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To prove the statement of Theorem 3.1, we first note a simplification of [33, Th. 4.1] which can be easily verified by considering block Toeplitz matrices for the special case of scalar block entries.
Let A ∈ ℝM×M, M ∈ ℕ, be a Toeplitz matrix generated by some function f ∈ L∞((−π, π), ℂ), that is,
Then the spectral norm of A is bounded from above by
Based on this result, Theorem 3.1 can be shown by some algebraic manipulations.
Proof of Theorem 3.1. First of all, let us consider the special case f2 = 0. Then the matrix F−1E coincides with f −1 1 E, which is well defined because f1 does not vanish by assumption of the theorem. Therefore, we have
Obviously, this matrix is positive semidefinite while the 12maximal eigenvalue is bounded from above by e+ e+2|e1e2| according to the Gershgorin circle theorem 22. Then the statement of the theorem directly follows by distinguishing between e1e2 ⩾ 0 and e1e2 < 0.
If f2 ≠ 0, we first note that the inverse of F reads
Therefore, the matrix F−1E is a Toeplitz matrix and generated by
Function f reaches its absolute extremum at x = 0 or x = π due to the fact that s1, s2, and s3 are real values.
Thus, the spectral norm of F−1E is bounded from above by
which can be easily verified by
This completes the proof of Theorem 3.1.
Proof of Theorem 5.3. First of all, we note that
Therefore, the value of ω0 is equal to 2/3 if and only if
and, particularly, ϑ ⩾ 1/2 is mandatory. If condition (A.3) is satisfied and d(ℓ) > dii, the first argument of the maximum in (5.5b) is bounded from above by 1/3 because
which is nonpositive for finite differences due to the fact that mii = m(ℓ) = 1, ϑ ⩾ 1/2, and d(ℓ) > dii while we have
in the context of linear finite elements. On the other hand, the second argument is not greater than 1/3 due to the fact that
by virtue of (2.7). Let us now assume that
Then the relaxation parameter ω0 attains the second argument of the maximum in (5.9) and, hence, the first expression in the definition of B(ℓ) is bounded from above by
Indeed, we have
and, on the other hand,
due to the fact that
Finally, the second argument of the maximum in (5.5b) satisfies
which proves the validity of inequality (5.10).
To prove identity (5.11), we have to show the validity of
whenever d(ℓ) ⩽ dii and ω ∈ (0, 1]. For this purpose, we first note that
due to the fact that
On the other hand, we have
which is nonnegative either for ζ = mii = m(ℓ) = 1 in case of finite differences or according to
for linear finite elements and ζ = 1/2.
Although the statement of Theorem 6.1 is true for finite element and finite difference discretizations, we prove the result by considering both spatial discretization techniques individually.
Proof of Theorem 6.1 for finite differences. To prove that spr(J(Cor,ℓ)(J(Jac,ℓ))ν) is smaller than 1 for all ℓ = 1, . . . , ̄N, we directly estimate the eigenvalues λ± ∈ ℂ of J(Cor,ℓ)(J(Jac,ℓ))ν which are the roots of the characteristic polynomial p : ℂ → ℂ
Here, the last identity is valid because
Therefore, the eigenvalues λ± satisfy
Let us now consider the special case (j(Jac,ℓ))ν(j(Jac,N+1−ℓ))ν ⩽ 0. Then the right hand side of (A.6) is obviously nonnegative and can be estimated by
Thus, both eigenvalues are real and satisfy
due to the reverse triangle inequality, Theorem 5.1, (A.5) and the fact that (j(Jac,ℓ))ν(j(Jac,N+1−ℓ))ν ⩽ 0.
On the other hand, for (j(Jac,ℓ))ν(j(Jac,N+1−ℓ))ν > 0, we can assume that
Otherwise, consider −(J(Jac,ℓ))ν instead of (J(Jac,ℓ))ν. Then estimate (A.8) can be exploited as in (A.7) to prove
and, hence, both eigenvalues are real and positive because
by (A.6) and (A.9). Furthermore, the maximal eigenvalue λ+ satisfies
by Theorem 5.1 and due to the fact that
because λ+ grows monotonically with 22respect to
Indeed, for instance, we have
by (A.9), which is nonnegative because
according to (A.8). This proves the statement of Theorem 6.1 for finite differences by exploiting (6.11).
Proof of Theorem 6.1 for finite elements. For finite elements, we first note that J(Cor,ℓ) is singular because
Therefore, the matrix J(Cor,ℓ)(J(Jac,ℓ))ν1+ν2 has a vanishing eigenvalue, too, and its spectral radius coincides with the absolute value of the trace, that is,
by virtue of (5.3) and (A.11). This proves the statement of Theorem 6.1 for finite elements by exploiting (6.11).
Proof of Lemma 6.1. To prove the inequalities, we first note that
due to the fact that
We now find upper bounds 22for the spectral norm of the submatrices by using Theorem 3.1, where different values for e1 and e2 are considered while
Indeed, the requirement |f2|< |f1| made in this theorem is valid because
which can be shown as in (5.8).
– Then, according to Theorem 43.1, the spectral norm of IK
is bounded from above by
using the quantities
This bound can be simplified by exploiting the identities
due to (6.5) and
where the numerator of the last fraction is nonpositive if ϑ ⩾ 1/2 by (6.4) or due to the fact that
for finite elements because
For ζ = 1 (and ϑ < 1/2), we observe
On the other hand, we have
according to (A.13).
– To estimate the spectral norm of
we can proceed similarly by replacing m(ℓ) and d(ℓ) by m(N+1−ℓ) and d(N+1−ℓ), respectively, while 22sℓhas to be substituted by cℓand vice versa. However, the numerator occurring in (A.16) does not have to be nonpositive for ζ = 1/2 and ϑ ⩾ 1/2 while the last inequality of (A.17) is not valid any more either. However, using the same ideas, we derive
because estimate (A.18) can be shown as in (A.16) and (A.17) while the first inequality 2in (A.19) is valid due to (A.14). For
the same inequality can be easily verified because
– Invoking Theorem 3.1 using e1 = m(ℓ) + ϑδtd(ℓ) and e2 = −m(ℓ) + (1 − ϑ)δtd(ℓ), an upper bound of
is given by
Furthermore, it is not necessary to take the absolute value of the first argument in the definition of the maximum in (A.20) because
by (2.13) and (A.13). On the other hand, the expression is bounded from above by
for a finite difference approximation, that is, ζ = 1, because
while, in case of finite elements, an upper bound is given by
by virtue of the fact that
because sℓ⩽ 1/2 for all ℓ = 1, . . . , ̄N.
– Finally, inequality (6.18) can be derived similarly by invoking
in case of a finite element approximation.