is injective and dim Pk,d = Nk,d ).
(2) Let us prove that there is c > 0, depending on S, k and l, such that
c kvkW l,p (S) ≤
l
X
m=k+1
|v|W m,p (S) +
X
|α|≤k
|fα (v)|,
∀v ∈ W k+1,p (S).
(8.9)
Reasoning by contradiction, let (vn )n∈N be a sequence such that
kvn kW l,p (S) = 1, lim |vn |W m,p (S) = 0, lim kΦk (vn )kℓ1 (RNk,d ) = 0,
n→∞
n→∞
(8.10)
for all m ∈ {k + 1: l}. Owing to the Rellich–Kondrachov Theorem B.104, we
infer that, up to a subsequence (not renumbered for simplicity), the sequence
(vn )n∈N converges strongly to a function v in W l−1,p (S). Moreover, (vn )n∈N
is a Cauchy-sequence in W l,p (S) since
kvn − vm kW l,p (S) ≤ kvn − vm kW l−1,p (S) + |vn − vm |W l,p (S) .
Hence, (vn )n∈N converges to v strongly in W l,p (S) (that the limit is indeed
v comes from the uniqueness of the limit in W l−1,p (S)). Owing to (8.10)
we infer that kvkW l,p (S) = 1, |v|W m,p (S) = 0 for all m ∈ {k + 1: l}, and
Φk (v) = 0. Repeated applications of Lemma B.50 (stating that in an open
connected set S, ∇v = 0 implies that v is constant in S) allows us to infer
from |v|W k+1,p (S) = 0 that v ∈ Pk,d . Since the restriction of Φk to Pk,d is an
isomorphism, this yields v = 0, which is a contradiction since kvkW l,p (S) = 1.
(3) Let v ∈ W l,p (S) and define π(v) ∈ Pk,d such that Φk (π(v)) = −Φk (v).
This is possible since the restriction of Φk to Pk,d is an isomorphism. Then,
inf kv + qkW l,p (S) ≤ kv + π(v)kW l,p (S) ≤ c
q∈Pk,d
l
X
m=k+1
|v + π(v)|W m,p (S)
+ c kΦk (v + π(v))kℓ1 (RNk,d ) = c (|v|W k+1,p (S) + . . . + |v|W l,p (S) )
since ∂ α π(v) = 0 for all α such that k + 1 ≤ |α|. This completes the proof.
⊓
⊔
94
Chapter 8. Local interpolation in affine meshes
Remark 8.9 (Peetre-Tartar Lemma). Step (2) in the above proof is similar to the Peetre–Tartar Lemma A.53. For instance, take l = k + 1 and define
X = W k+1,p (S), Y = [Lp (D)]Nk+1,d −Nk,d ×RNk,d , and Z = W k,p (S). Define
the operator
A : X ∋ v 7−→ ((∂ α v)|α|=k+1 , Φk (v)) ∈ Y.
Then, A is bounded and injective, and the embedding X ⊂ Z is compact.
Property (8.9) then results from the Peetre–Tartar Lemma.
⊓
⊔
Lemma 8.10 (Pk -Bramble–Hilbert for linear functionals). Under the
hypotheses of Lemma 8.8 (with l = k+1), there is c > 0 such that the following
holds for all f ∈ (W k+1,p (S))′ := L(W k+1,p (S); R) vanishing on Pk,d ,
|f (v)| ≤ c kf k(W k+1,p (S))′ |v|W k+1,p (S) ,
∀v ∈ W k+1,p (S).
(8.11)
Proof. Left as an exercise. The estimate (8.11) is proved in Bramble and
Hilbert [78, Thm. 2] and in Ciarlet and Raviart [149, Lem. 6].
⊓
⊔
Remark 8.11 (Terminology). It seems that there is some variability in
the literature regarding the exact statement of the Bramble–Hilbert Lemma.
For instance, Lemma 8.8 is attributed to Bramble and Hilbert in Brenner
and Scott [96, p. 102] and Ciarlet and Raviart [147, p. 219], whereas this
lemma is attributed to Deny and Lions in Ciarlet [146, p. 111], and it is not
given any name in Braess [75, p. 77]. Lemma 8.10 is called Bramble–Hilbert
Lemma in Ciarlet [146, p. 192] and Braess [75, p. 78]. Incidentally, there are
two additional Lemmas by Bramble and Hilbert that are the counterparts
of Lemma 8.8 and Lemma 8.10 for Qk,d polynomials, see Lemma 9.7 and
Lemma 9.8 below.
⊓
⊔
8.4 Local finite element interpolation
This section contains our main result on local finite element interpolation.
Recall the construction of §7.3 to generate a finite element and a local interpolation operator in each mesh cell K ∈ Th . Our goal is now to bound
the interpolation error v − IK v for a smooth function v. The key point is
that we want this bound to depend on K only through its size hK , assuming shape-regularity of the mesh sequence. We can see that the Bramble–
Hilbert/Deny–Lions Lemma 8.8 cannot be used directly on K since it will
lead to a constant depending on K. The crucial idea is then to use the fact
−1
◦ IKb ◦ ψK owing to Proposition 7.9 and to apply Lemma 8.8
that IK = ψK
b To allow for a bit more generality, we do not
on the fixed reference cell K.
assume that IKb is a finite element interpolation operator, and we assume that
−1
IK is defined by the relation IK = ψK
◦ IKb ◦ ψK instead of resulting from
the construction of §7.3.
Part II. Finite Element Interpolation
95
b Pb ) with Pb finite
Theorem 8.12 (Local interpolation). Let IKb ∈ L(V (K);
dimensional. Let p ∈ [1, ∞], let k, l ∈ N and assume that the following holds:
b Rq ).
(i) [Pk,d ]q ⊂ Pb ⊂ W k+1,p (K;
q
(ii) [Pk,d ] is pointwise invariant under IKb .
b Rq ) ֒→ V (K)
b (i.e., kb
b
(iii) W l,p (K;
v kV (K)
c kb
v kW l,p (K;R
b ∈ V (K)).
b ≤b
b q ) for all v
Let Th be a shape-regular sequence of affine meshes, let K ∈ Th , and define
−1
the operator IK := ψK
◦ IKb ◦ ψK , with ψK defined by (8.1), and assume that
−1
kAK kℓ2 kA−1
K kℓ2 ≤ γ kJK kℓ2 kJK kℓ2 ,
(8.12)
where γ > 0 is uniform with respect to K and h. Then, there exists c > 0 such
that the following local error estimate holds:
r−m
|v − IK v|W m,p (K;Rq ) ≤ c hK
(|v|W r,p (K;Rq ) + . . . + hs−r
K |v|W s,p (K;Rq ) ), (8.13)
for all r ∈ {0: k + 1}, all m ∈ {0: r}, s := max(l, r), all v ∈ W s,p (K; Rq ), and
all K ∈ Th .
Proof. Let r ∈ {0: k + 1}, m ∈ {0: r}, and set s = max(l, r). Let c denote a
generic constant whose value can change at each occurrence as long as it is
independent of K and v ∈ W s,p (K; Rq ).
b Rq ), we set G(w)
b The operator G is lin(1) For all w
b ∈ W s,p (K;
b := w
b −IKb (w).
b Rq ) ֒→
b is well-defined since s ≥ l and W l,p (K;
ear and well-defined (i.e., IKb (w)
b imply W s,p (K;
b Rq ) ֒→ V (K)).
b
b Rq ) boundV (K)
Moreover, G maps W s,p (K;
m,p b
q
s,p b
q
m,p b
q
edly to W
(K; R ), i.e., G ∈ L(W (K; R ); W
(K; R )). Assume first
that r ≥ 1. Then, the operator G vanishes on Pr−1,d , since r − 1 ≤ k and
Pr−1,d ⊂ Pk,d , which implies that [Pr−1,d ]q is pointwise invariant under IKb .
As a consequence, we infer that
b W m,p (K;R
b W m,p (K;R
|w
b − IKb w|
b q ) = |G(w)|
b q) =
≤ kGkL(W s,p (K;R
b q );W m,p (K;R
b q ))
≤c
inf
p
b∈[Pr−1,d ]q
inf
p
b∈[Pr−1,d ]q
inf
p
b∈[Pr−1,d ]q
|G(w
b + pb)|W m,p (K;R
b q)
kw
b + pbkW s,p (K;R
b q)
kw
b + pbkW s,p (K;R
b W r,p (K;R
b W s,p (K;R
b q ) ≤ c(|w|
b q ) + . . . + |w|
b q ) ),
b Rq ), where we used the estimate (8.8) from the Bramble–
for all w
b ∈ W s,p (K;
Hilbert/Deny–Lions Lemma 8.8 componentwise with with r−1 in lieu of k and
s in lieu of l (this is possible since s ≥ r). Furthermore, the above inequality
is trivial if r = 0.
(2) Now let v ∈ W s,p (K; Rq ). We infer that
1
−1 m
p
|v − IK v|W m,p (K;Rq ) ≤ c kA−1
b q)
K kℓ2 kJK kℓ2 |det(JK )| |ψK (v − IK v)|W m,p (K;R
1
−1 m
p
≤ c kA−1
b (ψK (v))|W m,p (K;R
b q)
K kℓ2 kJK kℓ2 |det(JK )| |ψK (v) − IK
1
−1 m
p
≤ c kA−1
b q ) + . . . + |ψK (v)|W s,p (K;R
b q ))
K kℓ2 kJK kℓ2 |det(JK )| (|ψK (v)|W r,p (K;R
s−r
r
m
≤ c kJ−1
K kℓ2 kJK kℓ2 (|v|W r,p (K;Rq ) + . . . + kJK kℓ2 |v|W s,p (K;Rq ) ),
96
Chapter 8. Local interpolation in affine meshes
where we have used the bound (8.6b) in the first line, the linearity of ψK and
−1
the assumption IK := ψK
◦ IKb ◦ ψK on the second line, the result of Step (1)
in the third line, and the bound (8.6a) together with (8.12) in the fourth
K
is
line. The estimate (8.13) follows by using (8.3) and the fact that σK = hρK
uniformly bounded owing to shape-regularity.
⊓
⊔
In the context of finite elements, assumption (i) in Theorem 8.12 is easy to
satisfy since Pb is in general composed of polynomial functions, and assumption (ii) follows from (i) since Pb is pointwise invariant under IKb .
Definition 8.13 (Degree of a finite element). The largest integer k such
b Rq ) is called the degree of the finite element.
that [Pk,d ]q ⊂ Pb ⊂ W k+1,p (K;
Remark 8.14 (Simpler version). Theorem 8.12 is usually formulated in
the literature under the assumption that the degree k of the finite element is
large enough so that k + 1 ≥ l, where l is the integer from assumption (iii). In
this case, the statement of the theorem can be simplified by taking r ≥ l so
that s = r. The estimate (8.7) from Lemma 8.8 is then sufficient to complete
the proof, and only one term is present in the error bound. The statement is
r−m
|v − IK v|W m,p (K;Rq ) ≤ c hK
|v|W r,p (K;Rq ) ,
(8.14)
for all r ∈ {l: k + 1}, all m ∈ {0: r}, all v ∈ W r,p (K; Rq ), and all K ∈ Th .
⊓
⊔
Example 8.15 (Lagrange elements). The choice (8.1) with AK = 1
for ψK is legitimate for scalar-valued Lagrange elements, see §7.4.1. Taking
b = C 0 (K),
b assumption (iii) is satisfied if we take l to be the smallest
V (K)
b ֒→ V (K)
b owing to
integer such that l > dp ; indeed, this implies that W l,p (K)
d
Theorem B.104. Assuming that k + 1 > p (so that k + 1 ≥ l), the simplified
statement (8.14) becomes
r−m
L
|v − IK
v|W m,p (K) ≤ chK
|v|W r,p (K) ,
(8.15)
for all r ∈ {l: k + 1}, all m ∈ {0: r}, all v ∈ W r,p (K), and all K ∈ Th . Instead,
the more general estimate (8.13) has to be used whenever k + 1 ≤ dp . For
instance, assume k = 1 and p ∈ [1, d2 ], d ≥ 2, so that k + 1 = 2 ≤ dp . For d = 2
and p = 1 or for d = 3 and p ∈ (1, 23 ], we can take l = 3 in assumption (iii)
(since l > dp ). For m = 0 and r = k + 1 = 2, we get s = max(l, r) = 3, which
L
yields kv − IK
vkLp (K) ≤ ch2K (|v|W 2,p (K) + hK |v|W 3,p (K) ).
⊓
⊔
Example 8.16 (Modal elements). In this case, it is legitimate to take
b = L1 (K;
b Rq ), see §7.4.2. Hence, the
l = 0 in assumption (iii) since V (K)
simplified bound (8.14) can always be used.
⊓
⊔
Example 8.17 (Piola transformations). The Piola transformations deT
fined in (7.12) and(7.13) satisfy (8.1) with AK := det(JK )J−1
K and AK := JK ,
respectively. Note that in both cases, the assumption (8.12) holds with c = 1.
The interpolation error analysis is detailed in Chapter 12.
⊓
⊔
Part II. Finite Element Interpolation
97
Exercises
Exercise 8.1 (High-order derivative). Let two integers m, d ≥ 2. Consider the map Φ : {1: d}m ∋ j 7−→ (Φ1 (j), . . . , Φd (j)) ∈ Nd where Φi (j) =
card{k ∈ {1: m} | jk = i} for all i ∈ {1: d}. Observe that |Φ(j)| = m by construction. Define Cm,d = maxα∈Nd ,|α|=m card{j ∈ {1: d}m | Φ(j) = α}. Let v
be a smooth (scalar-valued) function.
1
P
1
α 2 2
2
.
|∂
v|
(i) Show that kDm vkMm (Rd ,...,Rd ;R) ≤ Cm,d
d
α∈N m,|α|=m
m
(ii) Show that for d = 2, Cm,d = max0≤l≤m l = 2 .
(iii) Evaluate Cm,d for d = 3Pand m ∈ {2, 3}.
α
m
(iv) Show conversely that
α∈Nd ,|α|=m |∂ v| ≤ Čm,d kD vkMm (Rd ,...,Rd ;R)
with Čm,d = d+m−1
.
d−1
Exercise 8.2 (Flat triangle). Let K be a triangle with vertices (0, 0), (1, 0)
and (−1, ǫ) with 0 < ǫ ≪ 1. Consider the function v(x1 , x2 ) = x21 . Evaluate
L
L
the P1 Lagrange interpolant IK
v, see (7.11), and show that |v − IK
v|H 1 (K) ≥
−1
L
ǫ |v|H 2 (K) . (Hint: use a direct calculation of IK v.)
Exercise 8.3 (Barycentric coordinate). Let K be a simplex with barycentric coordinates {λi }0≤i≤d . Prove that |λi |W 1,∞ (K) ≤ ρ−1
K for all i ∈ {0: d}.
Exercise 8.4 (Lp -norm of shape functions). Let θK,i , i ∈ N , be a local
shape function. Let p ∈ [1, ∞]. Assume that the mesh sequence is shaped/p
regular. Prove that kθK,i kLp (K) is equivalent to hK uniformly with respect
to K.
Exercise 8.5 (Mapping Φk ). Let k ∈ N and set Nk,d = dim(Pk,d ) = k+d
d .
Let S be a connected, open, bounded Lipschitz set in Rd . Consider
the mapR
ping Φk : Pk,d ∋ q 7→ (fα (q))|α|≤k ∈ RNk,d , where fα (q) = S ∂ α v dx. Show
that Φk is an isomorphism.
Exercise 8.6 (Bramble–Hilbert). Prove Lemma 8.10.
Exercise 8.7 (Taylor polynomial). Let K be a convex cell. Consider a
Lagrange finite element of degree k ≥ 1 with nodes {ai }i∈N and associated
shape functions {θi }i∈N . Consider a sufficiently smooth function v. For all
x, y ∈ K, consider the Taylor polynomial of order k and the exact remainder
such that
Tk (x, y) = v(x) + Dv(x)(y − x) + . . . +
1 k
D v(x)(y − x, . . . , y − x),
{z
}
|
k!
k times
1
Rk (v)(x, y) =
Dk+1 v(ηx + (1 − η)y)(y − x, . . . , y − x),
{z
}
|
(k + 1)!
(k + 1) times
so that v(y) = Tk (x, y) + Rk (v)(x, y) for a certain η ∈ [0, 1].
98
Chapter 8. Local interpolation in affine meshes
P
L
L
(i) Prove that v(x) = IK
v(x) − i∈N Rk (v)(x, ai )θi (x), where IK
is the
Lagrange interpolant, see (7.11). (Hint:
interpolate
with
respect
to
y.)
Pnsh
L
(ii) Prove that Dm v(x) = Dm (IK
v)(x) − i=1
Rk (v)(x, ai )Dm θi (x) for all
m ≤ k.
L
m k+1−m
(iii) Infer that |v − IK
v|W m,∞ (K) ≤ cσK
hK
|v|W k+1,∞ (K) with constant
P
1
m
b
c = (k+1)! c∗ hKb i∈N |θi |W k+1,∞ (K)
b where c∗ stems from (8.6b) for s = m
and p = ∞.
Exercise 8.8 (Lp -stability of Lagrange interpolant). Let α ∈ (0, 1).
Consider the P1 finite element space spanned by θ1 (x) = 1 − x and θ2 (x) = x.
Consider the sequence of continuous functions {un }n>0 defined as follows over
the interval [0, 1]: un (x) = nα −1 if 0 ≤ x ≤ n1 and un (x) = x−α −1 otherwise.
(i) Prove that the sequence is uniformly bounded in Lp (0, 1) for all p such
that pα < 1.
L
L
(ii) Compute IK
un . Is the operator IK
stable in the Lp -norm?
L
(iii) Is the operator IK
stable in any Lr -norm with r ∈ [1, ∞)