LONG TIME BEHAVIOR OF BACKWARD

DIFFERENCE TYPE METHODS FOR PARABOLIC

EQUATIONS WITH MEMORY IN BANACH SPACE

N. Yu. Bakaev^{1}, S. Larsson^{2} and V. Thomée^{2}

May 6 1998

Abstract. We show stability in a Banach space framework of backward Euler and second order backward difference timestepping methods for a parabolic equation with memory. The results are applied to derive maximum norm stability estimates for piecewise linear finite element approximations in a plane spatial domain, which is accomplished by a new resolvent estimate for the discrete Laplacian. Error estimates are also given.

1. Introduction. In this work we study the behavior of the solutions of the initial value problem

(1.1)  u_{t} + Au = ∫_{0}^{t} b(t, τ)Au(τ) dτ + f(t), for t > 0, with u(0) = u_{0},

and of certain discrete analogues. The problem is considered in a Banach space X, u = u(t), u_{t} = du/dt, the kernel b(t, τ) is a smooth scalar function, and A is a linear not necessarily bounded operator such that −A generates a bounded holomorphic semigroup E(t) = e^{−tA} on X. More precisely we assume that A is (α, φ)-sectorial for some α > 0 and φ ∈ (0, π/2) in the sense that there is M ≥ 1 such that the resolvent R(λ) = (λI − A)^{−1} satisfies

(1.2) ∥R(λ)∥ ≤ \frac{M}{|λ − α|}, for λ ∈ C \ Σ_{αφ}, Σ_{αφ} = \{λ ∈ C : | arg(λ − α)| < φ\}.

The data of the problem, u_{0} and f(t), are assumed to be such that (1.1) has a solution for t ≥ 0 with the regularity properties needed in our analysis. Such problems occur in modelling heat conduction in materials with memory; see [10].

It is well-known that the semigroup E(t) may be defined by (cf. [5])

E(t) = \frac{1}{2πi} ∫_{Γ} e^{−tλ}R(λ) dλ, for t > 0, with Γ = Γ_{γφ} = ∂Σ_{γφ}, γ < α,

1991 Mathematics Subject Classification. 65R20, 65M60, 65J10.

Key words and phrases. Integro-differential equation, parabolic, Banach space, backward Euler method, backward difference method, sparse quadrature, finite element method, maximum norm, resolvent estimate, exponential decay.

This work was supported by the Swedish Research Council for Engineering Sciences (TFR).

Published in East-West J. Numer. Math. 6 (1998), 185–206.

where the contour Γ is oriented so that Im λ decreases along Γ. (Note that the domain of definition D(A) need not be dense in X [3].) It is also well-known that

(1.3)  ∥E(t)∥ ≤ CMe^{−αt}, for t ≥ 0,

and, hence, in the purely parabolic case (b(t, τ) ≡ 0), solutions of (1.1) satisfy

(1.4)  ∥u(t)∥ ≤ CM(e^{−αt}∥u_{0}∥ + ∫_{0}^{t} e^{−α(t−s)}∥f(s)∥ ds), for t ≥ 0.

The main purpose of the present work is to prove similar bounds for solutions of (1.1), under suitable assumptions on the kernel b(t, τ), and of the discrete counterparts studied.

In order to achieve this we will use the condition that for some γ ∈ [0, α), with Γ_{αφ} = {λ ∈ C : | arg(λ − α)| = φ},

(1.5)  sup_{t≥0} ∫_{0}^{t} e^{γ(t−τ)}|b(t, τ)| dτ < Λ_{αφγ} := inf_{λ∈Γ_{αφ}} \frac{Re λ − γ}{|λ|} .

The main result of Section 2, Theorem 2.1, shows that if (1.5) holds together with

(1.6)  sup_{t≥τ≥0}(e^{γ(t−τ)}|b(t, τ)|) < ∞,

then the solution of (1.1) satisfies (1.4) with α replaced by γ. In the special case when A is selfadjoint positive definite in a Hilbert space it suffices to assume (1.5) with φ = 0 and α equal to the smallest eigenvalue of A (we then have Λ_{α0γ} = 1 − γ/α) to obtain (1.4) with CM = 1 and γ instead of α, see Theorem 2.2. Note that the left side of (1.5) is the norm of the operator B, defined by (Bw)(t) = ∫_{0}^{t} b(t, τ)w(τ) dτ, with respect to the weighted norm |w|_{γ} = sup_{t>0}(e^{γt}∥w(t)∥), which will be used in our analysis.

In Section 3 we show similar properties for a discrete counterpart of (1.1) of backward Euler type. Let k > 0 be a time step, let t_{n} = nk, n = 0, 1, 2, …, and let U_{n} denote the approximation of u(t_{n}). We then replace the time derivative in (1.1) by the backward difference quotient ¯∂U_{n} = (U_{n} − U_{n−1})/k and the integral by a quadrature formula with nonnegative weights ω_{nj} = ω_{nj}(k),

(1.7)  (Q_{k}g)_{n} = ∑_{j=0}^{n−1} ω_{nj}g(t_{j}) ≈ ∫_{0}^{t_{n}} g(t) dt.

Setting f_{n} = f(t_{n}) our time discretization of (1.1) is then

¯∂U_{n} + AU_{n} = (B_{k}AU)_{n} + f_{n}, for n ≥ 1, with U_{0} = u_{0},

where, with b_{n}(s) = b(t_{n}, s), b_{nj} = b(t_{n}, t_{j}),

(1.8)  (B_{k}w)_{n} = (Q_{k}(b_{n}w))_{n} = ∑_{j=0}^{n−1} ω_{nj}b_{nj}w_{j}.

Under natural assumptions on b(t, τ) and the quadrature formula (1.7) we show discrete analogues of Theorems 2.1 and 2.2.

In Section 4 we replace the time derivative instead by a second order backward finite difference quotient D̄U_{n} = (\frac{3}{2}U_{n} − 2U_{n−1} + \frac{1}{2}U_{n−2})/k and show that the corresponding stability result holds also in this case. In Section 5 we show error estimates of orders O(k) and O(k^{2}), respectively, for the methods treated in Sections 3 and 4, under the appropriate accuracy assumptions on Q_{k}.

Our results are applied in Section 6 to derive maximum-norm bounds for piecewise linear finite element approximations of (1.1). In doing so we show a new resolvent estimate of the form (1.2) for the discrete Laplacian in two dimensions, where φ is arbitrary in (0, π/2) and M contains a logarithmic factor.

In the literature approximations of equations of the form (1.1) with more general operator integral kernels B(t, τ) have been considered in, e.g., [10], [11]. Our choice of B(t, τ) = b(t, τ)A is motivated by our wish to be able to use spectral representations to reduce our statements to analogous statements for scalar problems. Our results here generalize earlier results in [9] for the case of selfadjoint operators in Hilbert space, and also give more precise information about the exponential decay. For an analysis of backward Euler type methods for (1.1) in the present framework on a finite interval in time, see [1].

2. The continuous problem. Our main purpose here is to show the following analogue of (1.4) for our initial-value problem (1.1).

Theorem 2.1. Assume that A satisfies (1.2) with α > 0 and φ ∈ (0, π/2), and let 0 ≤ γ < α. Assume further that b(t, s) satisfies (1.5) and (1.6). Then

∥u(t)∥ ≤ CM(e^{−γt}∥u_{0}∥ + ∫_{0}^{t} e^{−γ(t−s)}∥f(s)∥ ds), for t ≥ 0.

With this purpose in mind we consider in addition to (1.1) the problems

(2.1)  u_{t} + Au = ∫_{s}^{t} b(t, τ)Au(τ) dτ, for t > s, with u(s) = ϕ,

where s ≥ 0 is arbitrary. Denoting the solution operators of (2.1) by F (t, s), so that u(t) = F (t, s)ϕ, it is not difficult to see that (cf. [10], Lemma 2.1) the solution of (1.1) may be written as

u(t) = F (t, 0)u_{0} + ∫_{0}^{t} F (t, s)f(s) ds.

To show Theorem 2.1 we shall therefore derive a bound of the form

(2.2)  ∥F (t, s)∥ ≤ CMe^{−γ(t−s)}.

By introducing the kernel b̃(s; t, τ) = b(t + s, τ + s), we may restrict our proof of (2.2) to the case s = 0. In fact, the kernel b̃(s; t, τ) satisfies (1.5), (1.6) since, e.g.,

∫_{0}^{t} |b̃(s; t, τ)|e^{γ(t−τ)} dτ = ∫_{s}^{t+s} |b(t + s, τ)|e^{γ(t+s−τ)} dτ ≤ sup_{t≥0} ∫_{0}^{t} |b(t, τ)|e^{γ(t−τ)} dτ.

The main part of our analysis will be carried out on the scalar problem obtained formally from (2.1) (with s = 0) by replacing A by λ ∈ C, and where thus v = v(λ; t) is the solution of

(2.3)  v_{t} + λv = λ ∫_{0}^{t} b(t, τ)v(τ) dτ, for t > 0, with v(0) = 1.

Before proving Theorem 2.1 we first consider the case that X is a Hilbert space and the operator A is selfadjoint, positive definite with a compact inverse so that there is associated with A an orthonormal basis of eigenvectors {φ_{j}}_{j=1}^{∞} and corresponding positive eigenvalues {λ_{j}}_{j=1}^{∞}. In this case we shall show the following slightly sharper result.

Theorem 2.2. Let X be a Hilbert space, let A be selfadjoint positive definite with compact inverse, and let α be its smallest eigenvalue. Let 0 ≤ γ ≤ α. Assume that

(2.4)  sup_{t≥0} ∫_{0}^{t} |b(t, s)|e^{γ(t−s)} ds ≤ 1 − \frac{γ}{α}.

Then

∥u(t)∥ ≤ e^{−γt}∥u_{0}∥ + ∫_{0}^{t} e^{−γ(t−s)}∥f(s)∥ ds, for t ≥ 0.

In view of the eigenvector expansion of u(t), the following lemma implies (2.2) with CM = 1, which proves Theorem 2.2.

Lemma 2.3. Assume that 0 ≤ γ ≤ α, and that (2.4) holds. Then

|v(λ; t)| ≤ e^{−γt}, for t > 0, λ ≥ α.

Proof. If γ = α there is no integral, and the result is obvious. We may therefore assume γ < α. By integration (2.3) may be written as the Volterra integral equation

(2.5) v(t) = e^{−λt} + ∫_{0}^{t} K(t, τ)v(τ) dτ, where K(t, τ) = λ ∫_{τ}^{t} e^{−λ(t−y)}b(y, τ) dy.

We consider first the case b(t, τ) ≥ 0. Then K(t, τ) ≥ 0, from which it easily follows that v ≥ 0. Introducing the nonnegative function w(t) = e^{γt}v(t) we want to show that w(t) ≤ 1 for t ≥ 0. Multiplying (2.3) by e^{γt} we find

w_{t} + (λ − γ)w = λ ∫_{0}^{t} b(t, τ)e^{γ(t−τ)}w(τ) dτ.

Since w_{t}(0) = −(λ − γ) < 0, w decreases in an interval to the right of 0 so that w(t) < 1 there. In order to show that w(t) ≤ 1 for t ≥ 0, assume to the contrary that w(t) = 1 for some t > 0 and let t_{0} be the first such point. We then have either b(t_{0}, τ) = 0 for τ ∈ (0, t_{0}), or, with strict inequality,

w_{t}(t_{0}) + λ − γ < λ ∫_{0}^{t_{0}} b(t_{0}, τ)e^{γ(t_{0}−τ)} dτ ≤ λ(1 − \frac{γ}{α}).

In the first case w_{t}(t_{0}) = −(λ − γ) < 0, and in the second w_{t}(t_{0}) < γ(1 − λ/α) ≤ 0, which contradicts the assumption that t_{0} is the first such point.

We now turn to the case of a general b(t, τ). Then (2.5) implies

|v(t)| ≤ e^{−λt} + ∫_{0}^{t} K̃(t, τ)|v(τ)| dτ, where K̃(t, τ) = λ ∫_{τ}^{t} e^{−λ(t−y)}|b(y, τ)| dy.

Defining V (t) by V (t) = e^{−λt} +∫_{0}^{t} K̃(t, τ)V (τ) dτ, we know from above that V (t) ≤ e^{−γt} for t ≥ 0, and to complete the proof it remains to show |v(t)| ≤ V (t). By subtraction this is equivalent to showing that if

z(t) ≥ ∫_{0}^{t} K̃(t, τ)z(τ) dτ = (Kz)(t), for t ≥ 0, with z(0) = 0,

then z(t) ≥ 0. But by repeated application z(t) ≥ (K^{n}z)(t) → 0, as n → ∞. □

We now turn to Theorem 2.1 in Banach space. We thus want to show (2.2) under the assumptions of Theorem 2.1, and as indicated above, it suffices to do this for s = 0. Considering (2.1) (with s = 0) as a parabolic equation with the integral as a forcing term, Duhamel’s principle shows

(2.6)  u(t) = E(t)u_{0} + ∫_{0}^{t} E(t − s) ∫_{0}^{s} b(s, τ)Au(τ) dτ ds.

The first term is immediately bounded as desired by (1.3), so it remains to consider the integral. For this purpose we shall first derive an estimate for the solution of the scalar problem (2.3).

Lemma 2.4. Under the assumptions of Theorem 2.1, the solution v(λ; t) of (2.3) is an entire holomorphic function of λ for fixed t > 0, and there are α_{1} ∈ (γ, α) and C such that

|v(λ; t) − e^{−λt}| ≤ \frac{C}{|λ|}e^{−γt}, λ ∈ Σ_{α_{1}φ}, t > 0.

Proof. The solution v(λ; t) of the Volterra equation (2.5) may be represented by a series whose terms are holomorphic in λ for fixed t. The series converges uniformly on compact sets of λ; hence v(λ; t) is holomorphic in λ.

With v_{λ}(t) = v(λ; t), e_{λ}(t) = e^{−λt}, we write (2.3) as

(2.7)  v_{λ} = e_{λ} + E_{λ}Bv_{λ},

where the operators E_{λ} and B are defined by

(E_{λ}w)(t) = λ ∫_{0}^{t} e^{−λ(t−s)}w(s) ds, and (Bw)(t) = ∫_{0}^{t} b(t, s)w(s) ds.

With | · |_{γ} denoting the weighted norm |w|_{γ} = sup_{t≥0} (e^{γt}|w(t)|), as well as the corresponding operator norm, we first note that assumption (1.5) shows that

(2.8)  |B|_{γ} = sup_{t≥0} ∫_{0}^{t} |b(t, s)|e^{γ(t−s)} ds ≤ βΛ_{αφγ}, with β < 1.

Straightforward calculations give

(2.9)  |E_{λ}|_{γ} = |λ| ∫_{0}^{∞} |e^{−(λ−γ)s}| ds = \frac{|λ|}{Re λ − γ} , for Re λ > γ,

and, using also (1.6),

(2.10) |Be_{λ}|_{γ} ≤ sup_{t≥0} ∫_{0}^{t} |b(t, s)|e^{γ(t−s)}|e^{−(λ−γ)s}| ds ≤ \frac{C}{Re λ − γ} , for Re λ > γ.

It follows from (2.8) and (2.9) that there are α_{1} ∈ (γ, α) and β_{1} such that, since Λ_{αφγ} depends continuously on α,

(2.11)  |E_{λ}B|_{γ} ≤ β \frac{Λ_{αφγ}}{Λ_{α_{1}φγ}} ≤ β_{1} < 1, for λ ∈ Σ_{α_{1}φ},

so that E_{λ}B is a contraction with respect to | · |_{γ} for such λ. Hence, since from (2.7) we have E_{λ}Bv_{λ} = E_{λ}Be_{λ} + (E_{λ}B)^{2}v_{λ}, it follows from (2.11), (2.9) and (2.10) that

|E_{λ}Bv_{λ}|_{γ} ≤ \frac{1}{1 − β_{1}} |E_{λ}Be_{λ}|_{γ} ≤ \frac{1}{1 − β_{1}} \frac{1}{Λ_{α_{1}φγ}} \frac{C}{Λ_{α_{1}φγ}|λ|} = \frac{C}{|λ|}, for λ ∈ Σ_{α_{1}φ}.

This completes the proof. □

We need the following well-known result which we include for completeness.

Lemma 2.5. Let α_{1} ∈ (γ, α). Assume that f, g are holomorphic in Σ_{α_{1}φ} and that

|f(λ)| + |g(λ)| ≤ \frac{C}{|λ|}, for λ ∈ Σ_{α_{1}φ}.

Define operators F, G, and H by

F = \frac{1}{2πi} ∫_{Γ} f(λ)R(λ) dλ, where Γ = Γ_{α_{1}φ},

and similarly with f replaced by g and h = fg. Then H = F G. Moreover,

(2.12)  AE(t) = \frac{1}{2πi} ∫_{Γ} λe^{−λt}R(λ) dλ.

Proof. Take α′ ∈ (α_{1}, α) and let Γ′ = Γ_{α′φ}. Then, by the resolvent identity R(λ) − R(μ) = (μ − λ)R(λ)R(μ), Fubini’s theorem, (1.2), and Cauchy’s integral formula,

F G = \frac{1}{(2πi)^{2}} ∫_{Γ′} ∫_{Γ} f(μ)g(λ)R(μ)R(λ) dλ dμ

= \frac{1}{2πi} ∫_{Γ} g(λ)\frac{1}{2πi} ∫_{Γ′} \frac{f(μ)}{μ − λ} dμ R(λ) dλ

− \frac{1}{2πi} ∫_{Γ′} f(μ)\frac{1}{2πi} ∫_{Γ} \frac{g(λ)}{μ − λ} dλ R(μ) dμ

= \frac{1}{2πi} ∫_{Γ} g(λ)f(λ)R(λ) dλ = H.

The identity (2.12) follows from AR(λ) = λR(λ) − I and (1.2). □

Proof of Theorem 2.1. With v(λ; t) the solution of (2.3) and Γ = Γ_{α_{1}φ} we define an operator V (t) by

(2.13) V (t) = \frac{1}{2πi} ∫_{Γ} v(λ; t)R(λ) dλ = E(t) + \frac{1}{2πi} ∫_{Γ} (v(λ; t) − e^{−λt})R(λ) dλ.

Using the bound in Lemma 2.4 we readily prove that

∥V (t)∥ ≤ CMe^{−γt}, for t ≥ 0.

We need to show that V (t) is equal to the solution operator F (t, 0) of (2.1) with s = 0. By uniqueness and (2.6) this follows if we prove that V (t) satisfies

(2.14)  V (t) = E(t) + ∫_{0}^{t} ∫_{τ}^{t} E(t − σ)Ab(σ, τ) dσ V (τ) dτ =: E(t) + J(t).

In order to do this we substitute (2.13) into J(t) and use Lemma 2.5 with f(λ) = λe^{−λ(t−σ)}, g(λ) = v(λ; t) to show

J(t) = ∫_{0}^{t} ∫_{τ}^{t} \frac{1}{2πi} ∫_{Γ} λe^{−λ(t−σ)}b(σ, τ)v(λ; τ)R(λ) dλ dσ dτ.

Since the norm of the integrand is bounded by Ce^{− Re λ(t−σ)}(e^{− Re λτ} + |λ|^{−1}) (cf. Lemma 2.4), we may use Fubini’s theorem and finally (2.5) to obtain

J(t) = \frac{1}{2πi} ∫_{Γ} ∫_{0}^{t} ∫_{τ}^{t} λe^{−λ(t−σ)}b(σ, τ)v(λ; τ) dσ dτ R(λ) dλ

= \frac{1}{2πi} ∫_{Γ} (v(λ; t) − e^{−λt})R(λ) dλ = V (t) − E(t).

This proves (2.14) and thus completes the proof of the theorem. □

3. Backward Euler type methods. We now turn to the stability of the time discrete problem

(3.1)  ¯∂U_{n} + AU_{n} = (B_{k}AU)_{n} + f_{n}, for n ≥ 1, with U_{0} = u_{0},

where B_{k} = Q_{k}b_{n} is defined in (1.7) and (1.8). In addition to our previous assumptions on the kernel b(t, τ) we shall now assume

(3.2)  sup_{t≥0} ∫_{0}^{t} e^{γ(t−τ)}|b′_{τ} (t, τ)| dτ < ∞.

We also assume that there are q > 0 and k_{0} > 0 such that the quadrature operator Q_{k} satisfies

(3.3)  0 ≤ ω_{nj} ≤ Ck^{q}, for 0 ≤ j ≤ n, k ≤ k_{0},

and

(3.4)  |(Q_{k}g)_{n} − ∫_{0}^{t_{n}} g dt| ≤ Ck^{q} ∫_{0}^{t_{n}} (|g| + |g′|) dt, for n ≥ 1, k ≤ k_{0}.

We first show a discrete variant of (1.5).

Lemma 3.1. Assume that (1.5), (3.2), and (3.4) hold. Then

sup_{n≥1} ∑_{j=0}^{n−1} ω_{nj}|b_{nj}|e^{γt_{n−j}} < Λ_{αφγ}, for k ≤ k_{0}.

Proof. This follows from (1.5), (3.2), and (3.4) with g(s) = |b(t_{n}, s)|e^{γ(t_{n}−s)}. □

We next introduce some notation that will be used in the sequel. The discrete analogue of the homogeneous problem (2.1) is

(3.5) ¯∂U_{n} + AU_{n} = (B_{s,k}AU)_{n} = ∑_{j=s}^{n−1} ω_{nj}b_{nj}AU_{j}, for n ≥ s + 1, with U_{s} = ϕ,

and we denote its solution operator by F_{ns} = F_{ns}(kA) so that U_{n} = F_{ns}ϕ. It is then easily verified that the solution of (3.1) may be represented as

(3.6)  U_{n} = F_{n0}u_{0} + kE_{k} ∑_{j=1}^{n} F_{nj}f_{j}, where E_{k} = (I + kA)^{−1}.

In order to show stability we thus want to bound F_{n0} and E_{k}F_{nj} for 1 ≤ j ≤ n, and in doing so we may restrict the consideration to j = 0. The main part of our analysis will be carried out on the scalar analogue of (3.5), which for s = 0 reads

(3.7)  ¯∂v_{n} + λv_{n} = λ(B_{k}v)_{n} = λ ∑_{j=0}^{n−1} ω_{nj}b_{nj}v_{j}, for n ≥ 1, with v_{0} = 1.

We begin with the following result in the case that A is a selfadjoint operator in a Hilbert space.

Theorem 3.2. Let X be a Hilbert space, let A be selfadjoint positive definite with compact inverse, and let α be its smallest eigenvalue. Let 0 ≤ γ < α. Assume that (1.5), (3.2), and (3.4) hold. Then there is k_{0} > 0 such that, for the solution of (3.1),

∥U_{n}∥ ≤ e^{−γt_{n}}∥u_{0}∥ + k ∑_{j=1}^{n} e^{−γt_{n−j}}∥f_{j}∥, for n ≥ 0, k ≤ k_{0}.

Proof. Since ∥E_{k}∥ ≤ 1 it suffices to show that ∥F_{ns}∥ ≤ e^{−γt_{n−s}}, and by above it suffices to bound F_{n} = F_{n0}. By spectral representation, this is achieved if we can show that for the solution of (3.7) we have |v_{n}| ≤ e^{−γt_{n}} for λ ≥ α. For this purpose we write (3.7) as

(3.8)  v_{n} = r(μ)v_{n−1} + μr(μ)(B_{k}v)_{n}, with r(μ) = (1 + μ)^{−1}, μ = kλ,

and, after multiplication by e^{γt_{n}},

(3.9)  e^{γt_{n}}v_{n} = e^{γk}r(μ)e^{γt_{n−1}}v_{n−1} + μr(μ) ∑_{j=0}^{n−1} ω_{nj}b_{nj}e^{γt_{n−j}}(e^{γt_{j}}v_{j}).

We note that as a result of Lemma 3.1 (with φ = 0) there are δ and k_{0} such that

∑_{j=0}^{n−1} ω_{nj}|b_{nj}|e^{γt_{n−j}} ≤ δ < 1 − \frac{γ}{α}, for n ≥ 1, k ≤ k_{0}.

Further, e^{γk} ≤ 1 + γ_{1}k for k small, if γ_{1} > γ, and hence, with γ_{1} = α(1 − δ),

e^{γk} + μδ ≤ 1 + α(1 − δ)k + kλδ ≤ 1 + kλ = 1 + μ, for λ ≥ α,

so that (e^{γk} + μδ)r(μ) ≤ 1. Hence by (3.9)

|e^{γt_{n}}v_{n}| ≤ max_{0≤j≤n−1} |e^{γt_{j}}v_{j}| ≤ ⋯ ≤ |v_{0}| = 1,

which completes the proof. □

We now turn to the Banach space case.

Theorem 3.3. Assume that (1.2) holds, and let 0 ≤ γ < α. Assume that (1.5), (1.6), and (3.2) hold, and further that the quadrature rule satisfies (3.3) and (3.4). Then there are k_{0} > 0 and C > 0 such that for the solution of (3.1) we have

∥U_{n}∥ ≤ CM(e^{−γt_{n}}∥u_{0}∥ + k ∑_{j=1}^{n} e^{−γt_{n−j}}∥f_{j}∥), for n ≥ 0, k ≤ k_{0}.

Proof. In view of (3.6) we now want to show

(3.10)  ∥F_{n0}∥ ≤ CMe^{−γt_{n}} and ∥E_{k}F_{nj}∥ ≤ CMe^{−γt_{n−j}}, for 1 ≤ j ≤ n.

As above it suffices to consider F_{n} = F_{n0}. We shall bound F_{n} by using the Dunford-Taylor functional calculus, in which for a rational function w(λ), holomorphic in a neighborhood of some Σ_{α_{1}φ} with α_{1} < α and with w(∞) = 0, we may write

(3.11)  w(A) = \frac{1}{2πi} ∫_{Γ} w(λ)R(λ) dλ, where Γ = Γ_{α_{1}φ}.

With μ = kλ we introduce the sequences v_{μ} = (v_{n}(μ))_{n=0}^{∞} and r_{μ} = (r(μ)^{n})_{n=0}^{∞}. After iteration (3.8) may then be written

(3.12)  v_{μ} = r_{μ} + E_{k,μ}B_{k}v_{μ},

where the operator B_{k} is defined by (1.8) with (B_{k}w)_{0} = 0 and E_{k,μ} by

(E_{k,μ}w)_{n} = r̃_{μ} ∑_{j=1}^{n} r(μ)^{n−j}w_{j}, where r̃_{μ} = μr(μ), for n ≥ 1,

with (E_{k,μ}w)_{0} = 0. Further, letting μ → ∞, we see that (3.12) holds with μ = ∞, where (E_{k,∞}w)_{n} = w_{n} for n ≥ 1, (E_{k,∞}w)_{0} = 0 and r_{∞} = (1, 0, 0, …). It follows by induction on n that v_{μ} is a rational function of μ. We now define the sequence w_{μ} by

(3.13)  w_{μ} = v_{μ} − r̃_{μ}v_{∞} − r_{μ} + r̃_{μ}r_{∞}.

Since w_{μ,n} = w_{n}(μ) = v_{n}(μ) − r̃_{μ}v_{n}(∞) − r(μ)^{n} is a rational function for n ≥ 1 which vanishes at μ = ∞, we have by (3.11)

(3.14)  F_{n} = v_{n}(kA) = kAE_{k}v_{n}(∞) + E_{k}^{n} + \frac{1}{2πi} ∫_{Γ} w_{n}(kλ) R(λ) dλ.

Similarly

(3.15)  E_{k}F_{n} = kAE_{k}^{2}v_{n}(∞) + E_{k}^{n+1} + \frac{1}{2πi} ∫_{Γ} r(kλ)w_{n}(kλ) R(λ) dλ.

Since kAE_{k} = I −E_{k} and kAE_{k}^{2} = E_{k} −E_{k}^{2}, the desired bounds in (3.10) will follow from the following two lemmas.

Lemma 3.4. For any γ ∈ [0, α) there are C and k_{0} such that, for k ≤ k_{0},

∥E_{k}^{n}∥ ≤ CMe^{−γt_{n}}, for n ≥ 0.

Proof. We have

E_{k}^{n} = k^{−n}(k^{−1}I + A)^{−n} = k^{−n} \frac{1}{(n − 1)!} ∫_{0}^{∞} t^{n−1}e^{−t/k}E(t) dt.

Using (1.3) with γ_{1} ∈ (γ, α) we obtain, since (1 + γ_{1}k)^{−1} ≤ e^{−γk} for k ≤ k_{0},

∥E_{k}^{n}∥ ≤ CMk^{−n} \frac{1}{(n − 1)!} ∫_{0}^{∞} t^{n−1}e^{−(k^{−1}+γ_{1})t} dt = CM(1 + kγ_{1})^{−n} ≤ CMe^{−γt_{n}}.

□

Lemma 3.5. Under the assumptions of Theorem 3.3, there are α_{1} ∈ (γ, α), k_{0} > 0, and C such that, for k ≤ k_{0}, n ≥ 1,

(3.16)  |v_{n}(∞)| ≤ Ce^{−γt_{n}},
(3.17)  |w_{n}(kλ)| ≤ Ce^{−γt_{n}}(\frac{1}{|λ|} + \frac{k^{q}}{1 + k|λ|}), for λ ∈ Σ_{α_{1}φ}.

Before we give the proof of this lemma we complete the proof of of Theorem 3.3 by noting that since r(kλ) is bounded on Γ it implies that the integrals in (3.14) and (3.15) are bounded by CM times

∫_{α}^{∞} (\frac{1}{y} + \frac{k^{q}}{1 + ky} ) \frac{dy}{y} ≤ C(1 + k^{q} log \frac{1}{k}) ≤ C.

This proves Theorem 3.3. □

LONG TIME BEHAVIOR OF BACKWARD DIFFERENCE TYPE METHODS 11

Proof of Lemma 3.5. The proof is modeled after the proof of Lemma 2.4. Using (3.12) we find that w_{μ} satisfies

\begin{aligned}w_{μ} & = E_{k,μ}B_{k}w_{μ} + r̃_{μ}(E_{k,μ} − E_{k,∞})B_{k}(v_{∞} − r_{∞}) \\ & + (E_{k,μ} − r̃_{μ}E_{k,∞})B_{k}r_{μ} + r̃_{μ}E_{k,∞}B_{k}(r_{μ} − r_{∞}) \\ & = E_{k,μ}B_{k}w_{μ} + I_{1} + I_{2} + I_{3}. \end{aligned} (3.18)

We shall first show that E_{k,μ}B_{k} is a contraction with respect to the weighted norm |w|_{γ} = sup_{n≥0}(e^{γt_{n}}|w_{n}|). To this end we shall show that for the corresponding operator norm (cf. (2.9))

(3.19)  |E_{k,μ}|_{γ} ≤ \frac{|λ|}{Re λ − γ_{1}} , for Re λ ≥ γ_{1} > γ.

Since by Lemma 3.1 there is β ∈ (0, 1) such that (cf. (2.8))

(3.20)  |B_{k}|_{γ} ≤ sup_{n≥1} ∑_{j=0}^{n−1} ω_{nj}|b_{nj}|e^{γt_{n−j}} ≤ βΛ_{αφγ} ≤ C, for k ≤ k_{0},

it follows that, for λ ∈ Σ_{α_{1}φ}, γ < γ_{1} < α_{1} < α, k ≤ k_{0},

(3.21)  |E_{k,μ}B_{k}|_{γ} ≤ \frac{|λ|}{Re λ − γ_{1}} βΛ_{αφγ} ≤ β \frac{Λ_{αφγ}}{Λ_{α_{1}φγ_{1}}} .

We may then choose α_{1}, γ_{1}, and then possibly a new k_{0}, so that (cf. (2.11))

(3.22)  |E_{k,μ}B_{k}|_{γ} ≤ β_{1} < 1, for λ ∈ Σ_{α_{1}φ}, k ≤ k_{0}.

Hence from (3.18)

|w_{μ}|_{γ} ≤ \frac{1}{1 − β_{1}} (|I_{1}|_{γ} + |I_{2}|_{γ} + |I_{3}|_{γ}).

In order to bound the right hand side we shall show that, for λ ∈ Σ_{α_{1}φ}, k ≤ k_{0},

|E_{k,μ} − r̃_{μ}E_{k,∞}|_{γ} + |E_{k,μ} − E_{k,∞}|_{γ} ≤ \frac{C}{1 + k|λ|}, (3.23)

|v_{∞} − r_{∞}|_{γ} ≤ Ck^{q}, (3.24)

|B_{k}r_{μ}|_{γ} ≤ C(\frac{1}{|λ|} + k^{q}), (3.25)

|r̃_{μ}B_{k}(r_{μ} − r_{∞})|_{γ} ≤ \frac{Ck^{q}}{1 + k|λ|}, (3.26)

Together these estimates yield

|I_{1}|_{γ} + |I_{2}|_{γ} + |I_{3}|_{γ} ≤ C(\frac{1}{|λ|} + \frac{k^{q}}{1 + k|λ|}),

which completes the proof of (3.17). Moreover, (3.16) follows from (3.24).

It remains to prove (3.19) and (3.23)–(3.26). For (3.19) we write

e^{γt_{n}}(E_{k,μ}w)_{n} = r̃_{μ} ∑_{j=1}^{n} (e^{γk}r(μ))^{n−j}e^{γt_{j}}w_{j}.

Here

(3.27) e^{γk}|r(μ)| ≤ \frac{e^{γk}}{1 + k Re λ} ≤ \frac{1}{1 + k(Re λ − γ_{1})}, for Re λ > γ_{1}, k ≤ k_{0},

and hence

(3.28)  |E_{k,μ}|_{γ} = |r̃_{μ}| ∑_{j=0}^{∞} (e^{γk}|r(μ)|)^{j} = |μ|\frac{e^{γk}|r(μ)|}{1 − e^{γk}|r(μ)|} ≤ \frac{|λ|}{Re λ − γ_{1}} .

We now prove (3.23). For n ≥ 2 we have

(E_{k,μ}w)_{n} = r̃_{μ}w_{n} + r̃_{μ} ∑_{j=1}^{n−1} r(μ)^{n−j}w_{j}

= r̃_{μ}(E_{k,∞}w)_{n} + r(μ)r̃_{μ} ∑_{j=2}^{n} r(μ)^{n−j}w_{j−1},

or (E_{k,μ} −r̃_{μ}E_{k,∞})w = r(μ)E_{k,μ}w̄, where w̄_{j} = w_{j−1} for j ≥ 2, w̄_{1} = w̄_{0} = 0. Hence, since Re λ ≥ c|λ| and |E_{k,μ}|_{γ} ≤ C,

|(E_{k,μ} − r̃_{μ}E_{k,∞})w|_{γ} ≤ |r(kλ)| |E_{k,μ}w̄|_{γ} ≤ \frac{C}{1 + k|λ|}|w|_{γ}, for λ ∈ Σ_{α_{1}φ}.

This proves the first bound in (3.23). The second bound follows from this and 1 = r̃_{μ} + r(μ), which implies E_{k,μ} − E_{k,∞} = (E_{k,μ} − r̃_{μ}E_{k,∞}) − r(μ)E_{k,∞}.

To prove (3.24) we use (3.12), (3.22), (3.3), and (1.6) to get

|v_{∞} − r_{∞}|_{γ} = |E_{k,∞}B_{k}v_{∞}|_{γ} ≤ \frac{1}{1 − β_{1}} |E_{k,∞}B_{k}r_{∞}|_{γ} = C sup_{n≥1} (e^{γt_{n}}ω_{n0}|b_{n0}|) ≤ Ck^{q}.

To prove (3.25) we use assumption (1.6) to get

(3.29) |B_{k}r_{μ}|_{γ} ≤ sup_{n≥1} ∑_{j=0}^{n−1} ω_{nj}|b_{nj}|e^{γt_{n−j}}(e^{γk}|r(μ)|)^{j} ≤ C sup_{n≥1} ∑_{j=0}^{n−1} ω_{nj}(e^{γk}|r(μ)|)^{j}.

With ν = Re λ − γ_{1} (3.27) shows e^{γk}|r(μ)| ≤ 1/2 for kν ≥ 1 and hence by (3.3)

(3.30)  ∑_{j=0}^{n−1} ω_{nj}(e^{γk}|r(μ)|)^{j} ≤ Ck^{q} ∑_{j=0}^{n−1} 2^{−j} ≤ Ck^{q}.

For kν ≤ 1, (e^{γk}|r(μ)|)^{j} ≤ e^{−cνt_{j}} with c < 1, and by (3.4)

(3.31) ∑_{j=0}^{n−1} ω_{nj}(e^{γk}|r(μ)|)^{j} ≤ ∑_{j=0}^{n−1} ω_{nj}e^{−cνt_{j}}

≤ ∫_{0}^{t_{n}} e^{−cνt} dt + Ck^{q} ∫_{0}^{t_{n}}(1 + ν)e^{−cνt} dt ≤ C(\frac{1}{ν} + k^{q}) ≤ C(\frac{1}{|λ|} + k^{q}).

Finally, by (1.6) and (3.3), and using (3.28), since (r_{μ} − r_{∞})_{n} = r(μ)^{n} for n ≥ 1 and (r_{μ} − r_{∞})_{0} = 0,

|r̃_{μ}B_{k}(r_{μ} − r_{∞})|_{γ} ≤ Ck^{q}|r(μ)| |r̃_{μ}| ∑_{j=0}^{∞} (e^{γk}|r(μ)|)^{j} ≤ Ck^{q}|r(μ)| ≤ \frac{Ck^{q}}{1 + k|λ|}.

This proves (3.26) and thus completes the proof of Lemma 3.5. □

4. Second order backward difference methods. In this section we shall treat second order backward difference methods of the form

(4.1)  D̄U_{n} + AU_{n} = (B_{k}AU)_{n} + f_{n}, for n ≥ 2,

where D̄U_{n} = (\frac{3}{2}U_{n} − 2U_{n−1} + \frac{1}{2}U_{n−2})/k and where again B_{k} = Q_{k}b_{n} is defined in (1.7) and (1.8), with start-up values U_{0} and U_{1} given by

(4.2)  ¯∂U_{1} + AU_{1} = (B_{k}AU)_{1} + f_{1}, U_{0} = u_{0}.

In analogy with (2.1) and (3.5) we consider the homogeneous problem

(4.3) D̄U_{n} + AU_{n} = (B_{s,k}AU)_{n}, for n ≥ s + 2, with U_{s} = ϕ_{0}, U_{s+1} = ϕ_{1},

and denote its solution operator by F_{ns} so that U_{n} = F_{ns}(ϕ_{0}, ϕ_{1}). The solution of (4.1) may then be represented as

(4.4) U_{n} = F_{n0}(U_{0}, U_{1}) + kẼ_{k} ∑_{s=2}^{n} F_{n,s−1}(0, f_{s}), where Ẽ_{k} = (\frac{3}{2}I + kA)^{−1},

and our aim is to bound F_{n0} and Ẽ_{k}F_{nj} with 1 ≤ j ≤ n − 1. As before we may set s = 0 and bound F_{n} = F_{n0} and Ẽ_{k}F_{n}, and then analyze the scalar version of (4.3) with s = 0, where v_{n} = v_{n}(kλ) satisfies

(\frac{3}{2} + kλ)v_{n} − 2v_{n−1} + \frac{1}{2}v_{n−2} = kλ(B_{k}v)_{n}, for n ≥ 2,

with v_{0}, v_{1} given and v = (v_{n})_{n=0}^{∞}, or, with μ = kλ,

(4.5)  v_{n} − \frac{4}{3 + 2μ}v_{n−1} + \frac{1}{3 + 2μ}v_{n−2} = \frac{2μ}{3 + 2μ}(B_{k}v)_{n}, for n ≥ 2.

We associate with this equation the characteristic polynomial

r^{2} − \frac{4}{3 + 2μ}r + \frac{1}{3 + 2μ} = (r − r_{+}(μ))(r − r_{−}(μ)),

with roots r_{±} = r_{±}(μ) = (2±\sqrt{1 − 2μ})/(3+2μ). The solution of (4.5) then satisfies

(4.6)  v_{n} = ρ̂_{n}(μ)v_{0} + ρ̄_{n}(μ)v_{1} + r̃_{μ} ∑_{j=2}^{n} ρ_{n−j}(μ)(B_{k}v)_{n},

where

ρ_{n} = \frac{r_{+}^{n+1} − r_{−}^{n+1}}{r_{+} − r_{−}} , ρ̄_{n} = ρ_{n−1}, ρ̂_{n} = −r_{+}r_{−} \frac{r_{+}^{n−1} − r_{−}^{n−1}}{r_{+} − r_{−}} , r̃_{μ} = \frac{2μ}{3 + 2μ},

with the obvious modification when r_{+} = r_{−}, i.e., when μ = 1/2.

In order to state our stability result we shall need to introduce the quantity

m_{φ} = sup_{| arg μ|≤φ} |\frac{2μ}{3 + 2μ}| ∑_{j=0}^{∞} |ρ_{j}(μ)|.

In the case φ = 0, this quantity occurred in the stability analysis in [9], where it was shown by computer calculation that 1 < m_{0} < 1.05.

Here we shall go directly to the Banach space case; with φ = 0 and M = 1 it covers the case of a selfadjoint operator in a Hilbert space as a special case.

Theorem 4.1. Assume that (1.2) holds, and let 0 ≤ γ < α. Assume further that b(t, τ) satisfies (1.6) and (3.2), and that

sup_{t≥0} ∫_{0}^{t} |b(t, s)|e^{γ(t−s)} ds < min(Λ_{αφγ}, m_{φ}^{−1}).

Assume finally that the quadrature rule satisfies (3.3) and (3.4). Then there are k_{0} > 0 and C > 0 such that for the solution of (4.1) and (4.2) we have

∥U_{n}∥ ≤ CM(e^{−γt_{n}}∥U_{0}∥ + k ∑_{j=1}^{n} e^{−γt_{n−j}}∥f_{j}∥), for n ≥ 0, k ≤ k_{0}.

Proof. In view of (4.4) and with F_{n} = F_{n0} the result will follow from

∥F_{n}(ϕ_{0}, ϕ_{1})∥ + ∥Ẽ_{k}F_{n}(ϕ_{0}, ϕ_{1})∥ ≤ CMe^{−γt_{n}}(∥ϕ_{0}∥ + ∥ϕ_{1}∥),

together with the obvious estimate for U_{1} (which follows from Theorem 3.3). Analogously to (3.12) we write for the solution (4.6) of the scalar problem (4.5)

(4.8)  v_{μ} = ρ̂_{μ}v_{0} + ρ̄_{μ}v_{1} + E_{k,μ}B_{k}v_{μ},

where v_{μ} = (v_{n}(μ))_{n=0}^{∞}, ρ̂_{μ} = (ρ̂_{n}(μ))_{n=0}^{∞}, ρ̄_{μ} = (ρ̄_{n}(μ))_{n=0}^{∞}, and where now

(4.9)  (E_{k,μ}w)_{n} = r̃_{μ} ∑_{j=2}^{n} ρ_{n−j}w_{j}, for n ≥ 2,

with (E_{k,μ}w)_{n} = 0 for n = 0, 1. Letting μ → ∞ (4.8) remains valid with v_{∞} = (v_{n}(∞))_{n=0}^{∞}, ρ̂_{∞} = (1, 0, 0, …), ρ̄_{∞} = (0, 1, 0, 0, …), and (E_{k,∞}w)_{n} = w_{n} for n ≥ 2, (E_{k,∞}w)_{n} = 0 for n = 0, 1. We define analogously to (3.13)

(4.10)  w_{μ} = v_{μ} − r̃_{μ}v_{∞} − (ρ̂_{μ} − r̃_{μ}ρ̂_{∞})v_{0} − (ρ̄_{μ} − r̃_{μ}ρ̄_{∞})v_{1}.

Since w_{μ,n} = w_{n}(μ) = v_{n}(μ)−r̃_{μ}v_{n}(∞)−ρ̂_{n}(μ)v_{0}−ρ̄_{n}(μ)v_{1} is a rational function of μ for n ≥ 2, which vanishes at μ = ∞, we may apply (3.11). We obtain for n ≥ 2

F_{n}(v_{0}, v_{1}) = v_{n}(kA) = kAẼ_{k}v_{n}(∞)

+ ρ̄_{n}(kA)v_{1} + ρ̂_{n}(kA)v_{0} + \frac{1}{2πi} ∫_{Γ} w_{n}(kλ) R(λ) dλ;

cf. (3.14), and for Ẽ_{k}F_{n}(v_{0}, v_{1}) we have the corresponding result with the integrand multiplied by (\frac{3}{2} + kλ)^{−1}. Since one easily sees that kAẼ_{k} and kAẼ_{k}^{2} are bounded as claimed, the desired bound (4.7) will follow once we have proved the appropriate bounds for v_{n}(∞), w_{n}(kλ), ρ̂_{n}(kA), and ρ̄_{n}(kA). These are the contents of the following two lemmas which are analogous to Lemmas 3.5 and 3.4, respectively. □

Lemma 4.2. Under the assumptions of Theorem 4.1 there are α_{1} ∈ (γ, α), k_{0} > 0, and C such that, for λ ∈ Σ_{α_{1}φ}, k ≤ k_{0}, n ≥ 1,

(4.11)  \begin{aligned}|v_{n}(∞)| & ≤ Ce^{−γt_{n}}, \\ |w_{n}(kλ)| & ≤ Ce^{−γt_{n}}(\frac{1}{|λ|} + \frac{k^{q}}{\sqrt{1 + k|λ|}})(|v_{0}| + |v_{1}|), \end{aligned}

Proof. The proof follows the lines of that of Lemma 3.5. Multiplying the case μ = ∞ of (4.8) by r̃_{μ} and subtracting the result from (4.8), using (4.10) shows

w_{μ} = E_{k,μ}B_{k}w_{μ} + r̃_{μ}(E_{k,μ} − E_{k,∞})B_{k}(v_{∞} − ρ̂_{∞}v_{0} − ρ̄_{∞}v_{1})
+ (E_{k,μ} − r̃_{μ}E_{k,∞})B_{k}(ρ̂_{μ}v_{0} + ρ̄_{μ}v_{1})
+ r̃_{μ}E_{k,∞}B_{k}((ρ̂_{μ} − ρ̂_{∞})v_{0} + (ρ̄_{μ} − ρ̄_{∞})v_{1})
= E_{k,μ}B_{k}w_{μ} + I_{1} + I_{2} + I_{3}.

We shall show that E_{k,μ}B_{k} is a contraction with respect to |·|_{γ}. For this purpose we show that for any δ > 0, γ_{1} > γ, and η ∈ (γ/γ_{1}, 1) there are ε > 0 and k_{0} such that for λ ∈ Σ_{γ_{1},φ}, k ≤ k_{0}, we have

(4.12)  |E_{k,μ}|_{γ} = |r̃_{μ}| ∑_{j=0}^{∞} e^{γt_{j}}|ρ_{j}| ≤ \{ \begin{matrix}η^{−2} \frac{|λ|}{Re λ − γ_{1}} + δ, & for k|λ| ≤ ε \\ m_{φ} + δ, & for k|λ| ≥ ε. \end{matrix}

As earlier, Lemma 3.1 implies (3.20) so that, in particular, using also (4.12) we may choose α_{1}, γ_{1}, and then k_{0}, so that |E_{k,μ}B_{k}|_{γ} ≤ β_{1} < 1 for λ ∈ Σ_{α_{1}φ}, k ≤ k_{0} (cf. (3.22)). To show (4.11) it remains to show

|I_{1}|_{γ} + |I_{2}|_{γ} + |I_{3}|_{γ} ≤ C(\frac{1}{|λ|} + \frac{k^{q}}{\sqrt{1 + k|λ|}})(|v_{0}| + |v_{1}|), for λ ∈ Σ_{α_{1}φ}.

For this we shall show that, for λ ∈ Σ_{α_{1}φ} and k ≤ k_{0},

|E_{k,μ} − r̃_{μ}E_{k,∞}|_{γ} + |E_{k,μ} − E_{k,∞}|_{γ} ≤ \frac{C}{\sqrt{1 + k|λ|}}, (4.13)

|v_{∞} − ρ̂_{∞}v_{0} − ρ̄_{∞}v_{1}|_{γ} ≤ Ck^{q}(|v_{0}| + |v_{1}|), (4.14)

|B_{k}ρ̂_{μ}|_{γ} + |B_{k}ρ̄_{μ}|_{γ} ≤ C(\frac{1}{|λ|} + k^{q}), (4.15)

|B_{k}(ρ̂_{μ} − ρ̂_{∞})|_{γ} + |B_{k}(ρ̄_{μ} − ρ̄_{∞})|_{γ} ≤ \frac{Ck^{q}}{\sqrt{1 + k|λ|}}. (4.16)

To prove (4.12) we first note that the equality follows at once from the definition of E_{k,μ} in (4.9). Since r_{+} − r_{−} = 2\sqrt{1 − 2μ}/(3 + 2μ) we conclude

|E_{k,μ}|_{γ} ≤ \frac{|μ|}{|\sqrt{1 − 2μ}|} ∑_{j=0}^{∞} e^{γt_{j}}(|r_{+}(μ)|^{j+1} + |r_{−}(μ)|^{j+1}).

We consider first small |μ| and begin with the contribution to the sum from r_{−}(μ). Since |r_{−}(μ)| ≤ κ < 1 for |μ| ≤ ε we have for such μ and k small

\frac{|μ|}{|\sqrt{1 − 2μ}|} ∑_{j=0}^{∞} e^{γt_{j}}|r_{−}(μ)|^{j+1} ≤ Ce^{−γk}|μ| ∑_{j=1}^{∞}(e^{γk}κ)^{j} ≤ C|μ| ≤ Cε, for |μ| ≤ ε,

which is as small as desired for ε small enough.

In order to consider the contribution from r_{+}(μ), let γ_{1} > γ. We note that r_{+}(μ) = 1 − μ + O(μ^{2}) as μ → 0, so that the bound in (3.27) holds also for r_{+}(μ) when μ ∈ Σ_{0φ} and |μ| ≤ ε. Reducing ε further, if necessary, we also have |(1 − 2μ)^{−1/2}| ≤ δ^{−1} for |μ| ≤ ε, and hence, as in (3.28),

|\frac{μr_{+}(μ)}{\sqrt{1 − 2μ}}| ∑_{j=0}^{∞}(e^{γk}|r_{+}(μ)|)^{j} ≤ \frac{1}{δ} \frac{|λ|}{Re λ − γ_{1}} , if λ ∈ Σ_{γ_{1}φ}, k|λ| ≤ ε.

For |μ| ≥ ε we write

|E_{k,μ}|_{γ} = |\frac{2μ}{3 + 2μ} ∑_{j=0}^{∞}(1 + (e^{γt_{j}} − 1))ρ_{j}| ≤ m_{φ} + |\frac{2μ}{3 + 2μ}| ∑_{j=1}^{∞}(1 − e^{−γt_{j}})e^{γt_{j}}|ρ_{j}|.

Here, since ρ_{j} = ∑_{l=0}^{j} r_{+}^{j−l}r_{−}^{l}, and e^{γk}|r_{±}(μ)| ≤ κ < 1, we have e^{γt_{j}}|ρ_{j}| ≤ (j +1)κ^{j}, so that

∑_{j=1}^{∞}(1 − e^{−γt_{j}})e^{γt_{j}}|ρ_{j}| ≤ ∑_{j=1}^{∞} γjk(j + 1)κ^{j} ≤ Ck,

which shows the desired result in this case.

We now show (4.13). Since |E_{k,μ}|_{γ} is bounded it suffices to consider large μ. We have

((E_{k,μ} − r̃_{μ}E_{k,∞})w)_{n} = r̃_{μ} ∑_{j=2}^{n−1} ρ_{n−j}w_{j},

and hence

|E_{k,μ} − r̃_{μ}E_{k,∞}|_{γ} = |r̃_{μ}| ∑_{j=1}^{∞} e^{γt_{j}}|ρ_{j}| ≤ ∑_{j=1}^{∞} e^{γt_{j}}|ρ_{j}| ≤ \frac{C}{\sqrt{|μ|}} ≤ \frac{C}{\sqrt{1 + k|λ|}},

where we have used that for large |μ|, e^{γk}|r_{±}(μ)| ≤ min(\frac{C}{\sqrt{|μ|}}, κ), κ < 1, so that

(4.17)  e^{γt_{j}}|ρ_{j}| ≤ C ∑_{l=0}^{j}(e^{γk}|r_{+}|)^{j−l}(e^{γk}|r_{−}|)^{l} ≤ \frac{C}{\sqrt{|μ|}}(j + 1)κ^{j−1}.

This proves the first bound in (4.13). The second bound follows from this and 1 = r̃_{μ} + 3/(3 + 2μ), which implies E_{k,μ} − E_{k,∞} = E_{k,μ} − r̃_{μ}E_{k,∞} − \frac{3}{3+2μ}E_{k,∞}.

To show (4.14) we use (4.8) with μ = ∞ as in the proof of (3.24).

For (4.15) we have using (1.6), cf. (3.29),

|B_{k}ρ̂_{μ}|_{γ} + |B_{k}ρ̄_{μ}|_{γ} ≤ C sup_{n≥1} ∑_{j=0}^{n−1} ω_{nj}e^{γt_{j}}(|ρ̂_{μj}| + |ρ̄_{μj}|).

For |μ| ≤ ε,

(4.18)  e^{γt_{j}}(|ρ̂_{j}(μ)| + |ρ̄_{j}(μ)|) ≤ C(e^{γk}|r_{+}(μ)|^{j} + κ^{j}), with κ < 1,

and we may use (3.30) and (3.31) for the desired bound. For |μ| ≥ ε, the left side of (4.18) is bounded by Cκ^{j} and again contributes as in (3.30).

We now prove (4.16). Since (ρ̂_{μ} − ρ̂_{∞})_{n} = ρ̂_{n}(μ) for n ≥ 2 and = 0 for n = 0, 1, we may use (3.3), (1.6) and (4.12) to get

|B_{k}(ρ̂_{μ} − ρ̂_{∞})|_{γ} + |B_{k}(ρ̄_{μ} − ρ̄_{∞})|_{γ} ≤ Ck^{q}|r̃_{μ}| ∑_{j=2}^{∞} e^{γt_{j}}(|ρ̂_{j}(μ)| + |ρ̄_{j}(μ)|)

≤ Ck^{q}|r̃_{μ}|(|r_{+}(μ)r_{−}(μ)| + ∑_{j=1}^{∞} e^{γt_{j}}|ρ_{j}|) ≤ C \frac{k^{q}}{\sqrt{1 + k|μ|}}.

The proof is now complete. □

Lemma 4.3. For any γ ∈ [0, α) there are C and k_{0} such that, for k ≤ k_{0},

∥ρ̂_{n}(kA)∥ + ∥ρ̄_{n}(kA)∥ ≤ CMe^{−γt_{n}}, for n ≥ 0.

Proof. Let F_{n}(μ) denote either ρ̂_{n}(μ) or ρ̄_{n}(μ). Let 0 ≤ γ < γ_{1} < α, set A_{γ_{1}} = A−γ_{1}I and note that ∥(λI −A_{γ_{1}})^{−1}∥ ≤ M/|λ−γ_{1}| ≤ CM/|λ| for λ ∉ Σ_{0φ}. Writing F_{n}(kA) = F_{n}(kA_{γ_{1}} + kγ_{1}I) we have

(4.19) e^{γt_{n}}F_{n}(kA) = \frac{1}{2πi} ∫_{Γ} e^{γt_{n}}F_{n}(kλ + kγ_{1})(λI − A_{γ_{1}})^{−1} dλ, with Γ = Γ_{0φ}.

We shall show that there are c, C, ε, k_{0} > 0 such that, for k ≤ k_{0},

(4.20)  e^{γt_{n}}|ρ_{n}(kλ + kγ_{1})| ≤ \{ \begin{matrix}Ce^{ct_{n}|λ|}, & for k|λ| ≤ ε, \\ Ce^{−ct_{n}|λ|}, & for k|λ| ≤ ε, λ ∈ Σ_{0φ}, \\ C(k|λ|)^{−1/2}, & for k|λ| ≥ ε, λ ∈ Σ_{0φ}. \end{matrix}

Since |r_{+}(μ)r_{−}(μ)| = |3 + 2μ|^{−1} ≤ C for Re μ ≥ 0, this implies the same bounds for e^{γt_{n}}|F_{n}(kλ + kγ_{1})| and the proof is then accomplished by deforming the contour Γ in (4.19) into the boundary of B(0, ε/t_{n})∪Σ_{0φ} and considering the cases |λ| = ε/t_{n}, ε/t_{n} ≤ |λ| ≤ ε/k, and |λ| ≥ ε/k separately.

Since |r_{+}(μ)| ≤ e^{−c Re μ} for |μ| ≤ 2ε, with c = γ/γ_{1}, we have for k ≤ ε/γ_{1} that e^{γk}|r_{+}(kλ + kγ_{1})| ≤ e^{−ck Re λ} for k|λ| ≤ ε. Since e^{γk}|r_{−}(μ)| ≤ κ ≤ e^{−c|μ|} for |μ| ≤ 2ε, and |r_{+}(μ) − r_{−}(μ)| is bounded below for |μ| ≤ 2ε, the first two of the bounds in (4.20) follow. The third bound follows readily from (4.17). □

5. Error estimates. So far we have only considered the stability of our discrete schemes for (1.1). We shall now briefly give some associated error estimates, and we then need to assume that the quadrature rules are accurate of orders O(k) and O(k^{2}) for the backward Euler and the second order backward difference methods, respectively. We assume thus that the quadrature rule defined in (1.7) satisfies, with m = 1, or 2 and J a positive integer, and with ∥g∥_{J} = ∑_{j=0}^{J} ∥g^{(j)}∥,

(5.1)  ∥q_{n}(g)∥ ≤ Ck^{m} ∫_{0}^{t_{n}} ∥g∥_{J} dt, where q_{n}(g) = (Q_{k}g)_{n} − ∫_{0}^{t_{n}} g dt.

Theorem 5.1. Under the assumptions of Theorem 3.3 and with (5.1) satisfied with m = 1 we have for the solutions of (3.1) and (1.1), for n ≥ 0, k ≤ k_{0},

∥U_{n} − u(t_{n})∥ ≤ Ck ∫_{0}^{t_{n}} e^{−γ(t_{n}−s)}(∥u_{tt}(s)∥ + (t_{n+1} − s)∥Au(s)∥_{J}) ds.

Proof. Setting e_{n} = U_{n} − u(t_{n}) we have

¯∂e_{n} + Ae_{n} = (B_{k}Ae)_{n} − τ_{n} + q_{n}(b_{n}Au), where τ_{n} = ¯∂u(t_{n}) − u_{t}(t_{n}),

and hence, by Theorem 3.3, since e_{0} = 0,

∥e_{n}∥ ≤ CMk ∑_{j=1}^{n} e^{−γt_{n−j}}(∥τ_{j}∥ + ∥q_{j}(b_{j}Au)∥), for n ≥ 0, k ≤ k_{0}.

Here

k ∑_{j=1}^{n} e^{−γt_{n−j}}∥τ_{j}∥ ≤ k ∑_{j=1}^{n} e^{−γt_{n−j}} ∫_{t_{j−1}}^{t_{j}} ∥u_{tt}∥ ds ≤ ke^{γk} ∫_{0}^{t_{n}} e^{−γ(t_{n}−s)}∥u_{tt}(s)∥ ds.

Further, using (5.1) and (1.6),

∥q_{j}(b_{j}Au)∥ ≤ Ck ∫_{0}^{t_{j}} ∥b(t_{j}, s)Au(s)∥_{J} ds ≤ Ck ∫_{0}^{t_{j}} e^{−γ(t_{j}−s)}∥Au(s)∥_{J} ds,

so that

k ∑_{j=1}^{n} e^{−γt_{n−j}}∥q_{j}(b_{j}Au)∥ ≤ Ck ∫_{0}^{t_{n}} e^{−γ(t_{n}−s)}(t_{n+1} − s)∥Au(s)∥_{J} ds.

Together these estimates complete the proof. □

We only state the corresponding result for the second order method.

Theorem 5.2. Under the assumptions of Theorem 4.1 and with (5.1) satisfied with m = 2 we have for the solutions of (4.1), (4.2), and (1.1), for n ≥ 0, k ≤ k_{0},

∥U_{n} − u(t_{n})∥ ≤ Ck^{2} ∫_{0}^{t_{n}} e^{−γ(t_{n}−s)}(∥u_{ttt}(s)∥ + (t_{n+1} − s)∥Au(s)∥_{J}) ds.

6. Maximum-norm stability of finite element approximations. In this final section we shall apply our results to the case that A is the discrete analogue of the Laplacian −Δ, with Dirichlet boundary conditions, in a convex plane domain Ω with smooth or polygonal boundary, based on piecewise linear finite elements. We first recall from [4] and [7] that for any φ ∈ (0, π/2) there is a positive M such that

∥(λI + Δ)^{−1}v∥_{L_{∞}} ≤ \frac{M}{|λ|}∥v∥_{L_{∞}}, for v ∈ C(Ω̄), λ ∈ C \ Σ_{0φ}.

In fact, with X = C(Ω̄), A = −Δ satisfies (1.2) for any φ ∈ (0, π/2) with α > 0 the smallest eigenvalue of −Δ (see the argument for the discrete analogue at the end of this section). Note that D(A) = \{u ∈ C(Ω̄) : u|_{∂Ω} = Δu|_{∂Ω} = 0\} with \overline{D(A)} ≠ X. Theorems 2.1 and 3.3 hence apply for any γ ∈ [0, α), and since Λ_{αφγ} may now be chosen arbitrarily close to Λ_{α0γ} = 1 − γ/α, condition (1.5) on b(t, s) essentially reduces to condition (2.4) in the selfadjoint case of Theorems 2.2 and 3.2.

Let now S_{h} denote the standard continuous piecewise linear functions associated with a member of a set of quasiuniform triangulations of Ω, and let −Δ_{h} : S_{h} → S_{h} be defined by (−Δ_{h}ψ, χ) = (∇ψ, ∇χ) for ψ, χ ∈ S_{h}. In order to apply our above results we need to know that −Δ_{h} satisfies (1.2) for suitable α and φ. It was shown in [6] that, with ℓ_{h} = log(1/h), c > 0,

∥(λI + Δ_{h})^{−1}χ∥_{L_{∞}} ≤ Cℓ_{h}^{2}|λ|^{−1}∥χ∥_{L_{∞}}, ∀χ ∈ S_{h}, λ ∈ C \ Σ_{0φ}, φ = \frac{π}{2} − cℓ_{h}^{−2}.

For our purposes it is desirable to have an estimate which is valid also for small values of | arg λ|. (In one dimension such a result, without the factor ℓ_{h}, is shown in [2].) With this in mind we shall show the following.

Lemma 6.1. For any φ ∈ (0, π/2) and with α the smallest eigenvalue of −Δ, there exists C = C_{φ} such that, for χ ∈ S_{h},

(6.1)  ∥(λI + Δ_{h})^{−1}χ∥_{L_{∞}} ≤ Cℓ_{h}|λ − α|^{−1}∥χ∥_{L_{∞}}, for λ ∈ C \ Σ_{αφ}.

Proof. In the first part of the proof we show (6.1) with α = 0. Let E_{h}(t) = e^{tΔ_{h}} be the semigroup on S_{h} generated by Δ_{h}. It is clear that E_{h}(z) = e^{zΔ_{h}} is holomorphic for all z ∈ C and we shall prove that for any φ_{1} ∈ (0, π/2) there is C such that

(6.2)  ∥E_{h}(z)χ∥_{L_{∞}} ≤ Cℓ_{h}∥χ∥_{L_{∞}}, ∀χ ∈ S_{h}, | arg z| ≤ φ_{1}.

Assume for a moment that this has been shown. We have

(λI + Δ_{h})^{−1} = − ∫_{0}^{∞} e^{λt}E_{h}(t) dt, for Re λ < 0.

For | arg λ| ∈ (φ, π] we may rotate the path of integration to any ray z = te^{iφ_{1}} with |φ_{1}| ≤ \frac{1}{2}(π −φ), say. In view of (6.2) this shows (6.1) with α = 0, because | arg(λz)| may be chosen in [\frac{1}{2}(π + φ), π], so that Re(λz) ≤ −ct|λ| with c = sin(φ/2).

The inequality (6.2) will follow, if we show that for any ε > 0 there is C_{ε} such that, for n ≥ 0, t > 0,

(6.3)  ∥E_{h}^{(n)}(t)χ∥_{L_{∞}} ≤ C_{ε}ℓ_{h}(1 + n^{1/2})n! t^{−n}(1 + ε)^{n}∥χ∥_{L_{∞}}, ∀χ ∈ S_{h}.

In fact, (6.3) implies that, for any δ ∈ (0, 1),

∥E_{h}(z)χ∥_{L_{∞}} = ∥ ∑_{n=0}^{∞} \frac{(z − t)^{n}}{n!} E_{h}^{(n)}(t)χ∥_{L_{∞}}

≤ C_{ε}ℓ_{h}∥χ∥_{L_{∞}} ∑_{n=0}^{∞} (\frac{|z − t|(1 + ε)}{t})^{n}(1 + n^{1/2}) ≤ C_{ε,δ}ℓ_{h}∥χ∥_{L_{∞}},

for all z in the union of the discs |z − t| ≤ δt/(1 + ε), t > 0, i.e., in the sector | arg z| ≤ arcsin(δ/(1 + ε)) = φ_{1}, where φ_{1} may be arbitrarily close to π/2.

The special cases n = 0, 1 of (6.3) were proved in [6] (cf. also [8]). In the following we extend the techniques of [6] to cover the general case n ≥ 1. We first define a discrete fundamental solution Γ = Γ_{h}^{x}(t) as Γ_{h}^{x}(t) = E_{h}(t)δ_{h}^{x} with δ_{h}^{x} ∈ S_{h} the discrete delta-function given by (δ_{h}^{x}, χ) = χ(x), ∀χ ∈ S_{h}. We then have

(E_{h}^{(n)}(t)χ)(x) = (Γ^{(n)}(t), χ) ≤ ∥Γ^{(n)}(t)∥_{L_{1}} ∥χ∥_{L_{∞}}.

Introducing also the modified distance function ρ(y) = ρ_{h}^{x}(y) = (|x − y|^{2} + h^{2})^{1/2}, we have (for simplicity we often write ∥ · ∥ for ∥ · ∥_{L_{2}} below)

∥Γ^{(n)}∥_{L_{1}} ≤ ∥ρ^{−1}∥ ∥ρΓ^{(n)}∥ ≤ Cℓ_{h}^{1/2}∥ρΓ^{(n)}∥,

and the estimate (6.3) will follow from the weighted L_{2}-bound

(6.4)  t^{n}∥ρΓ^{(n)}(t)∥ ≤ C_{ε}ℓ_{h}^{1/2}n! n^{1/2}(1 + ε)^{n}, for t > 0, n ≥ 1, ε > 0.

To show (6.4) we use the identity

(6.5)  (Γ^{(n+1)}, χ) + (∇Γ^{(n)}, ∇χ) = 0, ∀χ ∈ S_{h}, n ≥ 0,

to find

\frac{1}{2} \frac{d}{dt}(t^{2n}∥ρΓ^{(n)}∥^{2}) + t^{2n}∥ρ∇Γ^{(n)}∥^{2}

= t^{2n}(Γ^{(n+1)}, ρ^{2}Γ^{(n)} − χ) + t^{2n}(∇Γ^{(n)}, ∇(ρ^{2}Γ^{(n)} − χ))

− t^{2n}2(ρ∇Γ^{(n)}, Γ^{(n)}∇ρ) + nt^{2n−1}∥ρΓ^{(n)}∥^{2}, ∀χ ∈ S_{h}.

Taking χ = P_{h}(ρ^{2}Γ^{(n)}), with P_{h} the orthogonal L_{2}-projection onto S_{h}, then the first term on the right vanishes, and for the second term we obtain by Lemma 1.4 in [6], and the inverse inequality ∥∇χ∥ ≤ Ch^{−1}∥χ∥ for χ ∈ S_{h},

∥∇Γ^{(n)}∥ ∥∇(ρ^{2}Γ^{(n)} − χ)∥ ≤ Ch^{−1}∥Γ^{(n)}∥ Ch(∥Γ^{(n)}∥ + ∥ρ∇Γ^{(n)}∥)
≤ C(∥Γ^{(n)}∥^{2} + ∥Γ^{(n)}∥ ∥ρ∇Γ^{(n)}∥).

(6.6)

LONG TIME BEHAVIOR OF BACKWARD DIFFERENCE TYPE METHODS 21

Using also |∇ρ| ≤ 1, we find

\frac{1}{2} \frac{d}{dt}(t^{2n}∥ρΓ^{(n)}∥^{2}) + t^{2n}∥ρ∇Γ^{(n)}∥^{2}

≤ Ct^{2n}(∥Γ^{(n)}∥^{2} + ∥Γ^{(n)}∥ ∥ρ∇Γ^{(n)}∥) + nt^{2n−1}∥ρΓ^{(n)}∥^{2}.

This shows, for any ε > 0,

\frac{1}{2} \frac{d}{dt}(t^{2n}∥ρΓ^{(n)}∥^{2}) + (1 − ε)t^{2n}∥ρ∇Γ^{(n)}∥ ≤ C_{ε}t^{2n}∥Γ^{(n)}∥^{2} + nt^{2n−1}∥ρΓ^{(n)}∥^{2},

and hence by integration (6.7) t^{2n}∥ρΓ^{(n)}(t)∥^{2} + 2(1 − ε) ∫_{0}^{t} s^{2n}∥ρ∇Γ^{(n)}∥^{2} ds

≤ C_{ε} ∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2}ds + 2n ∫_{0}^{t} s^{2n−1}∥ρΓ^{(n)}∥^{2}ds.

We shall now bound the two terms on the right by the square of the right hand side of (6.4) and thus complete the proof.

Using (6.5) with n − 1 instead of n and χ = t^{2n}Γ^{(n)} we find

t^{2n}∥Γ^{(n)}∥^{2} + \frac{1}{2} \frac{d}{dt}(t^{2n}∥∇Γ^{(n−1)}∥^{2}) = nt^{2n−1}∥∇Γ^{(n−1)}∥^{2},

and hence after integration

∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds ≤ n ∫_{0}^{t} s^{2n−1}∥∇Γ^{(n−1)}∥^{2} ds.

Similarly, using instead χ = t^{2n−1}Γ^{(n−1)}, we obtain

\frac{1}{2} \frac{d}{dt}(t^{2n−1}∥Γ^{(n−1)}∥^{2}) + t^{2n−1}∥∇Γ^{(n−1)}∥^{2} = \frac{1}{2}(2n − 1)t^{2n−2}∥Γ^{(n−1)}∥^{2},

and after integration

∫_{0}^{t} s^{2n−1}∥∇Γ^{(n−1)}∥^{2} ds ≤ \frac{1}{2}(2n − 1) ∫_{0}^{t} s^{2n−2}∥Γ^{(n−1)}∥^{2} ds,

so that

∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds ≤ \frac{2n(2n − 1)}{4} ∫_{0}^{t} s^{2n−2}∥Γ^{(n−1)}∥^{2} ds.

By repeated application, and using ∫_{0}^{t} ∥Γ∥^{2} ds ≤ Cℓ_{h}, see (2.8) of [6],

(6.8)  ∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds ≤ \frac{(2n)!}{2^{2n}} ∫_{0}^{t} ∥Γ∥^{2} ds ≤ C \frac{(2n)!}{2^{2n}}ℓ_{h} ≤ C(n!)^{2}ℓ_{h}.

This shows the desired estimate for the first term in (6.7).

22 N. YU. BAKAEV, S. LARSSON AND V. THOMÉE

We now turn to the second term on the right in (6.7). We have

t^{2n−1}∥ρΓ^{(n)}∥^{2} + \frac{1}{2} \frac{d}{dt}(t^{2n−1}∥ρ∇Γ^{(n−1)}∥^{2})

= t^{2n−1}(Γ^{(n)}, ρ^{2}Γ^{(n)} − χ) + t^{2n−1}(∇Γ^{(n−1)}, ∇(ρ^{2}Γ^{(n)} − χ))

− 2t^{2n−1}(ρ∇Γ^{(n−1)}, Γ^{(n)}∇ρ) + \frac{1}{2}(2n − 1)t^{2n−2}∥ρ∇Γ^{(n−1)}∥^{2}

= I_{1} + I_{2} + I_{3} + I_{4}, (6.9)

Setting χ = P_{h}(ρ^{2}Γ^{(n)}), we get I_{1} = 0. By the same argument as in (6.6), the inverse inequality, and the fact that ∥ρ∇Γ^{(n)}∥ ≤ Ch^{−1}∥ρΓ^{(n)}∥ (see [6], p. 283), we derive

|I_{2}| ≤ Ct^{2n−1}∥∇Γ^{(n−1)}∥ h(∥Γ^{(n)}∥ + ∥ρ∇Γ^{(n)}∥)
≤ Ct^{2n−1}(∥Γ^{(n−1)}∥ ∥Γ^{(n)}∥ + ∥∇Γ^{(n−1)}∥ ∥ρΓ^{(n)}∥),

≤ C(t^{2n−2}∥Γ^{(n−1)}∥^{2} + t^{2n}∥Γ^{(n)}∥^{2}) + C_{ε}t^{2n−1}∥∇Γ^{(n−1)}∥^{2} + εt^{2n−1}∥ρΓ^{(n)}∥^{2}, and, using |∇ρ| ≤ 1,

|I_{3}| + |I_{4}| ≤ 2t^{2n−1}∥ρ∇Γ^{(n−1)}∥ ∥Γ^{(n)}∥ + \frac{1}{2}(2n − 1)t^{2n−2}∥ρ∇Γ^{(n−1)}∥^{2}
≤ Ct^{2n}∥Γ^{(n)}∥^{2} + nt^{2n−2}∥ρ∇Γ^{(n−1)}∥^{2}.

Inserting these estimates into (6.9) and integrating yields

(1 − ε) ∫_{0}^{t} s^{2n−1}∥ρΓ^{(n)}∥^{2} ds ≤ C ∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds + C ∫_{0}^{t} s^{2n−2}∥Γ^{(n−1)}∥^{2} ds

+ C_{ε} ∫_{0}^{t} s^{2n−1}∥∇Γ^{(n−1)}∥^{2} ds + n ∫_{0}^{t} s^{2n−2}∥ρ∇Γ^{(n−1)}∥^{2} ds.

To estimate the last term we recall from (6.7) that, for any ε > 0,

∫_{0}^{t} s^{2n}∥ρ∇Γ^{(n)}∥^{2} ds ≤ C_{ε} ∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds + n(1 + ε) ∫_{0}^{t} s^{2n−1}∥ρΓ^{(n)}∥^{2} ds. Using this with n replaced by n − 1 we thus find, again for any ε > 0,

∫_{0}^{t} s^{2n−1}∥ρΓ^{(n)}∥^{2} ds ≤ C_{ε} ∫_{0}^{t} s^{2n}∥Γ^{(n)}∥^{2} ds + C_{ε} ∫_{0}^{t} s^{2n−2}∥Γ^{(n−1)}∥^{2} ds
+ n^{2}(1 + ε) ∫_{0}^{t} s^{2n−3}∥ρΓ^{(n−1)}∥^{2} ds.

Hence, using also (6.8), ∫_{0}^{t} s^{2n−1}∥ρΓ^{(n)}∥^{2} ds ≤ Cℓ_{h}(n!)^{2} + (1 + ε)n^{2} ∫_{0}^{t} s^{2n−3}∥ρΓ^{(n−1)}∥^{2} ds.

By repeated application, and using that ∫_{0}^{t} s∥ρΓ′∥^{2} ds ≤ Cℓ_{h} (see [6], p. 284),

∫_{0}^{t} s^{2n−1}∥ρΓ(n)∥^{2} ds ≤ Cℓ_{h}(n!)^{2}(1+(1+ε)+⋯+(1+ε)^{n−1}) ≤ C_{ε}ℓ_{h}(n!)^{2}(1+ε)^{n},

which together with (6.8) and (6.7) completes the proof of (6.4).

It remains to show (6.1) with α equal to the smallest eigenvalue of −Δ. Since |λ − α| ≤ C|λ| for φ/2 ≤ | arg λ| ≤ π, |λ| ≥ ε > 0, it remains to show the bound for λ in the bounded set D = {λ : | arg(λ − α)| ≥ φ, | arg λ| ≤ φ/2} ∪ {λ : |λ| ≤ ε}. For this purpose we use the inequality

(6.10)  ∥χ∥_{L_{∞}} ≤ C∥Δ_{h}χ∥_{L_{2}}.

Setting T_{h} = (−Δ_{h})^{−1}, T = (−Δ)^{−1}, this follows easily with f = Δ_{h}χ from

∥T_{h}f∥_{L_{∞}} ≤ ∥P_{h}T f∥_{L_{∞}} + ∥(T_{h} − P_{h}T)f∥_{L_{∞}} ≤ C∥f∥_{L_{2}},

where we first used the boundedness of P_{h} in L_{∞}, then ∥T f∥_{L_{∞}} ≤ C∥T f∥_{H^{2}} ≤ C∥f∥_{L_{2}}, and finally, by standard arguments (cf. [8]),

\begin{aligned}∥(T_{h} − P_{h}T)f∥_{L_{∞}} & ≤ Ch^{−1}∥(T_{h} − P_{h}T)f∥_{L_{2}} \\ & ≤ Ch^{−1}(∥(T_{h} − T)f∥_{L_{2}} + ∥(P_{h} − I)T f∥_{L_{2}}) ≤ Ch∥f∥_{L_{2}}. \end{aligned}

Since Δ_{h}(λI + Δ_{h})^{−1} = I − λ(λI + Δ_{h})^{−1} we find from (6.10)

∥(λI + Δ_{h})^{−1}χ∥_{L_{∞}} ≤ C∥Δ_{h}(λI + Δ_{h})^{−1}χ∥_{L_{2}}
≤ C(1 + |λ| ∥(λI + Δ_{h})^{−1}∥_{L_{2}→L_{2}})∥χ∥_{L_{∞}}.

For λ ∈ D we have, since the eigenvalues of −Δ_{h} are bounded from below by α,

∥(λI + Δ_{h})^{−1}∥_{L_{2}→L_{2}} ≤ dist(λ, σ(−Δ_{h}))^{−1} ≤ C|λ − α|^{−1},

and since D is bounded this completes the proof. □

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1Department of Mathematics, Air Force Engineering Academy, Leningradskii Prospekt 40, Moscow 125190, Russia

2Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden

   E-mail address: bakaev@postman.ru, stig@math.chalmers.se, thomee@math.chalmers.se