This section summarises some known and some new algebraic properties of the model problem (3.1), such as the relation between the eigenvalue error and the error with respect to the norms |||.||| and ∥.∥ [105]
|||u − uℓ|||2 = λ∥u − uℓ∥2+ λℓ− λ. (3.2)
Throughout this section suppose that (λℓ, uℓ) ∈ R × Vℓ and (λℓ+m, uℓ+m) ∈ R × Vℓ+m are
discrete eigenpairs to the continuous eigenpair (λ, u) ∈ R × V on the levels ℓ and ℓ + m. Lemma 3.3.1 (Quasi-orthogonality). Let Tℓ+m be a refinement of the triangulation Tℓ
for some level ℓ such that Vℓ ⊂ Vℓ+m. Then, for eℓ := u − uℓ and eℓ+m := u − uℓ+m, the
quasi-orthogonality holds, i.e.
|||uℓ+m − uℓ|||2 = |||eℓ|||2− |||eℓ+m|||2− λ∥eℓ∥2 + λ∥eℓ+m∥2+ λℓ+m∥uℓ+m− uℓ∥2.
Proof. Since Tℓ+m is a refinement of Tℓ, (3.2) implies
|||uℓ+m− uℓ|||2 = λℓ+m∥uℓ+m− uℓ∥2+ λℓ− λℓ+m.
Hence,
|||uℓ+m− uℓ|||2 = λℓ+m∥uℓ+m− uℓ∥2+ λℓ− λ − (λℓ+m− λ)
= |||eℓ|||2− |||eℓ+m|||2− λ∥eℓ∥2+ λ∥eℓ+m∥2+ λℓ+m∥uℓ+m− uℓ∥2.
Let the residual Resℓ ∈ V∗ be defined by
Resℓ(v) := λℓb(uℓ, v) − a(uℓ, v) for all v ∈ V.
Notice that Vℓ ⊂ ker(Resℓ).
Lemma 3.3.2. Let Tℓ+m be a refinement of Tℓ such that Vℓ ⊂ Vℓ+m ⊆ V . Then it holds
that |||uℓ+m − uℓ||| ≤ |||Resℓ|||Vℓ+m∗ + (λℓ+m+ λℓ) 2 ∥uℓ+m− uℓ∥2 |||uℓ+m− uℓ||| .
Vℓ+m, show
|||uℓ+m− uℓ|||2 = λℓb(uℓ, uℓ+m− uℓ) − a(uℓ, uℓ+m− uℓ)
+ a(uℓ+m, uℓ+m− uℓ) − λℓb(uℓ, uℓ+m− uℓ) = Resℓ(uℓ+m− uℓ) + (λℓ+m + λℓ)(1 − b(uℓ+m, uℓ)) ≤ |||Resℓ|||Vℓ+m∗ |||uℓ+m− uℓ||| + (λℓ+m+ λℓ) 2 ∥uℓ+m− uℓ∥ 2 .
The remaining part of this section is devoted to show that the second term on the right hand side in Lemma 3.3.2 is of higher-order, namely
∥uℓ+m− uℓ∥ . Hℓr|||uℓ+m− uℓ|||.
Here and throughout this chapter, Hℓ := maxT ∈Tℓdiam(T ) is the maximal mesh-size and 0 < r ≤ 1 depends on the regularity of the solution of the corresponding boundary value problem. The first part follows the argumentation as in [105] for the case uℓ+m ≡ u. The
second part exploits regularity of the corresponding boundary value problem together with the Aubin-Nitsche technique. Let Gℓ : V → Vℓ denote the Galerkin projection onto
Vℓ such that for any v ∈ V it holds that
a(v − Gℓv, vℓ) = 0 for all vℓ ∈ Vℓ.
Suppose the i-th eigenvalue λ = λ∞,iis simple. Let the initial mesh-size H0be sufficiently
small such that there exist two separation bounds M and Mℓ+m, independent of Hℓ,
which satisfy for the index set Iℓ := {1, . . . , i − 1, i + 1, . . . , dim(Vℓ)}
0 < M := sup ℓ∈N0 max j∈Iℓ λ∞,i |λℓ,j − λ∞,i| < ∞ and 0 < Mℓ+m := max j∈Iℓ λℓ+m,i |λℓ,j − λℓ+m,i| < ∞.
Lemma 3.3.3. Let Tℓ+m be a refinement of Tℓ such that Vℓ ⊂ Vℓ+m ⊆ V , then for the
Galerkin projection Gℓ : V → Vℓ it holds that
∥uℓ+m− uℓ∥ ≤ 2(1 + Mℓ+m)∥uℓ+m− Gℓuℓ+m∥,
∥u − uℓ∥ ≤ 2(1 + M )∥u − Gℓu∥.
Proof. Note that for the Galerkin projection it holds that
(λℓ,j− λℓ+m,i)b(Gℓuℓ+m,i, uℓ,j) = λℓ+m,ib(uℓ+m,i− Gℓuℓ+m,i, uℓ,j).
Since uℓ,1, . . . , uℓ,Nℓ, for Nℓ = dim(Vℓ), forms an orthogonal basis for Vℓ, the Galerkin projection of uℓ+m,i can be written as
Gℓuℓ+m,i =
Nℓ
j=1
Let γ := b(Gℓuℓ+m,i, uℓ,i) be the coefficient for j = i in the previous formula. The
orthogonality of the discrete eigenfunctions uℓ,1, . . . , uℓ,Nℓ yield
∥Gℓuℓ+m,i− γuℓ,i∥2 =
Nℓ j=1 j̸=i b(Gℓuℓ+m,i, uℓ,j)2 = Nℓ j=1 j̸=i λℓ+m,i |λℓ,j− λℓ+m,i| 2
b(uℓ+m,i− Gℓuℓ+m,i, uℓ,j)2
≤ Mℓ+m2
Nℓ
j=1 j̸=i
b(uℓ+m,i− Gℓuℓ+m,i, uℓ,j)2
≤ M2
ℓ+m∥uℓ+m,i− Gℓuℓ+m,i∥2.
The triangle inequality shows that
∥uℓ+m,i∥ − ∥uℓ+m,i− γuℓ,i∥ ≤ ∥γuℓ,i∥ ≤ ∥uℓ+m,i∥ + ∥uℓ+m,i− γuℓ,i∥.
Since the eigenfunctions are normalized to one this implies |γ − 1| ≤ ∥uℓ+m,i− γuℓ,i∥.
Hence,
∥uℓ+m,i− uℓ,i∥ ≤ ∥uℓ+m,i− γuℓ,i∥ + ∥(γ − 1)uℓ,i∥ ≤ 2∥uℓ+m,i− γuℓ,i∥.
Thus,
∥uℓ+m,i− uℓ,i∥ ≤ 2∥uℓ+m,i− Gℓuℓ+m,i∥ + 2∥Gℓuℓ+m,i− γuℓ,i∥
≤ 2(1 + Mℓ+m)∥uℓ+m,i− Gℓuℓ+m,i∥.
The second inequality follows analogously since Vℓ ⊂ V .
Lemma 3.3.4. Let Tℓ+m be a refinement of Tℓ such that Vℓ ⊂ Vℓ+m ⊆ V . Suppose that
the solution of the corresponding boundary value problem to (3.1), seek z ∈ V such that
a(z, v) =
ˆ
Ω
f v dx for all v ∈ V,
is H1+r-regular for all f ∈ L2(Ω) and some 0 < r ≤ 1, i.e., z ∈ H1+r(Ω) ∩ V and
∥z∥H1+r(Ω) ≤ Creg∥f ∥L2(Ω). Then it holds that
∥uℓ+m− Gℓuℓ+m∥ ≤ CapxCregHℓr|||uℓ+m− uℓ|||,
∥u − Gℓu∥ ≤ CapxCregHℓr|||u − uℓ|||.
of z ∈ V
∥z − Gℓz∥H1(Ω) ≤ CapxHℓr∥z∥H1+r(Ω)
for some 0 < r ≤ 1 [27, Theorem 14.3.3]. The Aubin-Nitzsche duality technique for the dual boundary value problem, seek z ∈ V such that
a(z, v) = b(uℓ+m− Gℓuℓ+m, v) for all v ∈ V,
and the regularity assumption z ∈ H1+r(Ω) ∩ V ,
∥z∥H1+r(Ω)≤ Creg∥uℓ+m− Gℓuℓ+m∥,
lead to
∥uℓ+m− Gℓuℓ+m∥ ≤ CapxCregHℓr|||uℓ+m− Gℓuℓ+m|||
≤ CapxCregHℓr|||uℓ+m− uℓ|||.
The second inequality follows from formally taking m → ∞.
Lemma 3.3.5. Let Tℓ+m be a refinement of Tℓ such that Vℓ ⊂ Vℓ+m ⊆ V . For sufficiently
small initial mesh-size H0 there exists a constant C0 > 0 depending only on T0 such that
1 ≤ κ(Hℓ) < C0 with
|||uℓ+m − uℓ||| ≤ κ(Hℓ)|||Resℓ|||Vℓ+m∗ , |||u − uℓ||| ≤ κ(Hℓ)|||Resℓ|||V∗
and limHℓ→0κ(Hℓ) = 1.
Proof. Suppose that Hℓ is sufficiently small such that
δℓ := 2Capx2 C 2
reg(λℓ+m+ λℓ)(1 + max{M, Mℓ+m})2Hℓ2r ≪ 1.
Then Lemma 3.3.2, Lemma 3.3.3 together with Lemma 3.3.4 lead to |||uℓ+m− uℓ||| ≤ (1 − δℓ)−1|||Resℓ|||V∗
ℓ+m and |||u − uℓ||| ≤ (1 − δℓ)
−1|||Res ℓ|||V∗.
Notice that κ(Hℓ) := (1 − δℓ)−1 → 1 as the maximal mesh-size tends to zero and
C0 := (1 − δ0)−1.