RESOLVENT ESTIMATES FOR ELLIPTIC FINITE

ELEMENT OPERATORS IN ONE DIMENSION^{∗}

M. Crouzeix^{1}, S. Larsson^{2} and V. Thomée^{2}

Abstract. We prove the analyticity (uniform in h) of the semigroups generated on L_{p}(0, 1), 1 ≤ p ≤ ∞, by finite element analogues A_{h} of a one-dimensional second order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.

1. Introduction

Let A be an elliptic differential operator in one dimension defined by

Au = −(au′)′ + bu′ + cu, x ∈ Ω ≡ (0, 1), with u(0) = u(1) = 0,

where u′ = du/dx and the coefficients a, b, c are smooth and real-valued with a positive on Ω̄. Defining the corresponding sesquilinear form

A(u, v) = (au′, v′) + (bu′, v) + (cu, v), where (u, v) = ∫_{0}^{1} u\overline{v} dx,

we assume that there is a_{0} > 0 such that

(1.1)  Re A(v, v) ≥ a_{0}∥v′∥^{2},  v ∈ H_{0}^{1}(Ω),

where ∥u∥^{2} = (u, u). If this is not satisfied originally, it may be accomplished by adding a positive multiple of u to Au.

Let 0 = x_{0} < x_{1} < … < x_{N} < x_{N+1} = 1 define a partition of the interval Ω into subintervals Ω_{i} = (x_{i}, x_{i+1}) of lengths h_{i} = x_{i+1} − x_{i} and with h = max h_{i}. We shall think of this partition as a member of a family of such partitions, which is assumed to be quasi-uniform in the sense that for some c > 0 we have h_{i} ≥ ch, i = 1, …, N. For a fixed integer r ≥ 2 we define

S_{h} = {v ∈ H_{0}^{1}(Ω) : v|_{Ω_{i}} ∈ Π_{r−1}, i = 1, …, N},

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 65M60, 65M12.

Key words and phrases. Resolvent, sectorial, elliptic, analytic semigroup, Banach space, L_{p}, maximum norm, finite element method, rational approximation, stability, error estimate.

2Partly supported by the Swedish Research Council for Engineering Sciences (TFR).

∗Published in Math. Comp. 63 (1994), 121–140.

where Π_{r−1} denotes the space of all polynomials (with complex coefficients) of degree ≤ r − 1. We then define a discrete analogue A_{h} : S_{h} → S_{h} of the operator A by the relation

(A_{h}ψ, χ) = A(ψ, χ), ∀ψ, χ ∈ S_{h}.

Let R(λ, −A) = (λI+A)^{−1}, where λ is a complex parameter, denote the resolvent of −A. More precisely we define R(λ, −A)f for f ∈ H^{−1}(Ω), the dual space of H_{0}^{1}(Ω), as the solution u = u(·, λ) ∈ H_{0}^{1}(Ω) of the variational equation

(1.2)  A(u, χ) + λ(u, χ) = ⟨f, \overline{χ}⟩,  ∀χ ∈ H_{0}^{1}(Ω),

where ⟨·, ·⟩ denotes the duality pairing between H^{−1}(Ω) and H_{0}^{1}(Ω). Similarly, we note that R(λ, −A_{h})P_{h}f is the solution u_{h} = u_{h}(·, λ) ∈ S_{h} of

(1.3)  A(u_{h}, χ) + λ(u_{h}, χ) = ⟨f, \overline{χ}⟩,  ∀χ ∈ S_{h},

if we define the projection P_{h} : H^{−1}(Ω) → S_{h} by

(1.4)  (P_{h}f, χ) = ⟨f, \overline{χ}⟩,  ∀χ ∈ S_{h}.

Note that the restriction of P_{h} to L_{2}(Ω) is the standard orthogonal projection.

The main purpose of this work is to prove the following estimates of the resolvents of −A and −A_{h}. We use the notation ∥ · ∥_{p} for the standard norms in L_{p}(Ω), 1 ≤ p ≤ ∞.

Theorem 1.1. There are φ ∈ (\frac{1}{2}π, π) and C ≥ 1 such that the linear operators R(λ, −A) and R(λ, −A_{h})P_{h} are bounded from H^{−1}(Ω) into H_{0}^{1}(Ω), uniformly with respect to λ in the sector Σ_{φ} = {λ ∈ C : | arg λ| ≤ φ}, and, for 1 ≤ p ≤ ∞, we have

∥R(λ, −A)f∥_{p} + ∥R(λ, −A_{h})P_{h}f∥_{p} ≤ \frac{C}{1 + |λ|}∥f∥_{p}, f ∈ L_{p}(Ω), λ ∈ Σ_{φ}.

This theorem is proved in Section 2. The proof is based on estimates of the Green’s functions of (1.2) and (1.3), which we obtain by an energy argument. Since the relevant Green’s functions do not belong to H_{0}^{1}(Ω) in the multidimensional case, it is not clear whether this technique can be generalized; one-dimensionality is also used in various technical details below.

In the final Section 3 we discuss some applications of these resolvent estimates. The first application concerns stability and error estimates for spatially semidiscrete finite element approximations of the parabolic initial-boundary value problem

(1.5)  \begin{matrix}u_{t} + Au = f(x, t), & x ∈ Ω, t > 0, \\ u(0, t) = u(1, t) = 0, & t > 0, \\ u(x, 0) = u_{0}(x), & x ∈ Ω, \end{matrix}

where u = u(x, t), u_{t} = ∂u/∂t and where f and u_{0} are given. The semidiscrete finite element approximation u_{h}(t) ∈ S_{h} is defined by the equation

(1.6)  u_{h,t} + A_{h}u_{h} = P_{h}f(·, t), t > 0;  u_{h}(0) = u_{0h},

where u_{0h} ∈ S_{h} is an approximation of u_{0}.

We now think of A as an unbounded operator on L_{p}(Ω) with domain D(A) = W̊_{p}^{2}(Ω) = {u ∈ W_{p}^{2}(Ω) : u(0) = u(1) = 0}. Elementary arguments show that

(1.7)  ∥u∥_{m,p} ≤ C_{m}∥Au∥_{m−2,p},  u ∈ W_{p}^{m}(Ω) ∩ W̊_{p}^{2}(Ω), m ≥ 2,

where ∥u∥_{m,p} = (∑_{j=0}^{m} ∥u^{(j)}∥_{p}^{p})^{1/p} denotes the standard norm in W_{p}^{m}(Ω) (with the usual modification for p = ∞), and hence that A is a closed operator on L_{p}(Ω). Together with the resolvent estimates in Theorem 1.1 this shows that A and A_{h} generate analytic semigroups E(t) = e^{−tA} and E_{h}(t)P_{h} = e^{−tA_{h}}P_{h} in L_{p}(Ω), see for example [12, Theorem 2.5.2]. In the usual way we obtain the representation

(1.8)  E(t) = e^{−tA} = \frac{1}{2πi} ∫_{Γ} e^{tλ}R(λ, −A) dλ,

where the contour Γ is the boundary of Σ_{φ} oriented so that Im λ increases along Γ, and similarly

(1.9)  E_{h}(t) = e^{−tA_{h}} = \frac{1}{2πi} ∫_{Γ} e^{tλ}R(λ, −A_{h}) dλ.

Using (1.8), (1.9) and the resolvent estimates we obtain the bounds

(1.10)  ∥E(t)v∥_{p} + t∥AE(t)v∥_{p} ≤ C∥v∥_{p},  t > 0,

and

(1.11)  ∥E_{h}(t)P_{h}v∥_{p} + t∥A_{h}E_{h}(t)P_{h}v∥_{p} ≤ C∥v∥_{p},  t > 0,

by a standard argument. In a similar way we see that E(t) and E_{h}(t)P_{h} are bounded from H^{−1}(Ω) into H_{0}^{1}(Ω) for t > 0, in the latter case the bound is independent of h. From semigroup theory it also follows that u(t) = E(t)u_{0} and u_{h}(t) = E_{h}(t)u_{0h} are solutions of the homogeneous problems (1.5) and (1.6) with f(x, t) ≡ 0. Solutions of the nonhomogeneous problems can then be obtained by Duhamel’s principle: for (1.5) we have

(1.12)  u(t) = E(t)u_{0} + ∫_{0}^{t} E(t − s)f(·, s) ds,  t ≥ 0,

under suitable regularity assumptions on f. An analogous formula holds for (1.6).

In this connection we note that when p = ∞ we have \overline{D(A)} = C_{0}(Ω̄) = \{u ∈ C(Ω̄) : u(0) = u(1) = 0}, so that A is not densely defined in L_{∞}(Ω), and hence the standard theory of analytic semigroups does not apply when p = ∞. However, most of the semigroup theory can be developed without the assumption that D(A) is dense, see [13] and [6]. Note also that A is densely defined in C_{0}(Ω̄) with the maximum norm in which space the classical semigroup theory is thus applicable. But the existence theorems for (1.12) then require that f(t) ∈ C_{0}(Ω̄), which is not satisfactory, since it places an unnecessary boundary condition on f(t). We therefore prefer to work in L_{∞}(Ω) rather than in C_{0}(Ω̄).

Stability estimates of the form (1.11) with p = ∞, but with logarithmic factors in the right-hand sides, were obtained in [15] and [14]. Apart from removing the logarithm, our present proof of stability is simpler than those of [15, 14].

Using the stability estimate for E_{h}(t) in (1.11) we prove in Theorem 3.2 that for u_{0h} suitably chosen we have the error estimate

(1.13) ∥u_{h}(t) − u(t)∥_{p} ≤ Ch^{r}(∥u_{0}∥_{r,p} + ∫_{0}^{t} ∥u_{t}(s)∥_{r,p} ds),  1 ≤ p ≤ ∞.

Smooth solutions are thus approximated to optimal order.

In some applications it is important to allow nonsmooth initial data, such as discontinuous functions or even δ-functions. Let M(Ω) denote the set of finite measures μ with ∥μ∥_{M} = |μ|(Ω) = ∫_{Ω} d|μ|. With each μ ∈ M(Ω) we associate a distribution (also denoted μ) in the usual way: ⟨μ, χ⟩ = ∫_{Ω} χ(x) dμ(x) for χ ∈ C(Ω̄). Recall that P_{h} : M(Ω) → S_{h} is defined in (1.4). For solutions of the homogeneous equations, i.e., (1.5) and (1.6) with f(x, t) ≡ 0, and with u_{0h} = P_{h}u_{0} we show in Theorem 3.5 that, for t > 0,

(1.14)  \begin{aligned}∥u_{h}(t) − u(t)∥_{p} & ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p}, u_{0} ∈ L_{p}(Ω), 1 ≤ p ≤ ∞, \\ ∥u_{h}(t) − u(t)∥_{∞} & ≤ Ch^{r}t^{−(r+1)/2}∥u_{0}∥_{M}, u_{0} ∈ M(Ω). \end{aligned}

In addition to (1.10) and (1.11) the proof uses a stability property of E(t) and E_{h}(t) considered as operators from M(Ω) to L_{∞}(Ω), namely

∥E(t)v∥_{∞} + ∥E_{h}(t)P_{h}v∥_{∞} ≤ Ct^{−1/2}∥v∥_{M}, t > 0,

resulting from the corresponding resolvent estimates, which are also derived in Section 2.

We remark that our proofs of these error bounds differ slightly from those of [14], which introduce additional logarithmic factors.

Another application of the resolvent estimates in Theorem 1.1 concerns the stability and error analysis of fully discrete schemes based on rational approximations of the analytic semigroup E_{h}(t). Let r(λ) be a rational approximation of the exponential function exp(λ), which is accurate of order q ≥ 1, i.e.,

(1.15)  r(λ) = e^{λ} + O(λ^{q+1}),  λ → 0,

and A-acceptable, i.e.,

(1.16)  |r(λ)| ≤ 1,  Re λ ≤ 0.

Then E_{kh}^{n}u_{0h} = r(−kA_{h})^{n}u_{0h} is an approximation of E_{h}(t_{n})u_{0h}, where k is the timestep and t_{n} = nk for n = 1, 2, . . .. Using the resolvent estimate and contour integral representations one may then prove stability for the discrete evolution operator E_{kh}^{n},

(1.17)  ∥E_{kh}^{n}u_{0h}∥_{p} ≤ C∥u_{0h}∥_{p},  n ≥ 0, 1 ≤ p ≤ ∞,

see [3], [10], [11]. Stability may be combined with a local truncation error analysis in a similar way as in (1.13) to yield an error bound for certain completely discrete schemes. We demonstrate this in Theorem 3.3 for the backward Euler method:

(1.18)  \begin{matrix}U_{n} ∈ S_{h}, U_{0} = u_{0h}, \\ ¯∂_{t}U_{n} + A_{h}U_{n} = P_{h}f(t_{n}),  ¯∂_{t}U_{n} = (U_{n} − U_{n−1})/k,  n ≥ 1, \end{matrix}

and the Crank-Nicolson method:

(1.19)  \begin{matrix}U_{n} ∈ S_{h}, U_{0} = u_{0h}, \\ ¯∂_{t}U_{n} + A_{h}(U_{n} + U_{n−1})/2 = P_{h}f(t_{n−1/2}),  n ≥ 1. \end{matrix}

For general time discretization schemes it is necessary to introduce more complicated approximations of the source term f, see [2]. We refrain from addressing this difficulty here and restrict further discussion to the homogeneous problem (f ≡ 0). In this case we have U_{n} = E_{kh}^{n}u_{0h}, u_{h}(t) = E_{h}(t)u_{0h} and the resolvent estimates and contour integral representations yield the error bounds

(1.20) ∥U_{n} − u_{h}(t_{n})∥_{p} ≤ Ck^{j}∥A_{h}^{j}u_{0h}∥_{p},  t_{n} = nk ≥ 0, 1 ≤ p ≤ ∞, 1 ≤ j ≤ q,

and, if r is strongly A-acceptable, that is, if in addition |r(∞)| < 1, then

(1.21) ∥U_{n} − u_{h}(t_{n})∥_{p} ≤ Ck^{q}t_{n}^{−q}∥u_{0h}∥_{p}, t_{n} = nk > 0, 1 ≤ p ≤ ∞,

see for example [8]. Using the above bounds for u_{h}(t) − u(t) and some additional arguments, one may then obtain bounds for the total error U_{n}−u(t_{n}). For example, when u_{0h} = P_{h}u_{0}, the error bounds (1.14) and (1.21) immediately give

∥U_{n} − u(t_{n})∥_{p} ≤ C(h^{r}t_{n}^{−r/2} + k^{q}t_{n}^{−q})∥u_{0}∥_{p}, t_{n} = nk > 0, 1 ≤ p ≤ ∞,

because P_{h} is stable in L_{p}(Ω) by a result of [4]. We may also obtain error bounds which hold uniformly as t → 0, although an argument based directly on (1.20) is not satisfactory, since this applies a discrete norm to the discrete initial value. We show in Theorem 3.6 by a somewhat more involved argument that

(1.22) ∥U_{n} − u(t_{n})∥_{p} ≤ C(h^{r}∥u_{0}∥_{r,p} + k^{q}∥u_{0}∥_{2q,p}),  t_{n} = nk ≥ 0, 1 ≤ p ≤ ∞,

if f ≡ 0, u_{0} is sufficiently smooth and satisfies the appropriate compatibility conditions at x = 0, 1, and u_{0h} is suitably chosen.

2. Resolvent estimates

The main object of this section is to prove the bounds in Theorem 1.1 for the resolvents R(λ, −A) and R(λ, −A_{h}). The resolvent operators will be studied for λ in a sector

Σ_{κ,θ} = \{λ ∈ C : | arg (λ − κ)| ≤ θ\}, with κ ≥ 0, θ ∈ (\frac{1}{2}π, π).

For λ ∈ Σ_{κ,θ} it is convenient to write

(2.1) λ − κ = (ξ + iη)^{2} = ξ^{2} − η^{2} + 2iξη,  with ξ + iη ∈ Σ_{0,θ/2}, ξ, η ∈ R.

For future reference we note that for λ ∈ Σ_{κ,θ} we have

(2.2)  |η| ≤ δξ,  where δ = tan(\frac{1}{2}θ) > 1,

and

(2.3)  ξ^{2} ≤ |λ − κ| = ξ^{2} + η^{2} ≤ (1 + δ^{2})ξ^{2}.

The following estimate will be a basic tool.

Lemma 2.1. For any θ ∈ (\frac{1}{2}π, π) there are κ ≥ 0 and C ≥ 1 such that

∥v′∥^{2} + ξ^{2}∥v∥^{2} ≤ C|A(v, v) + λ∥v∥^{2}|, v ∈ H_{0}^{1}(Ω), λ ∈ Σ_{κ,θ},

where ξ is related to λ as in (2.1). If A is self-adjoint, i.e., b(x) ≡ 0, then we may choose κ = 0.

Proof. Setting

(2.5)  F = A(v, v) + λ∥v∥^{2},

we obtain, by taking the real part of (2.5) and using the transformation of variables in (2.1),

Re A(v, v) + (κ + ξ^{2} − η^{2})∥v∥^{2} = Re F,

and together with (1.1),

(2.6)  a_{0}∥v′∥^{2} + (κ + ξ^{2} − η^{2})∥v∥^{2} ≤ |F |.

By taking the imaginary part of (2.5) we obtain

Im A(v, v) + 2ξη∥v∥^{2} = Im F,

and, since Im A(v, v) = Im(bv′, v),

2ξ|η| ∥v∥^{2} ≤ |F | + ∥b∥_{∞}∥v′∥ ∥v∥.

Multiplying by \frac{1}{2}δ = \frac{1}{2} tan(\frac{1}{2}θ) we have, in view of (2.2),

η^{2}∥v∥^{2} ≤ δξ|η| ∥v∥^{2} ≤ \frac{1}{2}δ|F | + \frac{1}{2}δ∥b∥_{∞}∥v′∥ ∥v∥.

Adding this to (2.6) we obtain

a_{0}∥v′∥^{2} + (κ + ξ^{2})∥v∥^{2} ≤ (1 + \frac{1}{2}δ)|F | + \frac{1}{2}a_{0}∥v′∥^{2} + \frac{δ^{2}∥b∥_{∞}^{2}}{8a_{0}} ∥v∥^{2}

and (2.4) follows by taking κ = δ^{2}∥b∥_{∞}^{2}/(8a_{0}). □

If f ∈ L_{2}(Ω), then taking χ = u in (1.2) and applying the lemma shows

∥u′∥^{2} + ξ^{2}∥u∥^{2} ≤ C|(f, u)| ≤ C∥f∥ ∥u∥, λ ∈ Σ_{κ,θ}.

In particular, using (2.3), we have

∥R(λ, −A)f∥ ≤ \frac{C}{ξ^{2}} ∥f∥ ≤ \frac{C}{|λ − κ|}∥f∥, λ ∈ Σ_{κ,θ},

which shows that −A is a sectorial operator on L_{2}(Ω). The same argument applies to equation (1.3), so that −A_{h} is also sectorial, uniformly in h, with respect to the topology of L_{2}(Ω).

We next prove that −A is sectorial on L_{p}(Ω) for 1 ≤ p ≤ ∞. This result is wellknown, but we give an elementary proof based on the energy estimate of Lemma 2.1, which we shall then modify in Theorem 2.3 to show that −A_{h} is sectorial in the topology of L_{p}(Ω) uniformly in h.

Theorem 2.2. For any θ ∈ (\frac{1}{2}π, π) there are κ ≥ 0 and C ≥ 1 such that

∥R(λ, −A)f∥_{p} ≤ \frac{C}{|λ − κ|}∥f∥_{p}, f ∈ L_{p}(Ω), λ ∈ Σ_{κ,θ}, 1 ≤ p ≤ ∞.

Proof. Let θ ∈ (\frac{1}{2}π, π) be arbitrary and let κ be as in Lemma 2.1. Lemma 2.1 implies that the sesquilinear form B_{λ}(u, v) = A(u, v)+λ(u, v) is coercive on H_{0}^{1}(Ω), that is, |B_{λ}(u, u)| ≥ C^{−1}∥u′∥^{2} for all u ∈ H_{0}^{1}(Ω) and λ ∈ Σ_{κ,θ}. Hence, by the Lax-Milgram lemma we conclude that (1.2) has a unique solution and that

∥R(λ, −A)f∥_{H_{0}^{1}} ≤ C∥f∥_{H^{−1}}, f ∈ H^{−1}(Ω), λ ∈ Σ_{κ,θ}.

We shall prove the resolvent estimate for p = ∞. Since L_{1}(Ω)^{∗} = L_{∞}(Ω) and R(λ, −A)^{∗} = R(\overline{λ}, −A^{∗}), where the adjoint operator A^{∗} is of the same form as A, the estimate then follows for p = 1 by duality and for 1 < p < ∞ by interpolation.

To complete the proof it thus remains to show

(2.7)  ∥u(·, λ)∥_{∞} ≤ \frac{C}{|λ − κ|}∥f∥_{∞},  λ ∈ Σ_{κ,θ},

where u(·, λ) ∈ H_{0}^{1}(Ω) is the solution of equation (1.2) with f ∈ L_{∞}(Ω). Let g = g(·, y, λ) ∈ H_{0}^{1}(Ω) be the Green’s function of the adjoint operator A^{∗} + \overline{λ}I with singularity at the point y ∈ Ω, i.e.,

(2.8)  A(χ, g) + λ(χ, g) = χ(y),  ∀χ ∈ H_{0}^{1}(Ω).

Since by (1.2) and (2.8) we have u(y, λ) = ⟨f, \overline{g(·, y, λ)}⟩ = (f, g(·, y, λ)), the estimate (2.7) follows if we show

(2.9)  ∥g(·, y, λ)∥_{1} ≤ \frac{C}{|λ − κ|},  y ∈ Ω, λ ∈ Σ_{κ,θ}.

Our strategy for proving (2.9) is to introduce the function

(2.10)  v(x) = v(x, y, λ) = g(x, y, λ)e^{γξ|x−y|},

where ξ is related to λ as in (2.1) and γ is a positive number to be chosen below. We shall prove

(2.11)  ∥v(·, y, λ)∥_{∞} ≤ Cξ^{−1},  y ∈ Ω, λ ∈ Σ_{κ,θ},

which implies

|g(x, y, λ)| ≤ Cξ^{−1}e^{−γξ|x−y|}, x, y ∈ Ω, λ ∈ Σ_{κ,θ},

and (2.9) follows in view of (2.3). The Ansatz (2.10) is motivated by the observation that, in the special case when Au = −u′′ we have

0 ≤ g(x, y, ξ^{2}) ≤ \frac{1}{2}ξ^{−1}e^{−ξ|x−y|}, ξ > 0,

by the maximum principle, the bound being a fundamental solution. We also remark that g(x, y, λ) may be calculated explicitly in this case to give a direct proof of (2.9), see [5, Section 8.1].

To complete the proof it thus remains to prove (2.11). From (2.10) we have g(x) = v(x)e^{−γξ|x−y|}, so that

(2.12)  g′(x) = v′(x)e^{−γξ|x−y|} − γξv(x)e^{−γξ|x−y|}s(x − y),

where s(x) = ±1 according to the sign of x. Choosing χ(x) = v(x)e^{γξ|x−y|} for the test function in (2.8), we obtain by straightforward calculations that

\begin{aligned}A(v, v) + λ∥v∥^{2} & = v(y) + 2iγξ Im(as(· − y)v′, v) \\ & + γ^{2}ξ^{2}(av, v) − γξ(bs(· − y)v, v). \end{aligned} (2.13)

In order to apply Lemma 2.1 we need to bound the right-hand side. First we use the inequality

(2.14)  ∥v∥_{∞} ≤ \sqrt{∥v′∥ ∥v∥},  v ∈ H_{0}^{1}(Ω),

to get

|v(y)| ≤ \sqrt{∥v′∥ ∥v∥} ≤ \frac{1}{2}γ^{−1}ξ^{−1} + \frac{1}{4}∥v′∥^{2} + \frac{1}{4}γ^{2}ξ^{2}∥v∥^{2}.

Next we have

2γξ| Im(as(· − y)v′, v)| ≤ 2γξ∥a∥_{∞}∥v′∥ ∥v∥ ≤ \frac{1}{4}∥v′∥^{2} + Cγ^{2}ξ^{2}∥v∥^{2}.

Similarly,

γ^{2}ξ^{2}(av, v) ≤ γ^{2}ξ^{2}∥a∥_{∞}∥v∥^{2} = Cγ^{2}ξ^{2}∥v∥^{2},

and, using Poincaré’s inequality ∥v∥ ≤ ∥v′∥,

γξ|(bs(· − y)v, v)| ≤ γξ∥b∥_{∞}∥v′∥ ∥v∥ ≤ \frac{1}{4}∥v′∥^{2} + Cγ^{2}ξ^{2}∥v∥^{2}.

Thus the right-hand side of (2.13) can be bounded by

(2.15)  \frac{1}{2}γ^{−1}ξ^{−1} + \frac{3}{4}∥v′∥^{2} + Cγ^{2}ξ^{2}∥v∥^{2},

and Lemma 2.1 yields

\frac{1}{4}∥v′∥^{2} + (1 − Cγ^{2})ξ^{2}∥v∥^{2} ≤ \frac{1}{2}γ^{−1}ξ^{−1}, λ ∈ Σ_{κ,θ}.

By taking γ sufficiently small we arrive at

(2.16)  ∥v∥ ≤ Cξ^{−3/2}, ∥v′∥ ≤ Cξ^{−1/2},  λ ∈ Σ_{κ,θ},

which in view of (2.14) implies the desired bound (2.11). □

Remark. Since ∥g(·, y, λ)∥_{1} = ∥v(·, y, λ)e^{−γξ|·−y|}∥_{1} ≤ Cξ^{−1/2}∥v(·, y, λ)∥, we note that (2.9) actually follows directly from (2.16) without passing through (2.11).

We now turn to the corresponding result for the discrete problem (1.3).

Theorem 2.3. For any θ ∈ (\frac{1}{2}π, π) there are κ ≥ 0 and C ≥ 1 such that

∥R(λ, −A_{h})P_{h}f∥_{p} ≤ \frac{C}{|λ − κ|}∥f∥_{p}, f ∈ L_{p}(Ω), λ ∈ Σ_{κ,θ}, 1 ≤ p ≤ ∞.

Proof. Let θ ∈ (\frac{1}{2}π, π) be arbitrary and let κ be as in Lemma 2.1. Referring to the discussion at the beginning of the proof of Theorem 2.2 we note that it suffices to show the resolvent estimate for p = ∞.

Step 1. We first dispose of the case when λ is large compared to h^{−2} by noting that

(2.17)  ∥A_{h}χ∥_{∞} ≤ C_{1}h^{−2}∥χ∥_{∞},  χ ∈ S_{h}.

In fact, by inverse inequalities and the stability of P_{h} in L_{1}(Ω) (see [4]), we have

\begin{aligned}(A_{h}χ, ϕ) & = (A_{h}χ, P_{h}ϕ) = A(χ, P_{h}ϕ) ≤ C∥χ′∥_{∞}∥(P_{h}ϕ)′∥_{1} \\ & ≤ Ch^{−2}∥χ∥_{∞}∥P_{h}ϕ∥_{1} ≤ C_{1}h^{−2}∥χ∥_{∞}∥ϕ∥_{1}, \end{aligned}

for χ ∈ S_{h}, ϕ ∈ L_{1}(Ω), which shows (2.17). Thus if

(2.18)  |λ| ≥ C_{2}h^{−2},  where C_{2} = max(2C_{1}, κ),

then ∥λ^{−1}A_{h}χ∥_{∞} ≤ \frac{1}{2}∥χ∥_{∞}, so that by the stability of P_{h} in L_{∞}(Ω) (see [4]) (2.19)

∥(λI + A_{h})^{−1}P_{h}f∥_{∞} = ∥λ^{−1} ∑_{j=0}^{∞}(−λ^{−1}A_{h})^{j}P_{h}f∥_{∞} ≤ \frac{2}{|λ|}∥P_{h}f∥_{∞} ≤ \frac{C}{|λ|}∥f∥_{∞}.

Since (2.18) implies (note that h ≤ 1)

(2.20)  κ ≤ C_{2} ≤ C_{2}h^{−2},

so that |λ − κ| ≤ 2|λ|, we may conclude that the desired estimate holds for |λ| ≥ C_{2}h^{−2}.

Step 2. In this step of the proof we shall bound the solution of (1.3) at the nodal points and show

(2.21)  max_{1≤j≤N} |u_{h}(x_{j}, λ)| ≤ \frac{C}{|λ − κ|}∥f∥_{∞},  λ ∈ Σ_{κ,θ}, |λ| ≤ C_{2}h^{−2}.

For future reference we note that, by (2.3) and (2.20), the assumption |λ| ≤ C_{2}h^{−2} implies that ξ^{2} ≤ |λ − κ| ≤ |λ| + κ ≤ 2C_{2}h^{−2}, that is, ξh ≤ C.

Following the proof of Theorem 2.2 we let x_{j} be an arbitrary nodal point and introduce the discrete Green’s function g_{h} = g_{h}(·, x_{j}, λ) ∈ S_{h} defined by

A(χ_{h}, g_{h}) + λ(χ_{h}, g_{h}) = χ_{h}(x_{j}), ∀χ_{h} ∈ S_{h}.

With v(x) = v(x, x_{j}, λ) = g_{h}(x, x_{j}, λ)e^{γξ|x−x_{j}|}, we shall show for some γ ∈ (0, 1) that

(2.22)  ∥v(·, x_{j}, λ)∥_{∞} ≤ Cξ^{−1},  j = 1, …, N, λ ∈ Σ_{κ,θ}, |λ| ≤ C_{2}h^{−2},

which leads to (2.21) in the same way as in the proof of Theorem 2.2.

With χ(x) = v(x)e^{γξ|x−x_{j}|} the argument leading to (2.13) now yields

\begin{aligned}A(v, v) + λ∥v∥^{2} & = A(χ − χ_{h}, g_{h}) + λ(χ − χ_{h}, g_{h}) \\ & + χ_{h}(x_{j}) + 2iγξ Im(as(· − x_{j})v′, v) \\ & + γ^{2}ξ^{2}(av, v) − γξ(bs(· − x_{j})v, v), \end{aligned} (2.23)

where χ_{h} ∈ S_{h} is arbitrary. For χ_{h} we choose an “elliptic projection” of χ defined by

(χ′_{h}, ϕ′) = (χ′, ϕ′), ∀ϕ ∈ S_{h}.

It is well-known that χ_{h}(x_{j}) = χ(x_{j}), so that χ_{h}(x_{j}) = v(x_{j}). The last four terms are thus the same as in (2.13) and can be bounded as in (2.15). For the two remaining terms we have, by a well-known error estimate and an inverse inequality, with ∥u∥_{Ω_{i}}^{2} = ∫_{Ω_{i}} |u|^{2} dx,

|A(χ − χ_{h}, g_{h}) + λ(χ − χ_{h}, g_{h})|

≤ C ∑_{i=0}^{N}(∥χ′ − χ′_{h}∥_{Ω_{i}} ∥g′_{h}∥_{Ω_{i}} + |λ| ∥χ − χ_{h}∥_{Ω_{i}} ∥g_{h}∥_{Ω_{i}})

≤ C ∑_{i=0}^{N}(h_{i}^{r−1}∥χ^{(r)}∥_{Ω_{i}} ∥g′_{h}∥_{Ω_{i}} + |λ|h_{i}^{r}∥χ^{(r)}∥_{Ω_{i}} ∥g_{h}∥_{Ω_{i}})

≤ C ∑_{i=0}^{N} h_{i}^{r−2}∥χ^{(r)}∥_{Ω_{i}} ∥g_{h}∥_{Ω_{i}},

where we have also used the assumption that |λ| ≤ C_{2}h^{−2}. Recalling that χ(x) = g_{h}(x)e^{2γξ|x−x_{j}|}, where g_{h}^{(r)}(x) ≡ 0 on Ω_{i}, and that ξh ≤ C and γ ∈ (0, 1), we have

h_{i}^{r−2}∥χ^{(r)}∥_{Ω_{i}} ≤ Ce^{Cγ}h_{i}^{r−2} ∑_{l=0}^{r−1}(γξ)^{r−l}∥g_{h}^{(l)}∥_{Ω_{i}}e^{2γξ|x_{i}−x_{j}|}

≤ Ch_{i}^{r−2}((γξ)^{r}∥g_{h}∥_{Ω_{i}} + ∑_{l=1}^{r−1}(γξ)^{r−l}h_{i}^{1−l}∥g′_{h}∥_{Ω_{i}})e^{2γξ|x_{i}−x_{j}|}

≤ C(γξ∥g_{h}∥_{Ω_{i}} + ∥g′_{h}∥_{Ω_{i}})γξe^{2γξ|x_{i}−x_{j}|}.

Hence,

h_{i}^{r−2}∥χ^{(r)}∥_{Ω_{i}} ∥g_{h}∥_{Ω_{i}} ≤ C(∥g′_{h}∥_{Ω_{i}} + γξ∥g_{h}∥_{Ω_{i}})γξ∥g_{h}∥_{Ω_{i}}e^{2γξ|x_{i}−x_{j}|},

and since, cf. (2.12),

∥g′_{h}∥_{Ω_{i}} + γξ∥g_{h}∥_{Ω_{i}} ≤ C(∥v′∥_{Ω_{i}} + γξ∥v∥_{Ω_{i}})e^{−γξ|x_{i}−x_{j}|},

we may finally conclude that

|A(χ − χ_{h}, g_{h}) + λ(χ − χ_{h}, g_{h})| ≤ \frac{1}{8}∥v′∥^{2} + Cγ^{2}ξ^{2}∥v∥^{2}.

Thus the right-hand side of (2.23) may be bounded as in (2.15) and the bound (2.21) follows for γ sufficiently small by application of Lemma 2.1 as in the proof of Theorem 2.2.

Step 3. Having established the estimate (2.21) for u_{h} at the nodal points, we now want to show the same bound in the interiors of the subintervals. (Of course this is trivial if r = 2.) Assume that the maximum of |u_{h}| is attained in the subinterval Ω_{l}. We introduce the subspace

S_{h}(Ω_{l}) = {χ ∈ S_{h} : supp χ ⊂ Ω̄_{l}},

and define v_{h} ∈ S_{h}(Ω_{l}) by

(2.24)  A(v_{h} − u_{h}, χ) + κ(v_{h} − u_{h}, χ) = 0,  ∀χ ∈ S_{h}(Ω_{l}).

We shall first show for ũ_{h} = u_{h} − v_{h} ∈ S_{h} that with ∥u∥_{∞,Ω_{l}} = sup_{Ω_{l}} |u|,

(2.25)  ∥ũ_{h}∥_{∞,Ω_{l}} ≤ C max(|u_{h}(x_{l})|, |u_{h}(x_{l+1})|),

which in view of (2.21) implies

(2.26)  ∥ũ_{h}∥_{∞,Ω_{l}} ≤ \frac{C}{|λ − κ|}∥f∥_{∞},  λ ∈ Σ_{κ,θ}, |λ| ≤ C_{2}h^{−2}.

We shall then show

(2.27)  ∥v_{h}∥_{∞} ≤ \frac{C}{|λ − κ|}∥f∥_{∞},  λ ∈ Σ_{κ,θ}, |λ| ≤ C_{2}h^{−2},

which completes the proof, since u_{h} = ũ_{h} + v_{h}.

To show (2.25) we note that, with û_{h} equal to the linear interpolant of u_{h} on Ω_{l} and û_{h} = u_{h} elsewhere, equation (2.24) implies

A(û_{h} − ũ_{h}, χ) + κ(û_{h} − ũ_{h}, χ) = A(û_{h}, χ) + κ(û_{h}, χ), ∀χ ∈ S_{h}(Ω_{l}).

Since w_{h} = û_{h} − ũ_{h} ∈ S_{h}(Ω_{l}), this implies

A(w_{h}, w_{h}) + κ∥w_{h}∥^{2} = A(û_{h}, w_{h}) + κ(û_{h}, w_{h})

and an application of Lemma 2.1 (with λ = κ and with ∥u∥_{Ω_{l}}^{2} = ∫_{Ω_{l}} |u|^{2} dx) shows

∥w′_{h}∥^{2} ≤ C|A(û_{h}, w_{h}) + κ(û_{h}, w_{h})| ≤ C∥û′_{h}∥_{Ω_{l}}^{2} + C∥û_{h}∥_{Ω_{l}}^{2} + \frac{1}{2}∥w′_{h}∥^{2}.

Since ∥w_{h}∥_{∞} ≤ h^{1/2}∥w′_{h}∥, h^{−1/2}∥û_{h}∥_{Ω_{l}} +h^{1/2}∥û′_{h}∥_{Ω_{l}} ≤ C max(|u_{h}(x_{l})|, |u_{h}(x_{l+1})|), we may conclude that

∥w_{h}∥_{∞} ≤ C max(|u_{h}(x_{l})|, |u_{h}(x_{l+1})|),

and (2.25) follows, since ũ_{h} = û_{h} − w_{h} and ∥û_{h}∥_{∞,Ω_{l}} ≤ max(|u_{h}(x_{l})|, |u_{h}(x_{l+1})|). It remains to estimate ∥v_{h}∥_{∞}. From (2.24) and (1.3) we have

A(v_{h}, χ) + λ(v_{h}, χ) = (f, χ) − (λ − κ) (ũ_{h}, χ), ∀χ ∈ S_{h}(Ω_{l}).

Taking χ = v_{h} we obtain

A(v_{h}, v_{h}) + λ∥v_{h}∥^{2} = (f, v_{h}) − (λ − κ)(ũ_{h}, v_{h}).

An application of Lemma 2.1 yields, in view of (2.3) and (2.26),

\begin{aligned}∥v′_{h}∥^{2} + ξ^{2}∥v_{h}∥^{2} & ≤ C|(f, v_{h}) − (λ − κ)(ũ_{h}, v_{h})| \\ & ≤ C(∥f∥_{∞} + |λ − κ|∥ũ_{h}∥_{∞,Ω_{l}})∥v_{h}∥_{1} \\ & ≤ Ch^{1/2}∥f∥_{∞}∥v_{h}∥. \end{aligned} (2.28)

Hence

∥v_{h}∥ ≤ Ch^{1/2}ξ^{−2}∥f∥_{∞}.

Since ∥v_{h}∥_{∞} ≤ Ch^{−1/2}∥v_{h}∥ by an inverse inequality, we conclude that (2.27) holds. □

We may now prove our main result.

Proof of Theorem 1.1. We only consider R(λ, −A). The proof for R(λ, −A_{h}) is identical. Fix θ ∈ (\frac{1}{2}π, π). By Theorem 2.2 there are κ ≥ 0 and C ≥ 1 such that

(2.29)  ∥R(λ, −A)f∥_{p} ≤ \frac{C}{|λ − κ|}∥f∥_{p},  λ ∈ Σ_{κ,θ}.

The assumption (1.1) and Lemma 2.1 imply that B_{λ}(u, v) = A(u, v) + λ(u, v) is coercive uniformly for λ ∈ Σ_{−a_{0}/2,π/2} ∪ Σ_{κ,θ}, so that, by Lax-Milgram’s lemma, equation (1.2) has a unique solution and

∥R(λ, −A)f∥_{H_{0}^{1}} ≤ C∥f∥_{H^{−1}}, f ∈ H^{−1}(Ω), λ ∈ Σ_{−a_{0}/2,π/2} ∪ Σ_{κ,θ}.

Hence ∥R(λ, −A)f∥_{p} ≤ C∥f∥_{p}, and using also (2.29),

∥R(λ, −A)f∥_{p} ≤ \frac{C}{1 + |λ|}∥f∥_{p}, λ ∈ Σ_{−a_{0}/2,π/2} ∪ Σ_{κ,θ}.

We then choose φ ∈ (\frac{1}{2}π, θ], such that Σ_{0,φ} ⊂ Σ_{−a_{0}/2,π/2} ∪ Σ_{κ,θ}. Together these estimates show the required bounds for R(λ, −A) for λ ∈ Σ_{φ} = Σ_{0,φ}. □

We shall also need an estimate of the norms of the resolvents considered as operators from M(Ω) to L_{∞}(Ω), where M(Ω) denotes the set of finite measures μ with ∥μ∥_{M} = |μ|(Ω) = ∫_{Ω} d|μ|. With each μ ∈ M(Ω) we associate a distribution (also denoted μ) in the usual way: ⟨μ, χ⟩ = ∫_{Ω} χ(x) dμ(x) for χ ∈ C(Ω̄). Recall that P_{h} : M(Ω) → S_{h} is defined in (1.4).

Theorem 2.4. There is φ ∈ (\frac{1}{2}π, π) and C ≥ 1 such that

∥R(λ, −A)f∥_{∞} + ∥R(λ, −A_{h})P_{h}f∥_{∞} ≤ \frac{C}{\sqrt{1 + |λ|}}∥f∥_{M}, f ∈ M(Ω), λ ∈ Σ_{φ}.

Proof. Let θ ∈ (\frac{1}{2}π, π) be arbitrary and let κ be as in Lemma 2.1. Taking χ = u in (1.2) and applying the lemma shows

∥u′∥^{2} + ξ^{2}∥u∥^{2} ≤ C|⟨f, \overline{u}⟩| ≤ C∥f∥_{M} ∥u∥_{∞}, λ ∈ Σ_{κ,θ},

which in view of (2.14) and the arguments in the proof of Theorem 1.1 implies the desired result for R(λ, −A). The same argument applies to R(λ, −A_{h}). □

3. Error estimates

In view of the assumption (1.1) and the Lax-Milgram lemma we may define a bounded linear operator R_{h} : H_{0}^{1}(Ω) → S_{h} by the equation

(3.1)  A(R_{h}v − v, χ) = 0,  ∀χ ∈ S_{h}.

For this “Ritz projection” we have the following error estimates.

Lemma 3.1. There is a constant C such that for k = 1, …, r, 1 ≤ p ≤ ∞, we have

∥R_{h}v − v∥_{p} + h∥(R_{h}v − v)′∥_{p} ≤ Ch^{k}∥v∥_{k,p}, v ∈ H_{0}^{1}(Ω) ∩ W_{p}^{k}(Ω).

Proof. The case p = ∞ can be found in [14] or [7]. We prove the case p = 1 by a slight modification of the argument of [7]. The remaining case 1 < p < ∞ then follows by interpolation.

We shall show below that

(3.2)  ∥(R_{h}v − v)′∥_{1} ≤ Ch^{k−1}∥v∥_{k,1},  k = 1, …, r.

The desired bound for ∥R_{h}v − v∥_{1} then follows by the standard duality argument. In fact, let ϕ ∈ L_{∞}(Ω) be arbitrary and define ψ ∈ H_{0}^{1}(Ω) by

A(χ, ψ) = (χ, ϕ), ∀χ ∈ H_{0}^{1}(Ω).

Then

(R_{h}v − v, ϕ) = A(R_{h}v − v, ψ) = A(R_{h}v − v, ψ − R_{h}ψ)
≤ C∥(R_{h}v − v)′∥_{1}∥(R_{h}ψ − ψ)′∥_{∞}
≤ Ch^{k−1}∥v∥_{k,1}Ch∥ψ∥_{2,∞},

which implies ∥R_{h}v − v∥_{1} ≤ Ch^{k}∥v∥_{k,1}, since clearly ∥ψ∥_{2,∞} ≤ C∥ϕ∥_{∞}.

In order to show (3.2) we introduce the projection R̃_{h} : H_{0}^{1}(Ω) → S_{h} defined by

((R̃_{h}v − v)′, χ′) = 0, ∀χ ∈ S_{h}.

It follows that (R̃_{h}v)′ = P̃_{h}v′, where P̃_{h} denotes the L_{2} projection onto V_{h} = {χ ∈ L_{2}(Ω) : χ|_{Ω_{i}} ∈ Π_{r−2}}, see [7, Section 4]. Hence

(3.3)  ∥(R̃_{h}v − v)′∥_{1} ≤ Ch^{k−1}∥v∥_{k,1},  k = 1, …, r,

since P̃_{h} is stable in L_{1}(Ω) (this is trivial because P̃_{h} is a local projection).

It remains to estimate θ = R_{h}v − R̃_{h}v. We introduce the piecewise constant function ã defined by ã(x) = a(x_{k+1/2}) for x ∈ Ω_{k}, where x_{k+1/2} denotes the midpoint of Ω_{k}, and we note that

(ã(v − R̃_{h}v)′, χ′)_{Ω_{k}} = 0, ∀χ ∈ S_{h}.

Since θ ∈ S_{h}, we thus have

a_{0}∥θ′∥_{2}^{2} ≤ A(θ, θ) = A(R_{h}v − R̃_{h}v, θ) = A(v − R̃_{h}v, θ)
= (a(v − R̃_{h}v)′, θ′) + (b(v − R̃_{h}v)′, θ) + (c(v − R̃_{h}v), θ)
= ((a − ã)(v − R̃_{h}v)′, θ′) + (b(v − R̃_{h}v)′, θ) + (c(v − R̃_{h}v), θ).

Hence

∥θ′∥_{2}^{2} ≤ Ch∥(v − R̃_{h}v)′∥_{1}∥θ′∥_{∞} + C∥(v − R̃_{h}v)′∥_{1}∥θ∥_{∞} ≤ C∥(v − R̃_{h}v)′∥_{1}∥θ′∥_{2}.

Therefore

∥θ′∥_{1} ≤ ∥θ′∥_{2} ≤ C∥(v − R̃_{h}v)′∥_{1},

which together with (3.3) implies (3.2). □

We now turn to the proof of an error estimate in the case of smooth solutions of the nonhomogeneous problem.

Theorem 3.2. Let 1 ≤ p ≤ ∞, let u be a sufficiently smooth solution of (1.5) with u_{0} ∈ H_{0}^{1}(Ω) ∩ W_{p}^{r}(Ω) and let u_{h} be the corresponding solution of (1.6). Then

∥u_{h}(t) − u(t)∥_{p} ≤ C∥u_{0h} − u_{0}∥_{p} + Ch^{r}(∥u_{0}∥_{r,p} + ∫_{0}^{t} ∥u_{t}∥_{r,p} ds).

Proof. Following a standard practice we divide the error into two parts:

(3.4) e(t) ≡ u_{h}(t) − u(t) = (u_{h}(t) − R_{h}u(t)) + (R_{h}u(t) − u(t)) ≡ θ(t) + ρ(t).

In view of Lemma 3.1 we have

(3.5)  ∥D_{t}^{l}ρ(t)∥_{p} ≤ Ch^{k}∥D_{t}^{l}u(t)∥_{k,p},  l = 0, 1, k = 1, …, r,

where D_{t} = ∂/∂t. Since this estimates ∥ρ(t)∥_{p} in the appropriate way, it only remains to estimate θ(t), which belongs to S_{h}. In view of (1.6), the identity A_{h}R_{h} = P_{h}A and (1.5), we find that θ_{t} + A_{h}θ = −P_{h}ρ_{t}, and hence

(3.6)  θ(t) = E_{h}(t)θ(0) − ∫_{0}^{t} E_{h}(t − s)P_{h}ρ_{s}(s) ds.

The desired error bound now follows immediately by application of (3.5) and the stability estimate for E_{h}(t) in (1.11). □

The proof of Theorem 3.2 is based on the stability in L_{p}(Ω) of the discrete evolution operator E_{h}(t) together with the “truncation error estimate” of Lemma 3.1. Since L_{p}(Ω) stability is also available for completely discrete evolution operators E_{kh}^{n} = r(−kA_{h})^{n} based on A-acceptable rational functions, see (1.17), it is possible to obtain analogous error estimates for certain completely discrete schemes. We carry this out for the backward Euler and Crank-Nicolson methods in the following theorem.

Theorem 3.3. Let 1 ≤ p ≤ ∞, let u be a sufficiently smooth solution of (1.5) with u_{0} ∈ H_{0}^{1}(Ω) ∩ W_{p}^{r}(Ω) and let U_{n} be the corresponding approximation obtained by the backward Euler method (1.18). Then

∥U_{n} −u(t_{n})∥_{p} ≤ C∥u_{0h} −u_{0}∥_{p} +Ch^{r}(∥u_{0}∥_{r,p} +∫_{0}^{t_{n}} ∥u_{t}∥_{r,p} ds)+Ck ∫_{0}^{t_{n}} ∥u_{tt}∥_{p} ds.

For the Crank-Nicolson method (1.19) we have

\begin{aligned}∥U_{n} − u(t_{n})∥_{p} & ≤ C∥u_{0h} − u_{0}∥_{p} + Ch^{r}(∥u_{0}∥_{r,p} + ∫_{0}^{t_{n}} ∥u_{t}∥_{r,p} ds) \\ & + Ck^{2} ∫_{0}^{t_{n}}(∥u_{ttt}∥_{p} + ∥u_{tt}∥_{2,p}) ds. \end{aligned}

Proof. Using a splitting of the error analogous to (3.4), we have for the backward Euler method instead of (3.6)

θ_{n} = E_{kh}^{n}θ_{0} − k ∑_{j=1}^{n} E_{kh}^{n−j+1}P_{h}(¯∂_{t}ρ_{j} + (¯∂_{t}u(t_{j}) − u_{t}(t_{j}))),

where E_{kh} = (I + kA_{h})^{−1}, from which the proof proceeds using the stability property (1.17) with r(λ) = 1/(1 − λ) and standard estimates. The argument for the Crank-Nicolson argument is analogous. □

General time discretization schemes require more complicated approximations of the source term f, see [2]. Avoiding this difficulty, we shall content ourselves in the rest of this section with studying the homogeneous equation (f ≡ 0). We will then need smoothing properties, which are slightly more general than (1.10). In particular we need to deal with solutions which already possess some initial smoothness and compatibility. In order to express this we define for nonnegative integers m and for all p ∈ [1, ∞]

W̊_{p}^{m}(Ω) = {v ∈ W_{p}^{m}(Ω) : A^{j}v = 0 at x = 0, 1 for all integers j ∈ [0, m/2)}.

Lemma 3.4. Let m and k be nonnegative integers. Then

(3.7) ∥E(t)v∥_{m,p} + t∥E′(t)v∥_{m,p} ≤ C_{m}t^{−m/2}∥v∥_{p},  t > 0, v ∈ L_{p}(Ω),
(3.8) ∥E(t)v∥_{∞} + ∥E_{h}(t)P_{h}v∥_{∞} ≤ Ct^{−1/2}∥v∥_{M},  t > 0, v ∈ M(Ω).
(3.9)  ∥A^{k}E(t)v∥_{m,p} ≤ C_{k,m}t^{−k}∥v∥_{m,p},  t > 0, v ∈ W̊_{p}^{m}(Ω).

Proof. The regularity estimate (3.7) follows from the analyticity of E(t), see (1.10), together with (1.7) and the interpolation inequality

∥u′∥_{p} ≤ C\sqrt{∥u∥_{p}∥u′′∥_{p}}, 1 ≤ p ≤ ∞, u ∈ W̊_{p}^{2}(Ω),

which can be proved by elementary arguments. In fact, let m = 2l + i with l ≥ 0 and i = 0 or 1. Then

\begin{aligned}∥E(t)v∥_{m,p} & ≤ C_{m}∥A^{l}E(t)v∥_{i,p} ≤ C_{m}\sqrt{∥A^{l}E(t)v∥_{0,p}∥A^{l}E(t)v∥_{2i,p}} \\ & ≤ C_{m}\sqrt{∥A^{l}E(t)v∥_{0,p}∥A^{l+i}E(t)v∥_{0,p}} ≤ C_{m}t^{−m/2}∥v∥_{p}, \end{aligned}

and similarly for E′(t)v. The bounds in (3.8) follow from the resolvent estimates of Theorem 2.4 and the contour integral representation of the semigroups in the same way as (1.10), (1.11).

We now turn to the proof of (3.9) and first note that it suffices to consider the cases k = 0 and 1, since the case k ≥ 2 follows from these and the identity A^{k}E(t)v = (AE(t/k))^{k}v. We also note that the case m = 0 follows from (1.10). Assume now that that (3.9) has been proved also for m = 1. Then for m ≥ 2 we may write m = 2l + i with l ≥ 0, i = 0 or 1, and obtain for k = 0, 1

\begin{aligned}∥A^{k}E(t)v∥_{m,p} & ≤ C∥A^{k+l}E(t)v∥_{i,p} = C∥A^{k}E(t)A^{l}v∥_{i,p} \\ & ≤ Ct^{−k}∥A^{l}v∥_{i,p} ≤ C_{m}t^{−k}∥v∥_{m,p}, v ∈ W̊_{p}^{m}(Ω). \end{aligned}

It remains to consider m = 1. For k = 1 we have

∥AE(t)v∥_{1,p} ≤ C\sqrt{∥A^{2}E(t)v∥_{p}∥AE(t)v∥_{p}} ≤ Ct^{i−3/2}∥v∥_{2i,p}, v ∈ W̊_{p}^{2i}(Ω), i = 0, 1,

from which the desired result follows by interpolation. In fact, for each v ∈ W̊_{p}^{1}(Ω) and ϵ > 0 there is v_{ϵ} ∈ W̊_{p}^{2}(Ω) such that ∥v − v_{ϵ}∥_{p} ≤ Cϵ∥v∥_{1,p} and ∥v_{ϵ}∥_{2,p} ≤ Cϵ^{−1}∥v∥_{1,p}. For ϵ ≤ 1 this is achieved in a standard way by extension of v to an odd 2-periodic function and by convolution with an even mollifier on the scale ϵ. If ϵ > 1, then we take v_{ϵ} = 0. Hence with ϵ = t^{1/2} we obtain

\begin{aligned}∥AE(t)v∥_{1,p} & ≤ ∥AE(t)(v − v_{ϵ})∥_{1,p} + ∥AE(t)v_{ϵ}∥_{1,p} \\ & ≤ Ct^{−3/2}∥v − v_{ϵ}∥_{p} + Ct^{−1/2}∥v_{ϵ}∥_{2,p} ≤ Ct^{−1}∥v∥_{1,p}, v ∈ W̊_{p}^{1}(Ω). \end{aligned}

The final case m = 1, k = 0 follows from the Cauchy integral representation (1.8) once we have shown the resolvent estimate

(3.10)  ∥R(λ, −A)f∥_{1,p} ≤ \frac{C}{1 + |λ|}∥f∥_{1,p},  f ∈ W̊_{p}^{1}(Ω).

To prove this we set u = R(λ, −A)f and v = aBu′, where B(x) = exp(− ∫_{0}^{x} \frac{b(t)}{a(t)}dt). Then it is easy to show that v ∈ W_{p}^{2}(Ω) and v′ = B(λu + cu − f) so that

−(av′)′ + cv + λv = g, x ∈ Ω; v′(0) = v′(1) = 0,

where g = (aBf)′ − (aB)′λu − (caB)′u satisfies ∥g∥_{p} ≤ C∥f∥_{1,p}. An argument similar to the proof of Theorem 1.1 now gives ∥v∥_{p} ≤ \frac{C}{1+|λ|}∥g∥_{p}, which implies (3.10). The proof of Lemma 3.4 is now complete. □

Remark. The interpolation argument that we used in the proof of the special case m = 1, k = 1 of (3.9) may also be expressed by saying that W̊_{p}^{1}(Ω) is continuously imbedded in the space (L_{p}(Ω), W̊_{p}^{2}(Ω))_{1/2,∞} defined by the real interpolation method. The estimate (3.9) follows more directly from (1.10) when 1 < p < ∞, because W̊_{p}^{1}(Ω) = (L_{p}(Ω), W̊_{p}^{2}(Ω))_{1/2,∞} then. This is not true when p = 1, ∞ (cf. [9]), which is the reason for our indirect proof of the special case m = 1, k = 0 of (3.9).

We now consider the homogeneous equation with nonsmooth initial data.

Theorem 3.5. Assume that f(x, t) ≡ 0. Let u be the solution of (1.5) and u_{h} the solution of (1.6) with u_{0h} = P_{h}u_{0}. Then

∥u_{h}(t) − u(t)∥_{p} ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p},  t > 0, u_{0} ∈ L_{p}(Ω), 1 ≤ p ≤ ∞,
∥u_{h}(t) − u(t)∥_{∞} ≤ Ch^{r}t^{−(r+1)/2}∥u_{0}∥_{M},  t > 0, u_{0} ∈ M(Ω).

Proof. Since u(t) = E(t)u_{0} and u_{h}(t) = E_{h}(t)P_{h}u_{0}, we must estimate the norm of the error operator F_{h}(t) = E_{h}(t)P_{h} − E(t). We first consider (3.6) again and divide the interval of integration into (0, t/2) and (t/2, t). Integrating by parts in the first integral noting that P_{h}e(0) = 0 we get

θ(t) = −E_{h}(t/2)P_{h}ρ(t/2) − ∫_{0}^{t/2} E′_{h}(t − s)P_{h}ρ(s) ds − ∫_{t/2}^{t} E_{h}(t − s)P_{h}ρ_{s}(s) ds.

Using the stability estimates in (1.11) and the error estimates (3.5) we obtain

\begin{aligned}∥θ(t)∥_{p} & ≤ C∥ρ(t/2)∥_{p} + C ∫_{0}^{t/2}(t − s)^{−1}∥ρ(s)∥_{p} ds + C ∫_{t/2}^{t} ∥ρ_{s}(s)∥_{p} ds \\ & ≤ Ch^{k}(∥u(t/2)∥_{k,p} + t^{−1} ∫_{0}^{t/2} ∥u(s)∥_{k,p} ds + ∫_{t/2}^{t} ∥u_{s}(s)∥_{k,p} ds), \end{aligned}

for k = 1, …, r. In particular, it follows that

(3.11)  ∥F_{h}(t)u_{0}∥_{p} ≤ Ch^{r} sup_{0≤s≤t}(∥E(s)u_{0}∥_{r,p} + s∥E′(s)u_{0}∥_{r,p}).

Further, by (3.7) with m = 1 we obtain a preliminary estimate of low order:

(3.12) \begin{aligned}∥F_{h}(t)u_{0}∥_{p} & ≤ Cht^{−1/2} sup_{0<s<t}(s^{1/2}∥E(s)u_{0}∥_{1,p} + s^{3/2}∥E′(s)u_{0}∥_{1,p}) \\ & ≤ Cht^{−1/2}∥u_{0}∥_{p}. \end{aligned}

Writing

(3.13)  F_{h}(t)u_{0} = F_{h}(t/2)E(t/2)u_{0} + E(t/2)F_{h}(t/2)u_{0} + F_{h}(t/2)^{2}u_{0},

we have, by (3.11), (3.7) with m = r,

(3.14) \begin{aligned}∥F_{h}(t/2)E(t/2)u_{0}∥_{p} & ≤ Ch^{r} sup_{t/2≤s≤t}(∥E(s)u_{0}∥_{r,p} + s∥E′(s)u_{0}∥_{r,p}) \\ & ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p}, \end{aligned}

and in view of (3.8), using t/4 as an intermediate level,

(3.15)  ∥F_{h}(t/2)E(t/2)u_{0}∥_{∞} ≤ Ch^{r}t^{−(r+1)/2}∥u_{0}∥_{M}.

We further note that E(t/2)F_{h}(t/2) is the adjoint of the operator F_{h}(t/2)^{∗}E(t/2)^{∗}, where E(t)^{∗} = e^{−tA^{∗}} and similarly for F_{h}(t)^{∗}. Since the operator F_{h}(t/2)^{∗}E(t/2)^{∗} is bounded in L_{p}(Ω) for 1 ≤ p ≤ ∞ as in (3.14), its adjoint is similarly bounded in L_{p′}(Ω), p′ being the conjugate exponent to p. Hence

∥E(t/2)F_{h}(t/2)u_{0}∥_{p} ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p}.

For the third term in (3.13) we have by (3.12)

∥F_{h}(t/2)^{2}u_{0}∥_{p} ≤ Cht^{−1/2}∥F_{h}(t/2)u_{0}∥_{p},

and we may conclude that altogether

∥F_{h}(t)u_{0}∥_{p} ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p} + Cht^{−1/2}∥F_{h}(t/2)u_{0}∥_{p},

or, after r iterations,

∥F_{h}(t)u_{0}∥_{p} ≤ Ch^{r}t^{−r/2}(∥u_{0}∥_{p} + ∥F_{h}(t/2^{r})u_{0}∥_{p}).

Since, by stability ∥F_{h}(t/2^{r})u_{0}∥_{p} ≤ C∥u_{0}∥_{p}, we infer that

∥F_{h}(t)u_{0}∥_{p} ≤ Ch^{r}t^{−r/2}∥u_{0}∥_{p},

which proves the first inequality of the theorem.

Finally we consider the case when u_{0} ∈ M(Ω). For the second term on the right in (3.13) we then have by (3.8)

∥E(t/2)F_{h}(t/2)u_{0}∥_{∞} ≤ Ct^{−1/2}∥E(t/4)F_{h}(t/2)u_{0}∥_{M}.

Here E(t/4)F_{h}(t/2) is the adjoint of the operator F_{h}(t/2)^{∗}E(t/4)^{∗}. Since the operator F_{h}(t/2)^{∗}E(t/4)^{∗} : L_{∞}(Ω) → C(Ω̄) is bounded as in (3.14), its adjoint is similarly bounded in M(Ω). In fact,

(E(t/4)F_{h}(t/2)u_{0}, ϕ) = ⟨u_{0}, F_{h}^{∗}(t/2)E^{∗}(t/4)ϕ⟩ ≤ ∥u_{0}∥_{M}∥F_{h}^{∗}(t/2)E^{∗}(t/4)ϕ∥_{∞},

for all u_{0} ∈ M(Ω), ϕ ∈ C(Ω̄). Hence

∥E(t/2)F_{h}(t/2)u_{0}∥_{∞} ≤ Ch^{r}t^{−(r+1)/2}∥u_{0}∥_{M}.

The proof can now be completed in the same way as above, using the inequality

∥F_{h}(t/2^{r})u_{0}∥_{∞} ≤ Ct^{−1/2}∥u_{0}∥_{M},

which follows from (3.8). □

We conclude by proving the error bound (1.22) for a completely discrete approximation of the solution of the homogeneous problem, which we announced in the Introduction.

Theorem 3.6. Let u be the solution of (1.5) with f(x, t) ≡ 0 and let U_{n} = r(−kA_{h})^{n}u_{0h}. Assume that the rational function r is accurate of order q (1.15) and A-acceptable (1.16). Then, for 1 ≤ p ≤ ∞, we have

∥U_{n} − u(t_{n})∥_{p} ≤ C(∥u_{0h} − u_{0}∥_{p} + h^{r}∥u_{0}∥_{r,p} + k^{q}∥u_{0}∥_{2q,p}), t_{n} = nk ≥ 0,

if u_{0} ∈ W̊_{p}^{m}(Ω) with m = max(r, 2q).

Proof. By stability (cf. (1.17)) there is no loss of generality in assuming that u_{0h} = P_{h}u_{0}. Using (3.11) and (3.9) we obtain

∥E_{h}(t_{n})P_{h}u_{0} − E(t_{n})u_{0}∥_{p} ≤ Ch^{r}∥u_{0}∥_{r,p},

and it remains to estimate

G_{n}P_{h}u_{0} ≡ (r(−kA_{h})^{n} − E_{h}(t_{n}))P_{h}u_{0}.

Following [1] we use the identity

\begin{aligned}G_{n}P_{h}u_{0} & = ∑_{j=0}^{q−1} G_{n}A_{h}^{−j}(P_{h}A^{−1} − A_{h}^{−1}P_{h})A^{j+1}ũ_{0k} \\ & + G_{n}A_{h}^{−q}P_{h}A^{q}ũ_{0k} + G_{n}P_{h}(u_{0} − ũ_{0k}), \end{aligned} (3.16)

where ũ_{0k} is to be chosen so that

(3.17)  ∥u_{0} − ũ_{0k}∥_{p} ≤ Ck^{q}∥u_{0}∥_{2q,p},
(3.18)  ∥A^{q}ũ_{0k}∥_{p} ≤ C∥u_{0}∥_{2q,p},
(3.19)  ∥A^{j}ũ_{0k}∥_{r,p} ≤ Ck^{−j}∥u_{0}∥_{r,p},  0 ≤ j ≤ q − 1.

In [1] ũ_{0k} is defined by truncating the Fourier series of u_{0}, but this is not suitable here and we choose instead

ũ_{0k} = f(kA)E(kA)u_{0}, where f(λ) ≡ ∑_{n=0}^{q} (−λ)^{n}/n! = e^{−λ} + O(λ^{q+1}), λ → 0,

so that 1 − f(λ)e^{λ} = O(λ^{q+1}), as λ → 0. Hence, using the same contour Γ as in (1.8) and the resolvent estimate of Theorem 1.1, we have

∥u_{0} − ũ_{0k}∥_{p} = ∥A^{−q}(I − f(kA)E(kA))A^{q}u_{0}∥_{p}

= ∥\frac{1}{2πi} ∫_{Γ} λ^{−q}(1 − f(kλ)e^{kλ})R(λ, −A) dλ A^{q}u_{0}∥_{p}

= k^{q−1}∥\frac{1}{2πi} ∫_{Γ} λ^{−q}(1 − f(λ)e^{λ})R(k^{−1}λ, −A) dλ A^{q}u_{0}∥_{p}

≤ Ck^{q} ∫_{0}^{∞} ρ^{−q−1}|1 − f(ρe^{iφ}) exp(ρe^{iφ})| dρ ∥A^{q}u_{0}∥_{p}

≤ Ck^{q}∥u_{0}∥_{2q,p},

where we also used the fact that A is a second order differential operator. This is (3.17). Moreover, by the analyticity of E(t), see (1.10), we have

\begin{aligned}∥A^{q}ũ_{0k}∥_{p} & = ∥A^{q}f(kA)E(kA)u_{0}∥_{p} ≤ ∑_{n=0}^{q} \frac{1}{n!}∥(kA)^{n}E(kA)A^{q}u_{0}∥_{p} \\ & ≤ C∥A^{q}u_{0}∥_{p} ≤ C∥u_{0}∥_{2q,p}, \end{aligned}

which is (3.18). Using also the smoothing property (3.9) we obtain

\begin{aligned}∥A^{j}ũ_{0k}∥_{r,p} & = k^{−j}∥(kA)^{j}f(kA)E(kA)u_{0}∥_{r,p} ≤ k^{−j} ∑_{n=0}^{q} \frac{1}{n!}∥(kA)^{n+j}E(kA)u_{0}∥_{r,p} \\ & ≤ Ck^{−j}∥u_{0}∥_{r,p}, \end{aligned}

which is (3.19).

Returning now to the identity (3.16) we first note that (1.17), (1.20), and (1.11) imply

(3.20)  ∥G_{n}A_{h}^{−j}v_{h}∥_{p} ≤ Ck^{j}∥v_{h}∥_{p},  0 ≤ j ≤ q, v_{h} ∈ S_{h}.

Moreover, since A_{h}^{−1}P_{h} = R_{h}A^{−1}, Lemma 3.1 implies

∥(P_{h}A^{−1} − A_{h}^{−1}P_{h})v∥_{p} = ∥P_{h}(I − R_{h})A^{−1}v∥_{p} ≤ Ch^{r}∥A^{−1}v∥_{r,p}.

Hence, using also (3.19), we find

\begin{aligned}∥G_{n}A_{h}^{−j}(P_{h}A^{−1} − A_{h}^{−1}P_{h})A^{j+1}ũ_{0k}∥_{p} & ≤ Ck^{j}∥(P_{h}A^{−1} − A_{h}^{−1}P_{h})A^{j+1}ũ_{0k}∥_{p} \\ & ≤ Ck^{j}h^{r}∥A^{j}ũ_{0k}∥_{r,p} ≤ Ch^{r}∥u_{0}∥_{r,p}, \end{aligned}

for 0 ≤ j ≤ q − 1. Similarly, using (3.20) with j = q and (3.18), we have

∥G_{n}A_{h}^{−q}P_{h}A^{q}ũ_{0k}∥_{p} ≤ Ck^{q}∥A^{q}ũ_{0k}∥_{p} ≤ Ck^{q}∥u_{0}∥_{2q,p}.

Finally, by (3.20) with j = 0 and (3.17), we conclude that

∥G_{n}P_{h}(u_{0} − ũ_{0k})∥_{p} ≤ C∥u_{0} − ũ_{0k}∥_{p} ≤ Ck^{q}∥u_{0}∥_{2q,p},

and we have estimated all the terms in (3.16) in the desired way. □

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1IRMAR, Université de Rennes I, Campus de Beaulieu, F–35042 Rennes Cedex, France

E-mail address: crouzeix@irisa.fr

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

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