As we have seen in subsection 1.2.1, the weak formulation with S-bounded and S-coercive operators allows us to treat the equivalence of operators in an easy form, and also ensures well-posedness. Now we show that this concept also allows us to derive general mesh independent linear convergence results when no compact-equivalence is assumed, with no extra assumption. This is an advantage compared to the somewhat more general setting of Manteuffel et al. [52, 112], since our framework still covers all usual (Dirichlet, Neumann and Robin) boundary conditions, moreover, mesh independent linear convergence will be readily derived for general FEM discretizations. We follow our paper [19].
We note that such a uniform framework can only be given for FEM discretizations, owing to the Hilbert space background. Similar mesh independence results have also been given for FDM discretizations [37, 49, 52, 155], but these only concern rectangular domains where explicit calculations can be done, and are achieved (depending on the concrete pre-conditioner) with a case-by-case study.
1.3.1 Mesh independent linear convergence in Hilbert space
Let us consider the operator equation (1.1.1), whereL isS-bounded and S-coercive in the sense of Definition 1.2.1, and g ∈H. Using a Galerkin discretization, we want to solve the arising n×n system (1.2.23).
(a) Symmetric preconditioners
In general, whenLis nonsymmetric, we can take again the symmetric coercive operator S from Definition 1.2.1 and introduce the stiffness matrix ofSas preconditioner for system (1.2.23), i.e., Sh =
{⟨φi, φj⟩S
}n i,j=1
. To solve the preconditioned system
S−h1Lhc= ˜bh, (1.3.1)
one can apply the CG method using the Sh-inner product ⟨., .⟩Sh. As follows from (1.1.11) and (1.1.12), the convergence estimates depend on the bounds
λ0 =λ0(S−h1Lh) := inf{Lhc·c: Shc·c= 1}, Λ = Λ(S−h1Lh) := ∥S−h1Lh∥Sh, defined as in (1.1.8). Moreover, the convergence factor is determined by the ratio Λ/λ0. Proposition 1.3.1 If the operator L satisfies (1.2.2), then for any subspace Vh ⊂HS the stiffness matrix Lh satisfies
m(Shc·c)≤Lhc·c, |Lhc·d| ≤M∥c∥Sh∥d∥Sh (c,d∈Rn) (1.3.2) where m and M come from (1.2.2) and hence are independent of Vh.
Proof. Set u=
∑n i=1
ciφi ∈Vh and v =
∑n j=1
djφj ∈Vh in (1.2.2). Here
⟨LSu, v⟩S =
∑n i,j=1
⟨LSφi, φj⟩Scidj =
∑n i,j=1
(Lh)jicidj =Lhc·d and similarly
∥u∥2S =
∑n i,j=1
⟨φi, φj⟩Scicj =Shc·c=∥c∥2Sh, which show that (1.2.2) implies (1.3.2).
Thus we obtain that for any subspaceVh ⊂HS
Λ(S−h1Lh)≤M, λ0(S−h1Lh)≥m (1.3.3) independently of Vh. Then, using (1.1.11), we have proved
Theorem 1.3.1 Let the operator L satisfy (1.2.2). Then the GCG-LS method for for system (1.3.1) provides
(∥rk∥Sh
∥r0∥Sh
)1/k
≤(
1−(m M
)2)1/2
(k = 1,2, ..., n) (1.3.4) independently of Vh, and the CGN algorithm satisfies
(∥rk∥Sh
∥r0∥Sh
)1/k
≤21/k M −m
M +m (k= 1,2, ..., n) (1.3.5) independently of Vh.
We note that (1.3.4) holds as well for the GCR and Orthomin methods together with their truncated versions.
We mention as a special case when L itself is a symmetric operator. Then its S-coercivity and S-boundedness simply turns into the spectral equivalence relation
m∥u∥2S ≤ ⟨LSu, u⟩S ≤M∥u∥2S (u∈HS). (1.3.6) ThenLh is symmetric too. LetSbe the symmetric coercive operator from Definition 1.2.1, and introduce the stiffness matrix of S. It immediately follows, see e.g. [8], that
κ(S−h1Lh)≤ M
m. (1.3.7)
(b) Relation to previous conditions
Now we can clarify the relation of our setting to that by Manteuffel et al in [52].
Thereby they consider a more general situation than ours, similar to the Babuˇska lemma for well-posedness, which would mean with our terms that coercivity (the second inequality in (1.2.2)) can be replaced by the two weaker statements
sup
v∈HS
⟨LSu, v⟩S
∥v∥S ≥m∥u∥S (u∈HS), sup
u∈HS
⟨LSu, v⟩S >0 (v ∈HS). (1.3.8) However, in contrast to (1.2.2), the above inequalities are not automatically inherited in general subspacesVh with the same constants, i.e., no analogue of Proposition 1.3.1 holds.
Instead, the corresponding uniform relations for the discrete operators had to be assumed there, see (3.37)-(3.38) in [52]; with our notations, this means that one has to assume
sup
d∈Rn
Lhc·d
∥d∥Sh
≥m˜∥c∥Sh (c∈Vh), sup
c∈Rn
Lhc·d>0 (d∈Rn)
with a uniform constant ˜m >0 to obtain mesh independent linear convergence. (The first bound is an LBB type condition.) Although our assumptions (1.2.2) are more special, they hold for rather general elliptic operators as shown by Proposition 1.2.2, and provide
mesh independent linear convergence for arbitrary subspaces Vh ⊂HS without any further assumption.
(c) Nonsymmetric preconditioners
Let us consider a nonsymmetric preconditioning operator N for equation (1.1.1). We assume that N is S-bounded and S-coercive, i.e. N ∈ BCS(H) in the sense of Definition 1.2.1, for the same symmetric operator S as is L. Then we introduce the stiffness matrix of NS, i.e. Nh =
{⟨NSφj, φi⟩S
}n i,j=1
, as preconditioner for the discretized system (1.2.23).
To solve the preconditioned system
N−h1Lhc= ˜bh (1.3.9)
(with ˜bh = N−h1bh), we apply the CGN method under the Sh-inner product ⟨., .⟩Sh. By (1.1.12), this algorithm converges as
(∥rk∥Sh
∥r0∥Sh
)1/k
≤21/k κ(N−h1Lh)−1
κ(N−h1Lh) + 1 (k = 1,2, ..., n). (1.3.10) In the convergence analysis of nonsymmetric preconditioners, we must distinguish be-tween the bounds of Land N, i.e., (1.2.2) is replaced by
mL∥u∥2S ≤ ⟨LSu, u⟩S, |⟨LSu, v⟩S| ≤ML∥u∥S∥v∥S, mN∥u∥2S ≤ ⟨NSu, u⟩S, |⟨NSu, v⟩S| ≤MN∥u∥S∥v∥S
(1.3.11) for all u, v ∈HS.
Theorem 1.3.2 If the operatorsLandN satisfy (1.3.11), then for any subspaceVh ⊂HS
κ(N−h1Lh)≤ MLMN
mLmN and κ(N−h1Lh)≤(
1 + mL+mN
2mLmN ∥LS−NS∥)2
(1.3.12) independently of Vh.
Proof. (i) Let c ∈ Rn be arbitrary, d := N−h1Lhc, i.e. Nhd = Lhc, further, let u=
∑n j=1
cjφj ∈Vh and z =
∑n j=1
djφj ∈Vh. Then
mL∥u∥2S ≤ ⟨LSu, u⟩S =Lhc·c=Nhd·c=⟨NSz, u⟩S ≤ ∥NSz∥S∥u∥S,
hence mL∥u∥S ≤ ∥NSz∥S ≤ MN∥z∥S, and by exchanging L and N resp. u and z, we similarly obtain mN∥z∥S ≤ ∥LSu∥S ≤ML∥u∥S. Hence, altogether,
mL
MN ≤ ∥N−h1Lhc∥Sh
∥c∥Sh
= (Shd·d)1/2
(Shc·c)1/2 = ∥z∥S
∥u∥S ≤ ML
mN . (1.3.13)
(ii) We follow the proof of Proposition 1.3.2. Let c,d ∈ Rn and u, z ∈ Vh be as therein, k:=d−c and h:=
∑n j=1
kjφj =z−u. Then
mN∥h∥2S ≤ ⟨NSh, h⟩S =Nhk·k=Nhd·k−Nhc·k= (Lh−Nh)c·k
=⟨(LS−NS)u, h⟩S ≤ ∥LS−NS∥ ∥u∥S∥h∥S. Hence
∥z∥S ≤ ∥u∥S+∥h∥S ≤ ∥u∥S
( 1 + 1
mN∥LS−NS∥) . Exchanging L and N resp. uand z, we obtain ∥u∥S ≤ ∥z∥S
( 1 + m1
L∥LS−NS∥)
. In view of (1.3.13), the obtained bounds on the ratio ∥z∥S/∥u∥S imply
κ(N−h1Lh)≤(
1 + 1
mN∥LS −NS∥)(
1 + 1
mL∥LS−NS∥)
≤(
1 + mL+mN
2mLmN ∥LS−NS∥)2
where the second estimate uses the arithmetic-geometric mean inequality.
Hence, by (1.3.10), the CGN algorithm converges with a ratio bounded independently of Vh. Note that the above first estimate is a direct extension of the case of symmetric preconditioners: the latter is recovered by the case N = S, for which MN = mN = 1.
However, if both N and L have a large ratio M/m, then the upper bound in (1.3.12) becomes large even if N is an accurate approximation of L. In this case it is more useful to involve the difference of N and Lin the bound, as done in the second estimate above.
1.3.2 Mesh independent linear convergence for elliptic problems
Let us consider again the nonsymmetric elliptic problem (1.2.57), i.e., { Lu:=−div (A∇u) + b· ∇u+cu=g
u|ΓD = 0, ∂ν∂u
A +αu|ΓN = 0 (1.3.14)
on a bounded domain Ω ⊂ Rd, and we assume that L satisfies Assumptions 1.2.1. As a preconditioning operator, we consider in general a symmetric elliptic operatorSintroduced in (1.2.7):
Su≡ −div (G∇u) +σu for u|ΓD = 0, ∂ν∂u
G +βu|ΓN = 0, (1.3.15) assumed to satisfy Assumptions 1.2.2. Now, in contrast to section 1.2.4, we allow in general A ̸= G. We introduce the stiffness matrix Sh of S as preconditioner for system (1.2.58), and then solve the preconditioned system S−h1Lhc = ˜gh (with ˜gh = S−h1gh) with a CG algorithm. The basic conditioning estimate is as follows:
Proposition 1.3.2 For the system S−h1Lhc= ˜gh, the bounds (1.1.8) satisfy
Λ(S−h1Lh)≤M, λ0(S−h1Lh)≥m (1.3.16)
independently of Vh, where
M :=p1+CΩ,Sq−1/2∥b∥L∞(Ω)d+CΩ,S2 ∥c∥L∞(Ω)+CΓ2
N,S∥α∥L∞(ΓN) , m :=
(
p−01 +CΩ,L2 ∥σ∥L∞(Ω)+CΓ2
N,L∥β∥L∞(ΓN)