FINITE-ELEMENT METHODS FOR A STRONGLY

DAMPED WAVE EQUATION^{∗}

Stig Larsson^{1}, Vidar Thomée^{1} and Lars B. Wahlbin^{2}

Abstract. Error estimates of optimal order are proved for semidiscrete and completely discrete finite element methods for a linear wave equation with strong damping, arising in viscoelastic theory. It is demonstrated that the exact solution may be interpreted in terms of an analytic semigroup, and as a result that, although the solution has essentially the spatial regularity of its initial data, it is infinitely differentiable in time for t > 0. The estimates for the spatially discrete method are derived by energy arguments. Rational approximation of analytic semigroups is discussed in a general setting, by means of spectral representation, and the results are used to analyze the completely discrete schemes. Both smooth (and compatible) and less smooth data are considered.

1. Introduction. In this paper we consider the model initial-boundary value problem

(1.1)

u_{tt} + αAu_{t} + Au = 0, for (x, t) ∈ Ω × (0, ∞),
u = 0, on ∂Ω × (0, ∞),
u(·, 0) = ϕ, u_{t}(·, 0) = ψ, in Ω,

where Ω is a bounded domain in R^{d} with smooth boundary ∂Ω, A is a self-adjoint, positive definite, linear elliptic partial differential operator of second order with smooth coefficients, and α is a fixed positive constant.

The differential equation in (1.1) is sometimes referred to as a strongly damped wave equation, and boundary value problems for it, and nonlinear versions thereof, occur in internally damped systems, e.g., in viscoelastic theory (cf. Fitzgibbon [3], Webb [15] and Massatt [10]). Our analysis below applies also, with only trivial modifications, to the equation

u_{tt} + (αA + β)u_{t} + Au = 0,

i.e., in the presence of a mixture of internal and external damping.

For the purpose of numerical solution of (1.1) by the Galerkin finite element method we assume for now that we are given a family S_{h} of finite dimensional subspaces of H_{0}^{1} = H_{0}^{1}(Ω) with the standard approximation property that, for some r ≥ 2,

inf_{χ∈S_{h}}(∥v − χ∥ + h∥v − χ∥_{1}) ≤ Ch^{r}∥v∥_{r}, for v ∈ H^{r} ∩ H_{0}^{1},

1991 Mathematics Subject Classification. 65M15, 65M60. Key words and phrases. Strongly damped wave equation, finite element method, analytic semi-

group, rational approximation, A-stability, error estimate..

1Supported by the Swedish National Board for Technical Development (STUF). 2Supported by the Army Research Office through the Mathematical Sciences Institute of Cornell

University, and by the National Science Foundation.

∗Published in IMA J. Numer. Anal. 11 (1991), 115–142.

Typeset by AMS-TEX

where ∥ · ∥ is the norm in L_{2} = L_{2}(Ω) and ∥ · ∥_{s} is that in H^{s} = H^{s}(Ω). In the technical work below this assumption will be somewhat weakened.

With A(·, ·) denoting the standard Dirichlet bilinear form corresponding to the differential operator A, we may then first pose the semidiscrete initial-boundary value problem to find u_{h}(t) ∈ S_{h} for t ≥ 0 such that

(1.2)  \begin{matrix}(u_{h,tt}, χ) + αA(u_{h,t}, χ) + A(u_{h}, χ) = 0,  ∀χ ∈ S_{h}, t > 0, \\ u_{h}(0) = ϕ_{h}, u_{h,t}(0) = ψ_{h}, \end{matrix}

where ϕ_{h} and ψ_{h} are approximations of ϕ and ψ in S_{h}, and (·, ·) denotes the inner product in L_{2}.

The error analysis for this method requires some information about the regularity properties of the solution of (1.1). Introducing, in addition to the norm in H^{s}, that in the subspace Ḣ^{s} = Ḣ^{s}(Ω) defined by |v|_{s} = ∥A^{s/2}v∥, we show that, for any j ≥ 0 and s ≥ 0,

|u(t)|_{s} + t^{j}|D_{t}^{j+1}u(t)|_{s} ≤ C(|ϕ|_{s} + |ψ|_{s}), for t > 0.

This shows that if ϕ, ψ ∈ Ḣ^{s}, then u and u_{t} are also in Ḣ^{s}, and further that the solution has an infinite number of time derivatives in Ḣ^{s} for t > 0. A mild smoothing property also with respect to the space variable is expressed by the inequality

|D_{t}^{j}u(t)|_{s} ≤ Ct^{−j}(|ϕ|_{s} + |ψ|_{s−σ}), for t > 0, when 0 ≤ σ ≤ 2,

but further smoothing with respect to x does not hold.

The space Ḣ^{s} used above is the subspace of H^{s} defined essentially by the boundary conditions A^{j}u = 0 for j < s/2 (when s is a positive integer), and the results just described thus assume certain nontrivial boundary conditions on ϕ and ψ. A more careful analysis shows that these assumptions may be somewhat reduced. We demonstrate, for instance, that D_{t}^{j}u(t) is in H^{r} for t > 0, if (and only if) ϕ ∈ H^{r}, ψ ∈ H^{r−2}. Normally, in this case, ∥u(t)∥_{r} will be unbounded as t tends to 0, except when ϕ ∈ H^{r} ∩ Ḣ^{r−2}, ψ ∈ Ḣ^{r−2}.

As a result of these regularity properties we will be able to show that, for the solutions of (1.2) and (1.1),

(1.3) ∥u_{h}(t) − u(t)∥ + min(t, 1)∥u_{h,t}(t) − u_{t}(t)∥ ≤ Ch^{r}(∥ϕ∥_{r} + |ψ|_{r−2}),  for t ≥ 0,

if ϕ ∈ H^{r} ∩ Ḣ^{max(r−2,1)}, ψ ∈ Ḣ^{r−2}, and if ϕ_{h} is an arbitrary approximation of ϕ of optimal order and ψ_{h} is more specifically chosen as the L_{2}-projection of ψ onto S_{h}. Error bounds containing negative powers of t are also deduced for higher order time derivatives under the same regularity assumptions. Note that for r ≤ 4 the assumptions on ϕ and ψ contain no other boundary conditions than ϕ = ψ = 0 on ∂Ω. For r = 5 we will also be able to show an optimal order estimate for u_{h}(t) − u(t) when t is strictly positive, under the conditions ϕ ∈ H^{5} ∩ H_{0}^{1}, ψ ∈ H^{3} ∩ H_{0}^{1}.

Further we show that

(1.4)  ∥u_{h}(t) − u(t)∥ + ∥u_{h,t}(t) − u_{t}(t)∥ ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}),  for t ≥ 0,

if ϕ, ψ ∈ Ḣ^{r}, and both ϕ_{h} and ψ_{h} are arbitrary initial approximations of optimal order. These results are obtained by the energy method.

The error bound (1.4) shows that it is sufficient to have initial data in Ḣ^{r} in order to have optimal order convergence, for both u_{h} and u_{h,t}, uniformly for t ≥ 0. This is in contrast to the situation for the wave equation (α = 0 in (1.1)), where it is known that one extra derivative of initial regularity is needed, see Rauch [12]. On the other hand, (1.3) implies that the regularity of ϕ and ψ can be reduced to a limited extent, while retaining the optimal order of convergence for u_{h} for t ≥ 0. But our regularity analysis shows that the initial regularity cannot be further reduced, even if we restrict our attention to strictly positive t. This differs from the situation for parabolic problems, where no initial regularity is needed to prove optimal order convergence for positive times. Thus, the behavior of the strongly damped wave equation is, in this respect, intermediate between the behaviors of hyperbolic and parabolic equations. For other investigations of finite element methods in situations with limited smoothing, see Crouzeix, Thomée and Wahlbin [2], Genis [6], and Thomée and Zhang [14].

Our regularity analysis is accomplished after introducing w = (u, u_{t})^{T} and noting that (1.1) may be written as the system

\begin{matrix}w_{t} + Bw = 0, t > 0, \\ w(0) = w_{0} = (ϕ, ψ)^{T} , \end{matrix}

where

(1.5)  B = [ \begin{matrix}0 & −I \\ A & αA \end{matrix} ] .

One may then show that −B generates an analytic semigroup in Ḣ^{s} × Ḣ^{s−σ} for 0 ≤ σ ≤ 2. This was done in special cases in [15] and [10].

The semidiscrete equation (1.2) can be written similarly as

(1.6)  \begin{matrix}w_{h,t} + B_{h}w_{h} = 0,  t > 0, \\ w_{h}(0) = w_{0h} = (ϕ_{h}, ψ_{h})^{T} , \end{matrix}

where w_{h} = (u_{h}, u_{h,t})^{T} ∈ S_{h} × S_{h}, and B_{h} is the analogue of (1.5) with A replaced by its discrete counterpart A_{h} : S_{h} → S_{h} defined by

(A_{h}φ, χ) = A(φ, χ), ∀φ, χ ∈ S_{h}.

The formulation (1.6) may also be used as a basis for discretization in time. Thus let r(z) be a rational function approximating e^{−z} to order O(z^{p+1}) for small z and such that |r(z)| ≤ 1 for Re z ≥ 0. We may then define an approximation W_{n} = (U_{n}, V_{n})^{T} to the solution of (1.6) at time t_{n} = nk, where k is the time step, by

\begin{aligned}W_{n} & = r(kB_{h})W_{n−1}, n ≥ 1, \\ W_{0} & = w_{0h} = (ϕ_{h}, ψ_{h})^{T} . \end{aligned}

For example, using the backward Euler method,

(1.7)  (I + kB_{h})W_{n} = W_{n−1},  n ≥ 1,

corresponds to r(z) = 1/(1 + z). After some manipulation, (1.7) may also be written as

[I + (αk + k^{2})A_{h}]V_{n} = V_{n−1} − kA_{h}U_{n−1},
U_{n} = U_{n−1} + kV_{n}.

It is easy to see that, with ¯∂_{t}U_{n} = (U_{n} − U_{n−1})/k, the discrete solution satisfies the following totally implicit three level Galerkin approximation of (1.1), namely

(¯∂_{t}^{2}U_{n}, χ) + αA(¯∂_{t}U_{n}, χ) + A(U_{n}, χ) = 0, ∀χ ∈ S_{h}, n ≥ 2,

with the appropriate choice of U_{1} from above.

Another standard example is provided by the Crank-Nicolson scheme

(I + \frac{k}{2} B_{h})W_{n} = (I − \frac{k}{2} B_{h})W_{n−1}, n ≥ 1,

where r(z) = (1 − z/2)/(1 + z/2). We have similarly to above

[I + (α\frac{k}{2} + \frac{k^{2}}{4} )A_{h}]V_{n} = [I − (α\frac{k}{2} + \frac{k^{2}}{4} )A_{h}]V_{n−1} − kA_{h}U_{n−1},

U_{n} = U_{n−1} + \frac{k}{2} (V_{n} + V_{n−1}),

and

(¯∂_{t}^{2}U_{n}, χ) + αA(˜∂_{t}U_{n}, χ) + A(Û_{n}, χ) = 0, ∀χ ∈ S_{h}, n ≥ 2,

where ˜∂_{t}U_{n} = (U_{n} − U_{n−2})/2k and Û_{n} = (U_{n} + 2U_{n−1} + U_{n−2})/4.

For the general totally discrete method thus described we show that, with optimal order initial approximations, we have the smooth data error estimate

∥U_{n} − u(t_{n})∥ + ∥V_{n} − u_{t}(t_{n})∥ ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}) + Ck^{p}(|ϕ|_{2p} + |ψ|_{2p}),

for ϕ, ψ ∈ Ḣ^{s}, s = max(r, 2p). The proof depends on spectral arguments using the analyticity of the semigroup generated by −B_{h}, and a representation of initial data, used in a parabolic case by Baker, Bramble and Thomée [1]. An error estimate for U_{n} only is also obtained with two orders lower regularity requirement on ψ.

For methods with |r(∞)| < 1 it is possible to show a nonsmooth data error estimate for the semidiscrete solution of the form

(1.8) ∥U_{n} − u_{h}(t_{n})∥ + ∥V_{n} − u_{h,t}(t_{n})∥ ≤ Ck^{p}t_{n}^{−p}(∥ϕ_{h}∥ + ∥ψ_{h}∥),  for t_{n} > 0.

In spite of this, since the smoothing properties of (1.1) with respect to x are limited as described above, we are only able to use (1.8) to obtain optimal order error estimates for the completely discrete scheme in situations when such estimates are available for the semidiscrete problem. In particular, combination of (1.8) and (1.3) yields that if ϕ ∈ H^{r} ∩ Ḣ^{max(r−2,1)}, ψ ∈ Ḣ^{r−2}, and with ϕ_{h} an optimal order approximation to ϕ, and ψ_{h} chosen as the L_{2}-projection of ψ, then we have, for t_{n} positive,

∥U_{n} − u(t_{n})∥ + ∥V_{n} − u_{t}(t_{n})∥ ≤ C(t_{n})(h^{r} + k^{p})(∥ϕ∥_{r} + |ψ|_{r−2}).

For r = 5 an optimal order result for t_{n} > 0 is obtained when ϕ ∈ H^{5} ∩ H_{0}^{1}, ψ ∈ H^{3} ∩ H_{0}^{1}.

Error estimates in the case of a smooth solution have been given earlier in Lin, Thomée and Wahlbin [9] for a semidiscrete version of the equation

(1.9)  u_{tt} + A(t)u_{t} + B(t)u = f(t),  for x ∈ Ω, t > 0,

where A(t) and B(t) are second order differential operators, with A(t) self-adjoint positive definite elliptic. Our equation is a special case of (1.9), with the additional feature that B(t) = A(t) = A. This causes the problem to be associated with a contractive analytic semigroup and also permits a precise analysis by the spectral method.

The plan of the paper is as follows. In Section 2 we present the initial-boundary value problem to be studied, introduce a more general class of finite element methods than the one described above for the discretization with respect to the spatial variable, and discuss existence and regularity of the exact solution of (1.1).

In Section 3 we estimate the error in the spatially semidiscrete approximation. In Section 4 we discuss, in a general setting, approximation of an analytic semigroup by a rational function of its generator. In Section 5 we combine results from Sections 3 and 4 and obtain error bounds for a class of completely discrete schemes.

In the following we let C and c denote various positive constants, not necessarily the same at each occurrence, that are independent of the functions and discretization parameters. In particular, they may depend on α.

2. The strongly damped wave equation. We will thus consider the initialboundary value problem (1.1). We assume that A is a second order elliptic operator given by

Au = − ∑_{i,j=1}^{d} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂u}{∂x_{j}} ) + a_{0}(x)u,

where the coefficients are smooth, a_{0}(x) ≥ 0, and {a_{ij}} is uniformly positive definite. We may then think of A as a self-adjoint, positive definite, unbounded linear operator in L_{2} = L_{2}(Ω) with dense domain D(A) = H^{2}(Ω) ∩ H_{0}^{1}(Ω), and (1.1) can be written as an evolution problem in L_{2}:

(2.1)  \begin{matrix}u_{tt} + αAu_{t} + Au = 0,  t > 0, \\ u(0) = ϕ, u_{t}(0) = ψ. \end{matrix}

In a standard way we may write (2.1) as a 2 × 2 system,

(2.2)  \begin{matrix}w_{t} + Bw = 0,  t > 0, \\ w(0) = w_{0}, \end{matrix}

where w = (u, u_{t})^{T} , w_{0} = (ϕ, ψ)^{T} and

B = [ \begin{matrix}0 & −I \\ A & αA \end{matrix} ] .

We will show below that −B generates an analytic semigroup in the Hilbert space L_{2} ×L_{2}, so that (2.2) has a unique solution w(t) = exp(−tB)w_{0} for any w_{0} ∈ L_{2} ×L_{2}.

We next introduce a class of finite element methods for discretization with respect to the spatial variable that is more general than the standard Galerkin finite element method described in the Introduction.

The inverse of A is a self-adjoint, positive definite, compact, linear operator on L_{2}, which we denote by T. For the approximation of the solution of (1.1) we assume that we are given a family {S_{h}}_{h>0} of finite-dimensional subspaces of L_{2} and corresponding linear operators T_{h} : L_{2} → S_{h} with the following properties:

(i) T_{h} is self-adjoint, positive semi-definite on L_{2}, and positive definite on S_{h}; (ii) for some integer r ≥ 2 we have

(2.3)  ∥(T_{h} − T)f∥ ≤ C h^{r} ∥f∥_{r−2},  for f ∈ H^{r−2}.

Here and in the following ∥ · ∥ and (·, ·) denote the norm and inner product in L_{2}, or L_{2} × L_{2}, unless otherwise stated, and ∥ · ∥_{s} is the norm in H^{s} = H^{s}(Ω).

We note that (ii) implies that

\begin{aligned}inf_{χ∈S_{h}} ∥χ − v∥ & ≤ ∥T_{h}Av − v∥ = ∥(T_{h} − T)Av∥ \\ & ≤ Ch^{r}∥Av∥_{r−2} ≤ Ch^{r}∥v∥_{r}, for v ∈ H^{r} ∩ H_{0}^{1}. \end{aligned}

In other words, for the orthogonal projection P_{h} : L_{2} → S_{h} we have

(2.4)  ∥P_{h}v − v∥ ≤ Ch^{r}∥v∥_{r},  for v ∈ H^{r} ∩ H_{0}^{1}.

Since T_{h} is invertible on S_{h}, we may define A_{h} = T_{h}^{−1} : S_{h} → S_{h}, and a semidiscrete approximation u_{h}(t) ∈ S_{h} by

(2.5)  \begin{matrix}u_{h,tt} + αA_{h}u_{h,t} + A_{h}u_{h} = 0,  t > 0, \\ u_{h}(0) = ϕ_{h}, u_{h,t}(0) = ψ_{h}, \end{matrix}

where ϕ_{h}, ψ_{h} ∈ S_{h} are suitable approximations of ϕ, ψ. We can also write this as

(2.6)  \begin{matrix}w_{h,t} + B_{h}w_{h} = 0,  t > 0, \\ w_{h}(0) = w_{0h}, \end{matrix}

where w_{h}(t) = (u_{h}(t), u_{h,t}(t))^{T} ∈ S_{h} × S_{h}, w_{0h} = (ϕ_{h}, ψ_{h})^{T} and

B_{h} = [ \begin{matrix}0 & −I \\ A_{h} & αA_{h} \end{matrix} ] .

An example of the situation described is provided by the standard Galerkin finite element method (1.2), but our assumptions also allow isoparametric elements as well as more general finite element methods, where the S_{h} do not need to satisfy the boundary condition in (1.1), see Thomée [13, Chapter 2].

Some of our analysis will take place in the framework of the spaces Ḣ^{s} = Ḣ^{s}(Ω) defined for real s by Ḣ^{s} = D(A^{s/2}) with norms

(2.7)  |v|_{s} = ∥A^{s/2}v∥ = (∑_{j=1}^{∞} λ_{j}^{s}|(v, ω_{j})|^{2})^{1/2},

where {λ_{j}}_{j=1}^{∞} are the eigenvalues and {ω_{j}}_{j=1}^{∞} is the corresponding normalized eigensystem of A. It is well-known (cf. Grisvard [7] or Fujiwara [5]) that, for s ≥ 0, \frac{1}{2}s − \frac{1}{4} not an integer, Ḣ^{s} is the subspace of H^{s} characterized by the boundary conditions A^{j}v = 0 on ∂Ω for j < \frac{1}{2}s − \frac{1}{4}, and that the norm of Ḣ^{s} is equivalent to that of H^{s}. In particular, we have Ḣ^{1} = H_{0}^{1}, Ḣ^{2} = H^{2} ∩ H_{0}^{1} with equivalent norms.

Lemma 2.1. The operators −B and −B_{h} generate analytic semigroups exp(−tB) in L_{2} × L_{2} and exp(−tB_{h}) in S_{h} × S_{h} (supplied with the norm of L_{2} × L_{2}), respectively. Moreover, exp(−tB) is an analytic semigroup in Ḣ^{s} × Ḣ^{s−σ} for s ∈ R, 0 ≤ σ ≤ 2.

Proof. The desired result follows from the following lemma, see, e.g., Pazy [11, Theorem 2.5.2].

Lemma 2.2. Let s ∈ R, 0 ≤ σ ≤ 2. Then there is an angle θ_{0} ∈ (0, π/2) and a positive number C such that the sector

Σ = {z ∈ C : θ_{0} ≤ | arg z| ≤ π} ∪ {0}

is contained in the resolvent sets of B and B_{h} for h > 0, and

∥(zI − B)^{−1}∥ ≤ \frac{C}{|z|}, for z ∈ Σ, z ≠ 0, (2.8)

∥(zI − B_{h})^{−1}∥ ≤ \frac{C}{|z|}, for z ∈ Σ, z ≠ 0, h > 0, (2.9)

where in (2.8) and (2.9) ∥ · ∥ denotes the norm of bounded linear operators in Ḣ^{s} × Ḣ^{s−σ} and S_{h} × S_{h}, respectively.

Proof. Let c_{0} > 0 be a lower bound for the eigenvalues of A and A_{h}. (It follows from (2.3) that there is an upper bound for ∥T_{h}∥, and hence a positive lower bound for the spectrum of A_{h}, independently of h.) We prove the statements about B; the argument depends only on c_{0} and therefore applies equally well to B_{h}.

We want to estimate the solution of the equation

(zI − B)w = f,

for any f ∈ Ḣ^{s} × Ḣ^{s−σ}. With f = ∑_{j} f̂_{j}ω_{j} the eigenfunction expansion of f, where f̂_{j} ∈ C^{2}, it follows that w = ∑_{j} ŵ_{j}ω_{j} with ŵ_{j} = (zI − B(λ_{j}))^{−1}f̂_{j}, where

B(λ) = [ \begin{matrix}0 & −1 \\ λ & αλ \end{matrix} ] , (2.10)

(zI − B(λ))^{−1} = (z^{2} − αλz + λ)^{−1} [ \begin{matrix}z − αλ & −1 \\ λ & z \end{matrix} ] .

In view of Parseval’s relation and (2.7) we must show that

(2.11) | [ \begin{matrix}λ^{s} & 0 \\ 0 & λ^{s−σ} \end{matrix} ] (zI − B(λ))^{−1} [ \begin{matrix}λ^{s} & 0 \\ 0 & λ^{s−σ} \end{matrix} ]^{−1} |

= |(z^{2} − αλz + λ)^{−1} [ \begin{matrix}z − αλ & −λ^{σ/2} \\ λ^{1−σ/2} & z \end{matrix} ] | ≤ \frac{C}{|z|}, z ∈ Σ, λ ≥ c_{0},

for a suitable sector Σ, where | · | on the left denotes the norm of the 2 × 2 matrix.

We first locate the zeros of z^{2} − αλz + λ (i.e., the eigenvalues of B(λ)) for λ ≥ c_{0}. They are

(2.12)  τ_{±}(λ) = \frac{αλ}{2} ± \sqrt{(\frac{αλ}{2})^{2} − λ}

and it is easily seen that they lie in the set

{z ∈ C : |z − α^{−1}| = α^{−1}, Re z ≥ αc_{0}/2} ∪ {z ∈ C : Re z > α^{−1}, Im z = 0}.

Now let θ_{0} = arctan(2/(α^{2}c_{0})). This defines a sector Σ of the desired form and a constant c_{1} > 0, such that

|z − τ_{±}(λ)| ≥ c_{1}|z|, z ∈ Σ, λ ≥ c_{0}.

Since, in addition,

|z − αλ| ≤ c_{2}|z − τ_{+}(λ)|, z ∈ Σ, λ ≥ c_{0},

we hence find that

\begin{aligned}|\frac{z − αλ}{z^{2} − αλz + λ}| & = |\frac{z − αλ}{(z − τ_{+}(λ))(z − τ_{−}(λ))}| \\ & ≤ \frac{c_{2}}{|z − τ_{−}(λ)|} ≤ \frac{c_{2}}{c_{1}|z|}, z ∈ Σ, λ ≥ c_{0}, \end{aligned}

and similarly for the other entries on the left-hand side of (2.11). This shows (2.11) and the proof is complete.

The special case 0 ≤ σ ≤ s < 2 of Lemma 2.1 was shown in [10] in a more general context. The case s = σ = 2 was obtained in [15] by a simple argument.

We shall now turn to a discussion of the regularity of the solution of the initial boundary value problem (1.1). The first result, which refers to the scale of spaces Ḣ^{s}, follows directly from the analyticity of the semigroup as described in Lemma 2.1.

Lemma 2.3. For any integer j ≥ 0 and real numbers s and σ with 0 ≤ σ ≤ 2 there is a positive number C such that for the solution u of (2.1) we have

|D_{t}^{j}u(t)|_{s} + |D_{t}^{j+1}u(t)|_{s−σ} ≤ Ct^{−j}(|ϕ|_{s} + |ψ|_{s−σ}), for t > 0.

Proof. Since exp(−tB) is an analytic semigroup in Ḣ^{s} × Ḣ^{s−σ}, we have, in the corresponding norm, for w(t) = exp(−tB)w_{0},

|D_{t}^{j}w(t)|_{s×(s−σ)} ≤ Ct^{−j}|w_{0}|_{s×(s−σ)}, t > 0,

which is the desired result.

This result can also be easily derived directly from the eigenfunction expansion of the solution of (1.1), namely

u(t) = ∑_{i=1}^{∞} û_{i}(t)ω_{i},

where, with τ_{±}(λ) defined in (2.12),

(2.13) û_{i}(t) = \frac{τ_{+}(λ_{i})e^{−tτ_{−}(λ_{i})} − τ_{−}(λ_{i})e^{−tτ_{+}(λ_{i})}}{τ_{+}(λ_{i}) − τ_{−}(λ_{i})} ϕ̂_{i} + \frac{e^{−tτ_{−}(λ_{i})} − e^{−tτ_{+}(λ_{i})}}{τ_{+}(λ_{i}) − τ_{−}(λ_{i})} ψ̂_{i},

with the obvious modification if λ_{i} = 4/α^{2}, i.e., when τ_{+}(λ_{i}) = τ_{−}(λ_{i}). This representation may be obtained directly from (1.1) by the standard separation of variables argument.

We shall now use this representation to demonstrate that, in contrast to the situation in the case of a parabolic equation, there is no more smoothing in x with respect to the scale of spaces Ḣ^{s} than that exhibited in Lemma 2.3. In fact, as λ → ∞ we have by (2.12) that τ_{−}(λ) ∼ 1/α, τ_{+}(λ) ∼ αλ, and hence, for any t > 0,

(2.14)  û_{i}(t) ∼ e^{−t/α}(ϕ̂_{i} + \frac{1}{αλ_{i}} ψ̂_{i}),  as i → ∞.

Thus the least conditions required on ϕ and ψ in order to guarantee that u(t) belongs to Ḣ^{s} are ϕ ∈ Ḣ^{s} and ψ ∈ Ḣ^{s−2}.

It is clear from (2.13) that

(2.15)  |D_{t}^{j}û_{i}(t)| ≤ C(t)(|ϕ̂_{i}| + |ψ̂_{i}|),  for i ≥ 1, t > 0,

and hence u(t) = ∑_{i=1}^{∞} û_{i}(t)ω_{i} is well defined in L_{2} for ϕ, ψ ∈ L_{2} together with its time derivatives for t > 0. We shall think of this as a generalized solution of (1.1); when the û_{i}(t) are such that u and u_{t} are in D(A) = Ḣ^{2}, this is a genuine solution.

A natural question to ask is now what regularity the solution possesses in the situation that ϕ and ψ are smooth without satisfying the higher order boundary conditions of the Ḣ^{s} spaces. It will turn out that if ϕ ∈ H^{s} and ψ ∈ H^{s−2}, then the generalized solution and its time derivatives belong to H^{s} for t > 0. This is expressed in the following lemma, which shows, in particular, that if ϕ and ψ are smooth and vanish on ∂Ω, then they are the initial values of a smooth genuine solution.

Lemma 2.4. Let s ≥ 2, σ, τ ≥ 0, and assume that ϕ ∈ H^{s} ∩Ḣ^{σ} and ψ ∈ H^{s−2} ∩Ḣ^{τ}. Then D_{t}^{j}u(t) ∈ H^{s} for t > 0, and, with [y]_{+} = max(y, 0), we have

∥D_{t}^{j}u(t)∥_{s} ≤ C(∥ϕ∥_{s} + ∥ψ∥_{s−2} + t^{−[(s−σ)/2+j−1]_{+}}|ϕ|_{σ} + t^{−[(s−τ)/2+j−1]_{+}}|ψ|_{τ} ).

Proof. Since ω_{i} ∈ Ḣ^{s} for any s ≥ 0, the smoothness properties of u(t) depend only on the behavior of û_{i}(t) for large i. We write (2.13) in the form

û_{i}(t) = [a_{i}^{−}(t) − a_{i}^{+}(t)]ϕ̂_{i} + [b_{i}^{−}(t) − b_{i}^{+}(t)]ψ̂_{i},

where + corresponds to exp(−tτ_{+}(λ_{i})) and correspondingly for −. We set, with κ = 8/α^{2},

u_{±}^{ϕ}(t) = ∑_{λ_{i}>κ} a_{i}^{±}ϕ̂_{i}ω_{i}, u_{±}^{ψ}(t) = ∑_{λ_{i}>κ} b_{i}^{±}ψ̂_{i}ω_{i}.

Clearly then, by (2.15), we have for any s ≥ 0, with C = C_{s},

(2.16) |D_{t}^{j}(u(t) − [u_{−}^{ϕ}(t) − u_{+}^{ϕ}(t)] − [u_{−}^{ψ}(t) − u_{+}^{ψ}(t)])|_{s}

= | ∑_{λ_{i}≤κ} D_{t}^{j}û_{i}(t)ω_{i}|_{s} = ( ∑_{λ_{i}≤κ} λ_{i}^{s}|D_{t}^{j}û_{i}(t)|^{2})^{1/2} ≤ C(∥ϕ∥ + ∥ψ∥).

We shall also show

(2.17)  ∥D_{t}^{j}u_{−}^{ϕ}(t)∥_{s} ≤ C∥ϕ∥_{s},
(2.18)  ∥D_{t}^{j}u_{−}^{ψ}(t)∥_{s} ≤ C∥ψ∥_{s−2},
(2.19)  |D_{t}^{j}u_{+}^{ϕ}(t)|_{s} ≤ Ct^{−[(s−σ)/2+j−1]_{+}}|ϕ|_{σ},
(2.20)  |D_{t}^{j}u_{+}^{ψ}(t)|_{s} ≤ Ct^{−[(s−τ)/2+j−1]_{+}}|ψ|_{τ} ,

which estimates together then will complete the proof.

We start with (2.19) and note that

D_{t}^{j}u_{+}^{ϕ}(t) = ∑_{λ_{i}>κ}(−1)^{j}τ_{+}(λ_{i})^{j}a_{i}^{+}(t)ϕ̂_{i}ω_{i}.

Here

\begin{aligned}λ_{i}^{s/2}|τ_{+}(λ_{i})^{j}a_{i}^{+}(t)| & = λ_{i}^{s/2}τ_{+}(λ_{i})^{j}τ_{−}(λ_{i})[τ_{+}(λ_{i}) − τ_{−}(λ_{i})]^{−1}e^{−tτ_{+}(λ_{i})} \\ & ≤ Cλ_{i}^{s/2+j−1}e^{−ctλ_{i}} ≤ Ct^{−[(s−σ)/2+j−1]_{+}}λ_{i}^{σ/2}, \end{aligned}

so that

|D_{t}^{j}u_{+}^{ϕ}(t)|_{s} ≤ Ct^{−[(s−σ)/2+j−1]_{+}}( ∑_{λ_{i}>κ} λ_{i}^{σ}|ϕ̂_{i}|^{2})^{1/2} ≤ Ct^{−[(s−σ)/2+j−1]_{+}}|ϕ|_{σ}.

The proof of (2.20) is analogous.

We now turn to the proof of (2.17) and (2.18) and note first that

D_{t}^{j}u_{−}^{ϕ}(t) = ∑_{λ_{i}>κ}(−1)^{j}τ_{−}(λ_{i})^{j}a_{i}^{−}(t)ϕ̂_{i}ω_{i}.

We shall show below that, for any j ≥ 0 and N > 0, there exist a polynomial P_{j,N}(·, ·) and a constant C = C_{j,N} such that

(2.21)  |τ_{−}(λ_{i})^{j}a_{i}^{−}(t) − e^{−t/α}P_{j,N}(t, λ_{i}^{−1})| ≤ Cλ_{i}^{−N},  for t > 0, λ_{i} ≥ κ.

Assuming this for a moment we may define, with T = A^{−1} as above,

ũ_{−}^{ϕ,j}(t) = e^{−t/α}P_{j,N}(t, T)ϕ = e^{−t/α} ∑_{i=1}^{∞} P_{j,N}(t, λ_{i}^{−1})ϕ̂_{i}ω_{i}.

Clearly then, if N ≥ s/2,

(2.22)  |D_{t}^{j}u_{−}^{ϕ}(t) − ũ_{−}^{ϕ,j}(t)|_{s} ≤ C(∑_{i=1}^{∞} λ_{i}^{s}λ_{i}^{−2N}|ϕ̂_{i}|^{2})^{1/2} ≤ C∥ϕ∥.

Further, since T^{l}ϕ ∈ H^{s+2l} ⊂ H^{s} for l ≥ 0, we have

∥ũ_{−}^{ϕ,j}(t)∥_{s} ≤ C∥ϕ∥_{s}.

Together these estimates show (2.17).

To show (2.21) we first note that 0 ≤ τ_{−}(λ) − 1/α ≤ Cλ^{−1} for λ ≥ κ and that hence by Taylor expansion of exp(−t(τ_{−}(λ) − 1/α)),

|e^{−tτ_{−}(λ)} − e^{−t/α} ∑_{j=0}^{N−1} \frac{(−t(τ_{−}(λ) − 1/α))^{j}}{j!} | ≤ C(t/λ)^{N}e^{−t/α} ≤ Cλ^{−N}, λ ≥ κ.

Thus by expanding τ_{−}(λ) in powers of λ^{−1} we find for a suitable polynomial P_{N}(·, ·),

|e^{−tτ_{−}(λ)} − e^{−t/α}P_{N}(t, λ^{−1})| ≤ Cλ^{−N}, λ ≥ κ.

Expanding similarly τ_{−}(λ)^{j}τ_{+}(λ)[τ_{+}(λ) − τ_{−}(λ)]^{−1} in powers of λ^{−1} completes the proof of (2.21). We note, in particular, for later use that P_{0,N}(t, 0) = P_{N}(t, 0) ≡ 1.

To show (2.18) we need instead of (2.21) the existence of polynomials Q_{j,N} such that

(2.23) |τ_{−}(λ_{i})^{j}b_{i}^{−}(t) − e^{−t/α}λ_{i}^{−1}Q_{j,N}(t, λ_{i}^{−1})| ≤ Cλ_{i}^{−N},  for t > 0, λ_{i} ≥ κ,

which is demonstrated in the analogous way. The additional factor λ_{i}^{−1} in front of the polynomial in (2.23) comes from the fact that the coefficient [τ_{+}(λ_{i}) − τ_{−}(λ_{i})]^{−1} of exp(−tτ_{−}(λ_{i})) in b_{i}^{−}(t) behaves like (αλ_{i})^{−1} (cf. (2.14)). In this case we define

ũ_{−}^{ψ,j}(t) = e^{−t/α}T Q_{j,N}(t, T)ψ = e^{−t/α} ∑_{i=1}^{∞} λ_{i}^{−1}Q_{j,N}(t, λ_{i}^{−1})ψ̂_{i}ω_{i}.

FINITE-ELEMENT METHODS FOR A STRONGLY DAMPED WAVE EQUATION 11

and find, similarly to above, since T^{l}ψ ∈ H^{s−2+2l} ⊂ H^{s} for l ≥ 1, that

∥ũ_{−}^{ψ,j}(t)∥_{s} ≤ C∥ψ∥_{s−2}.

The proof of Lemma 2.4 is now complete.

The results of Lemma 2.4 are also sharp. To indicate a proof of this, let us first show that if ψ = 0 and u(t) is in H^{s} for some positive t, then ϕ ∈ H^{s}. To see this we write

ũ_{−}^{ϕ,0} = u_{+}^{ϕ} + u − [u − (u_{−}^{ϕ} − u_{+}^{ϕ})] − [u_{−}^{ϕ} − ũ_{−}^{ϕ,0}].

Since u_{±}^{ψ}(t) = 0, (2.16) shows that u(t)−(u_{−}^{ϕ}(t)−u_{+}^{ϕ}(t)) is in Ḣ^{s}. By (2.22) the same is true for u_{−}^{ϕ}(t)−ũ_{−}^{ϕ,0}(t). But by (2.19) we have u_{+}^{ϕ}(t) ∈ Ḣ^{s}, and hence ũ_{−}^{ϕ,0}(t) ∈ H^{s}. This function is of the form

(2.24)  ũ_{−}^{ϕ,0}(t) = e^{−t/α} ∑_{l=0}^{M} a_{l}(t)T^{l}ϕ,  with a_{0}(t) = P_{0,N}(t, 0) ≡ 1,

where the a_{l}(t) are polynomials in t. Hence, since T^{l}ϕ is smoother than ϕ for any l > 0, we conclude easily that ϕ ∈ H^{s}.

We next show that if ψ = 0 and u(t) is bounded in H^{s}, uniformly for t ≥ 0, then ϕ ∈ Ḣ^{s−2}. For this we first note that by (2.24) ũ_{−}^{ϕ,0}(t) is bounded in H^{s} uniformly for t ≥ 0, so that we now know that this holds for u_{+}^{ϕ}(t). But since this function belongs to Ḣ^{s}, we have by equivalence of norms that u_{+}^{ϕ}(t) is bounded in Ḣ^{s} so that, in particular, for t = 0,

∑_{λ_{i}>κ} λ_{i}^{s}a_{i}^{+}(0)^{2}|ϕ̂_{i}|^{2} < ∞.

Since

a_{i}^{+}(0) = τ_{−}(λ_{i})[τ_{+}(λ_{i}) − τ_{−}(λ_{i})]^{−1} ∼ \frac{1}{α^{2}λ_{i}} , as i → ∞,

we finally conclude that

|ϕ|_{s−2}^{2} = ∑_{i=1}^{∞} λ_{i}^{s−2}|ϕ̂_{i}|^{2} < ∞.

A similar discussion shows that, if ϕ = 0 and u(t) ∈ H^{s} for some positive t, then ψ ∈ H^{s−2} and, if in addition u(t) is bounded in H^{s} for small t, then ψ ∈ Ḣ^{s−2}.

We end this section with the following stability result for the semidiscrete version of our model problem.

Lemma 2.5. For any integer j ≥ 0 we have for the solution of (2.5)

∥D_{t}^{j}u_{h}(t)∥ + ∥D_{t}^{j+1}u_{h}(t)∥ ≤ Ct^{−j}(∥ϕ_{h}∥ + ∥ψ_{h}∥), for t > 0.

Proof. This follows at once by the analyticity of exp(−tB_{h}) in S_{h} × S_{h}.

3. Semidiscretization with respect to space. In this section we estimate the error between the solution of (1.1) and its semidiscrete approximation defined by (2.5).

The following is our main result. In addition to giving a uniform error bound for u_{h} for all positive times, it shows that, as a result of the regularity of u in time, the time derivatives of the error may be bounded, with error bounds which contain negative powers of t for small t, but which are uniform for large t.

Theorem 3.1. Assume that ϕ ∈ H^{r} ∩ Ḣ^{max(r−2,1)}, ψ ∈ Ḣ^{r−2}, and let

∥ϕ_{h} − ϕ∥ ≤ Ch^{r}∥ϕ∥_{r}, ψ_{h} = P_{h}ψ.

Then we have, with t_{−} = min(t, 1),

(3.1)  ∥D_{t}^{j}(u_{h}(t) − u(t))∥ ≤ Ch^{r}t_{−}^{−j}(∥ϕ∥_{r} + |ψ|_{r−2}),  for t > 0 .

Proof. We begin by showing the result for t ≤ 1. We note that without restricting the generality we may assume that ϕ_{h} = P_{h}ϕ, since, by the stability result of Lemma 2.5 and (2.4), and since ϕ ∈ H^{r} ∩ H_{0}^{1}, the additional error caused by the general choice of ϕ_{h} is bounded by

Ct^{−j}∥ϕ_{h} − P_{h}ϕ∥ ≤ Ch^{r}t^{−j}∥ϕ∥_{r}.

Applying the operator T = A^{−1} to the differential equation in (2.1) and integrating with respect to t, we have

(3.2)  T u_{t} + αu + ∫_{0}^{t} u dτ = T ψ + αϕ.

Performing the analogous operations to (2.5) and subtracting (3.2) from the resulting equation, we obtain for the error e = u_{h} − u,

(3.3) \begin{matrix}T_{h}e_{t} + αe + ∫_{0}^{t} e dτ = η ≡ −(T_{h} − T)u_{t} + (T_{h} − T)ψ + α(P_{h} − I)ϕ, t > 0, \\ T_{h}e(0) = 0. \end{matrix}

Here we have used the fact that T_{h}P_{h} = T_{h}, which follows from

(T_{h}P_{h}f, g) = (P_{h}f, T_{h}g) = (f, T_{h}g) = (T_{h}f, g), ∀f, g ∈ L_{2}.

The core of the proof of (3.1) for small t is the following lemma, which is deduced by an energy argument.

Lemma 3.2. For each t^{∗} > 0, β ≥ 0 and ϵ > 0 there is C = C(βt^{∗}, ϵ) such that, if e satisfies

T_{h}e_{t} + αe + β ∫_{0}^{t} e(τ) dτ = η, 0 < t < t^{∗}, (3.4)

T_{h}e(0) = 0,

then

(3.5)  ∥e(t)∥ ≤ C sup_{0<τ<t} ∥η(τ)∥ + ϵ sup_{0<τ<t}(τ∥η_{t}(τ)∥),  0 < t < t^{∗}.

Proof. Writing ẽ(t) = ∫_{0}^{t} e(τ) dτ and taking the inner product of (3.4) with e_{t}, we have

(T_{h}e_{t}, e_{t}) + \frac{α}{2} \frac{d}{dt}∥e∥^{2} = −β(ẽ, e_{t}) + (η, e_{t}).

Multiplication by t leads to

\begin{matrix}t(T_{h}e_{t}, e_{t}) + \frac{α}{2} \frac{d}{dt}(t∥e∥^{2}) = \frac{α}{2}∥e∥^{2} − βt(ẽ, e_{t}) + t(η, e_{t}) \\ = \frac{α}{2}∥e∥^{2} + β \frac{d}{dt}(\frac{1}{2}∥ẽ∥^{2} − t(ẽ, e)) + βt∥e∥^{2} + \frac{d}{dt}(t(η, e)) − (η, e) − t(η_{t}, e), \end{matrix}

and since T_{h} is positive semi-definite, we obtain by simple estimates

\frac{d}{dt}(t∥e∥^{2}) ≤ C (1 + βt^{∗})∥e∥^{2} + C \frac{d}{dt}(\frac{β}{2} ∥ẽ∥^{2} − βt(ẽ, e) + t(η, e)) + C∥η∥^{2} + \frac{ϵ^{2}}{2} t^{2}∥η_{t}∥^{2}

for 0 < t < t^{∗}, and after integration with respect to t,

t∥e(t)∥^{2} ≤ C(1 + βt^{∗}) ∫_{0}^{t} ∥e∥^{2} dτ + C(\frac{β}{2} ∥ẽ∥^{2} − βt(ẽ, e) + t(η, e))

+ C ∫_{0}^{t} ∥η∥^{2} dτ + \frac{ϵ^{2}}{2} ∫_{0}^{t} τ^{2}∥η_{t}∥^{2} dτ

≤ C(1 + βt^{∗} + β^{2}t^{∗}^{2}) ∫_{0}^{t} ∥e∥^{2} dτ + C ∫_{0}^{t} ∥η∥^{2} dτ + \frac{ϵ^{2}}{2} ∫_{0}^{t} τ^{2}∥η_{t}∥^{2} dτ

+ Ct∥η(t)∥^{2} + \frac{1}{2}t∥e(t)∥^{2}, 0 < t < t^{∗},

where we have used the fact that ẽ(0) = 0 and ∥ẽ(t)∥^{2} ≤ t^{∗} ∫_{0}^{t} ∥e∥^{2} dτ for 0 < t < t^{∗}. Hence, absorbing βt^{∗} into the constant,

\begin{aligned}t∥e(t)∥^{2} & ≤ C ∫_{0}^{t} ∥e∥^{2} dτ + C ∫_{0}^{t} ∥η∥^{2} dτ \\ & + ϵ^{2} ∫_{0}^{t} τ^{2}∥η_{t}∥^{2} dτ + Ct∥η(t)∥^{2}, 0 < t < t^{∗}. \end{aligned} (3.6)

In order to estimate ∫_{0}^{t} ∥e∥^{2} dτ we take the inner product of (3.4) with e to obtain

(T_{h}e_{t}, e) + α∥e∥^{2} + β(ẽ, e) = (η, e).

Since T_{h} is self-adjoint, this leads to

\frac{d}{dt}(T_{h}e, e) + 2α∥e∥^{2} + β \frac{d}{dt}∥ẽ∥^{2} ≤ \frac{1}{α}∥η∥^{2} + α∥e∥^{2},

and, upon integration,

(T_{h}e, e) + α ∫_{0}^{t} ∥e∥^{2} dτ + β∥ẽ∥^{2} ≤ C ∫_{0}^{t} ∥η∥^{2} dτ,

since T_{h}e(0) = 0, ẽ(0) = 0. Together with (3.6) this shows

(3.7) t∥e(t)∥^{2} ≤ C ∫_{0}^{t} ∥η∥^{2} dτ + ϵ^{2} ∫_{0}^{t} τ^{2}∥η_{t}∥^{2} dτ + Ct∥η(t)∥^{2},  0 < t < t^{∗},

from which (3.5) follows. The proof of the lemma is now complete.

We now show (3.1) for j = 0. We have by Lemma 3.2, using also (3.3), (2.3), and (2.4),

\begin{aligned}∥e(t)∥ & ≤ C sup_{0<τ<t}(∥η(τ)∥ + τ∥η_{t}(τ)∥) \\ & ≤ Ch^{r}(∥ϕ∥_{r} + |ψ|_{r−2} + sup_{0<τ<t}(∥u_{t}(τ)∥_{r−2} + τ∥u_{tt}(τ)∥_{r−2})), 0 < t ≤ 1. \end{aligned}

Here, by Lemma 2.3 (or Lemma 2.4),

∥u_{t}(τ)∥_{r−2} + τ∥u_{tt}(τ)∥_{r−2} ≤ C(|ϕ|_{r−2} + |ψ|_{r−2}),

which completes the proof for j = 0.

To show the result for general j we proceed by induction to demonstrate that

(3.8)  t^{j}∥D_{t}^{j}e(t)∥ ≤ C sup_{0<τ<t}(∑_{l=0}^{j+1} τ^{l}∥D_{t}^{l}η(τ)∥),  for 0 < t ≤ 1.

Assuming this for a moment we have now

t^{j}∥D_{t}^{j}e(t)∥ ≤ Ch^{r}{∥ϕ∥_{r} + |ψ|_{r−2} + sup_{0<τ<t}(∑_{l=0}^{j+1} τ^{l}∥D_{t}^{l+1}u(τ)∥_{r−2})}, 0 < t ≤ 1,

from which the desired estimate again follows by Lemma 2.3 (or Lemma 2.4). For the proof of (3.8) we differentiate (3.3) j times and multiply by t^{j} to obtain

T_{h}(t^{j}e^{(j)})_{t} + αt^{j}e^{(j)} = −t^{j}e^{(j−1)} + t^{j}η^{(j)} + jt^{j−1}T_{h}e^{(j)},

where e^{(j)} = D_{t}^{j}e and η is as above. Now, by (3.3),

T_{h}e^{(j)} = −αe^{(j−1)} − ẽ^{(j−1)} + η^{(j−1)},

so that

T_{h}(t^{j}e^{(j)})_{t} + αt^{j}e^{(j)} = −t^{j}e^{(j−1)} − jt^{j−1}ẽ^{(j−1)} − αjt^{j−1}e^{(j−1)} + t^{j}η^{(j)} + jt^{j−1}η^{(j−1)}.

Hence, by Lemma 3.2 (with β = 0) and using t ≤ 1, we have

sup_{0<τ<t}(τ^{j}∥e^{(j)}∥) ≤ C sup_{0<τ<t}(τ^{j−1}∥e^{(j−1)}∥ + τ^{j−1}∥ẽ^{(j−1)}∥)

+ C sup_{0<τ<t}(τ^{j+1}∥η^{(j+1)}∥ + τ^{j}∥η^{(j)}∥ + τ^{j−1}∥η^{(j−1)}∥)

+ 2ϵ sup_{0<τ<t}(τ^{j}∥e^{(j)}∥).

Here ẽ^{(j−1)} = e^{(j−2)} for j ≥ 2, and ∥ẽ^{(j−1)}(τ)∥ ≤ τ sup_{0<σ<τ} ∥e(σ)∥ for j = 1. Hence, choosing ϵ < 1/2, (3.8) follows by induction. This completes the proof of Theorem 3.1 for t ≤ 1.

We now turn to t > 1. This time we make the special choice ϕ_{h} = R_{h}ϕ, where R_{h} denotes the ”elliptic projection” defined by R_{h} = T_{h}A, which as above shows the general result by stability, since ϕ ∈ H^{r} ∩ H_{0}^{1}, and

(3.9) ∥R_{h}v − v∥ = ∥(T_{h} − T)Av∥ ≤ Ch^{r}∥Av∥_{r−2} ≤ Ch^{r}∥v∥_{r},  v ∈ H^{r} ∩ H_{0}^{1}.

We split the error as

e = u_{h} − u = (u_{h} − R_{h}u) + (R_{h}u − u) ≡ θ + ρ,

and note that, by our assumptions, θ ∈ S_{h} and

(3.10)  θ(0) = 0, θ_{t}(0) = P_{h}ψ − R_{h}ψ = −P_{h}ρ_{t}(0).

By (3.9) and Lemma 2.4 we have

\begin{aligned}∥D_{t}^{j}ρ(t)∥ & = ∥(R_{h} − I)D_{t}^{j}u(t)∥ ≤ Ch^{r}∥D_{t}^{j}u(t)∥_{r} \\ & ≤ Ch^{r}t^{−j}(∥ϕ∥_{r} + |ϕ|_{r−2} + |ψ|_{r−2}), for t > 0, \end{aligned} (3.11)

and it remains to estimate D_{t}^{j}θ.

From (2.1), (2.5) and the definition of R_{h} we obtain

θ_{tt} + αA_{h}θ_{t} + A_{h}θ = −P_{h}(R_{h} − I)u_{tt} = −P_{h}ρ_{tt}.

Integrating with respect to t and using also (3.10) we find that θ satisfies

(3.12)  \begin{matrix}θ_{t} + αA_{h}θ + A_{h}θ̃ = −P_{h}ρ_{t},  t > 0, \\ θ(0) = 0, \end{matrix}

where θ̃(t) = ∫_{0}^{t} θ(τ) dτ. We will show by induction that, with |θ|_{1,h} = ∥Ah^{\frac{1}{2}} θ∥ a discrete analogue of the H_{0}^{1} norm, we have

(3.13)  ∥D_{t}^{j}θ(t)∥^{2} + ∫_{1}^{t} |D_{t}^{j}θ|_{1,h}^{2} dτ ≤ Ch^{2r}(∥ϕ∥_{r} + |ψ|_{r−2})^{2},  for t ≥ 1.

Together with (3.11) this will complete the proof.

We begin with a lemma which implies (3.13) for j = 0.

Lemma 3.3. The solution of (3.12) satisfies, with e = θ + ρ,

∥θ(t)∥^{2} + α ∫_{1}^{t} |θ|_{1,h}^{2} dτ ≤ C sup_{0<τ<1}(∥e(τ)∥^{2} + ∥ρ(τ)∥^{2}) + C ∫_{1}^{t} ∥ρ_{t}∥^{2} dτ, t ≥ 1.

Proof. Taking the inner product of (3.12) by θ, and using the facts that A_{h} is selfadjoint, positive definite and that T_{h} = A_{h}^{−1} is bounded independently of h, we get

\frac{1}{2} \frac{d}{dt}∥θ∥^{2} + α|θ|_{1,h}^{2} + \frac{1}{2} \frac{d}{dt}|θ̃|_{1,h}^{2} ≤ ∥ρ_{t}∥∥θ∥ ≤ C∥ρ_{t}∥^{2} + \frac{α}{2}|θ|_{1,h}^{2}.

Integrating with respect to t we obtain by simple estimates

∥θ(t)∥^{2} + α ∫_{1}^{t} |θ|_{1,h}^{2} dτ + |θ̃(t)|_{1,h}^{2} ≤ C(∥θ(1)∥^{2} + |θ̃(1)|_{1,h}^{2}) + C ∫_{1}^{t} ∥ρ_{t}∥^{2} dτ.

In order to estimate |θ̃(1)|_{1,h} we now take the inner product of (3.12) by θ̃ and find

\begin{aligned}\frac{α}{2} \frac{d}{dt}|θ̃|_{1,h}^{2} + |θ̃|_{1,h}^{2} & = −(θ_{t}, θ̃) − (ρ_{t}, θ̃) = −(e_{t}, θ̃) \\ & = −\frac{d}{dt}(e, θ̃) + (e, θ) ≤ −\frac{d}{dt}(e, θ̃) + C(∥e∥^{2} + ∥ρ∥^{2}). \end{aligned}

Hence

|θ̃(1)|_{1,h}^{2} ≤ C∥e(1)∥ ∥θ̃(1)∥ + C ∫_{0}^{1} (∥e∥^{2} + ∥ρ∥^{2}) dτ,

which yields

|θ̃(1)|_{1,h}^{2} ≤ C sup_{0<τ<1}(∥e(τ)∥^{2} + ∥ρ(τ)∥^{2}).

Since ∥θ(1)∥ is obviously estimated by the same quantity, this completes the proof of the lemma.

To show (3.13) for j = 0 we note that e(t) and ρ(t) have already been bounded as desired for t ≤ 1 and that by Lemma 2.4,

∫_{1}^{t} ∥ρ_{t}∥^{2} dτ ≤ Ch^{2r} ∫_{1}^{t} ∥u_{t}∥_{r}^{2} dτ ≤ Ch^{2r} ∫_{1}^{t} τ^{−2} dτ (∥ϕ∥_{r} + |ψ|_{r−2})^{2}.

There remains only the induction step in the proof of (3.13). For this purpose we differentiate (3.12) j times with respect to t and multiply by θ^{(j)} = D_{t}^{j}θ to obtain for j ≥ 1

\begin{aligned}\frac{1}{2} \frac{d}{dt}∥θ^{(j)}∥^{2} + α|θ^{(j)}|_{1,h}^{2} & = −(A_{h}θ^{(j−1)}, θ^{(j)}) − (ρ^{(j+1)}, θ^{(j)}) \\ & ≤ C(|θ^{(j−1)}|_{1,h}^{2} + ∥ρ^{(j+1)}∥^{2}) + \frac{α}{2}|θ^{(j)}|_{1,h}^{2}. \end{aligned}

Hence, assuming (3.13) for j − 1, we have by the estimates for t ≤ 1 and by (3.11) that

\begin{aligned}∥θ^{(j)}(t)∥^{2} + ∫_{1}^{t} |θ^{(j)}|_{1,h}^{2} dτ & ≤ ∥θ^{(j)}(1)∥^{2} + C ∫_{1}^{t}(|θ^{(j−1)}|_{1,h}^{2} + ∥ρ^{(j+1)}∥^{2}) dτ \\ & ≤ Ch^{2r}(∥ϕ∥_{r} + |ψ|_{r−2})^{2}, for t ≥ 1. \end{aligned}

The proof of Theorem 3.1 is now complete.

We note that the assumptions of Theorem 3.1 are exactly those which are required in Lemma 2.4 to guarantee that u(t) is bounded in H^{r}, uniformly for t ≥ 0, and weaker than what is needed for u(t) to be bounded in Ḣ^{r} by Lemma 2.3. In particular, for r ≤ 4, none of the higher order boundary conditions of the Ḣ^{s} spaces are required, but only ϕ = 0, and, if r = 3, 4 also ψ = 0, on ∂Ω.

For r = 5 the boundary conditions Aϕ = Aψ = 0 on ∂Ω are also needed since ϕ, ψ ∈ Ḣ^{3} is assumed in that case. By a slightly more careful analysis we are able to omit these extra artificial boundary conditions, at the expense of a weak singularity for t small. We shall show that in this case, when ϕ ∈ H^{5} ∩ H_{0}^{1} and ψ ∈ H^{3} ∩ H_{0}^{1} we have for any ϵ ∈ (0, 1/2)

(3.14)  ∥e(t)∥ ≤ C_{ϵ}h^{5}t^{−(1+ϵ)/4}(∥ϕ∥_{5} + ∥ψ∥_{3}),  for 0 < t ≤ t^{∗}.

To see this we note from (3.7) of the proof of Lemma 3.2 that

∥e(t)∥^{2} ≤ Ct^{−1} ∫_{0}^{t}(∥η∥^{2} + τ^{2}∥η_{t}∥^{2}) dτ.

(The last term on the right in (3.7) may easily be estimated by the first two terms.) Hence, by the definition of η, cf. (3.3), we find

(3.15)  ∥e(t)∥^{2} ≤ Ch^{10}(∥ϕ∥_{5}^{2} + ∥ψ∥_{3}^{2} + t^{−1} ∫_{0}^{t}(∥u_{t}∥_{3}^{2} + τ^{2}∥u_{tt}∥_{3}^{2}) dτ).

But under our assumptions on ϕ and ψ, we have ϕ, ψ ∈ Ḣ^{(5−ϵ)/2} and hence, by Lemma 2.4,

\begin{aligned}∥u_{t}(τ)∥_{3} + τ∥u_{tt}(τ)∥_{3} & ≤ C_{ϵ}(∥ϕ∥_{3} + ∥ψ∥_{1} + τ^{−(1+ϵ)/4}(|ϕ|_{(5−ϵ)/2} + |ψ|_{(5−ϵ)/2})) \\ & ≤ Cτ^{−(1+ϵ)/4}(∥ϕ∥_{5} + ∥ψ∥_{3}), \end{aligned}

which, inserted into (3.15) gives (3.14).

In view of the system formulation (2.2) of our model problem (1.1) and the estimates of Lemma 2.3, it appears natural to attempt to derive an error estimate for u_{h,t}, which is uniform down to t = 0. In the following theorem such an estimate is derived which requires ϕ, ψ ∈ Ḣ^{r}, thus with higher regularity for ψ and (if r > 2) more boundary conditions on ϕ than in Theorem 3.1. Comparing with Lemmas 2.3 and 2.4 these extra assumptions are not surprising.

Theorem 3.4. Let ϕ, ψ ∈ Ḣ^{r} and assume that

∥ϕ_{h} − ϕ∥ ≤ Ch^{r}|ϕ|_{r}, ∥ψ_{h} − ψ∥ ≤ Ch^{r}|ψ|_{r}.

Then there is a constant C such that

∥u_{h}(t) − u(t)∥ + ∥u_{h,t}(t) − u_{t}(t)∥ ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}), for t ≥ 0.

Proof. We again note that, without restricting generality, we may assume that ϕ_{h} = P_{h}ϕ, ψ_{h} = P_{h}ψ. In fact, by Lemma 2.5, the additional error caused by the general choice may be bounded by

C(∥ϕ_{h} − P_{h}ϕ∥ + ∥ψ_{h} − P_{h}ψ∥) ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}).

Theorem 3.1 already bounds ∥e∥ in the desired way. In order to estimate ∥e_{t}∥ we differentiate (3.3) once to obtain

T_{h}e_{tt} + αe_{t} = −(T_{h} − T)u_{tt} − e, t > 0,

and note also that T_{h}e_{t}(0) = 0. Hence by Lemma 3.2 (with β = 0, so that C is independent of the length of the time interval)

∥e_{t}(t)∥ ≤ C sup_{0<τ<t}(∥(T_{h} − T)u_{tt}(τ)∥ + τ∥(T_{h} − T)u_{ttt}(τ)∥ + ∥e(τ)∥ + τ∥e_{t}(τ)∥).

By Theorem 3.1 we have

∥e(τ)∥ + τ∥e_{t}(τ)∥ ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r−2}).

Further, using the differential equation, and Lemma 2.3,

\begin{aligned}∥(T_{h} − T)u_{tt}(τ)∥ & ≤ Ch^{r}∥u_{tt}(τ)∥_{r−2} ≤ Ch^{r}(∥u_{t}(τ)∥_{r} + ∥u(τ)∥_{r}) \\ & ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}), \end{aligned}

and similarly

τ∥(T_{h} − T)u_{ttt}(τ)∥ ≤ Ch^{r}τ(∥u_{tt}(τ)∥_{r} + ∥u_{t}(τ)∥_{r}) ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}),

which completes the proof.

We finally remark that the higher order time derivatives of the error can also be estimated under the same regularity assumptions as in Theorem 3.4, but with negative powers of t occurring in the bounds.

4. Rational approximation of analytic semigroups. In this section we discuss the approximation of an analytic semigroup by a rational function of its generator. In the following section we shall combine this with our previous discussion of the spatially semidiscrete problem (2.5) and obtain error bounds for a class of completely discrete schemes for the strongly damped wave equation (1.1).

The technique that we employ here is similar that that of Le Roux [8]. The results of [8], however, do not apply directly to the non-parabolic problem (1.1). For earlier related work see also Fujita and Mizutani [4].

Let A be a densely defined linear operator in a Banach space X, with norm ∥ · ∥, such that the resolvent set ρ(A) of A satisfies, for some θ_{0} ∈ (0, \frac{π}{2} ),

(4.1)  ρ(A) ⊃ Σ = {z ∈ C : θ_{0} ≤ | arg z| ≤ π} ∪ {0},

and such that

(4.2)  ∥(zI − A)^{−1}∥ ≤ \frac{M}{|z|},  for z ∈ Σ, z ≠ 0.

Then −A is the infinitesimal generator of an analytic semigroup {exp(−tA)}_{t≥0}, see Pazy [11, Theorem 2.5.2 and Section 2.6].

Let r(z) be a rational function approximating e^{−z} to order p + 1 ≥ 2, i.e.,

(4.3)  r(z) = e^{−z} + O(z^{p+1}),  as z → 0.

We also assume that r(z) satisfies the stability requirement

(4.4)  |r(z)| ≤ 1,  for Re z ≥ 0.

Under the latter assumption we can write

r(z) = γ + ∑_{l,m} γ_{l,m}(α_{m} − z)^{−l},

where the sum is finite, γ = r(∞) and the poles α_{m} lie in the left half-plane. We define for k > 0

r(kA) = γI + ∑_{l,m} γ_{l,m}(α_{m}I − kA)^{−l}.

This is a bounded linear operator on X, since α_{m}/k ∈ Σ. We want to show that the iterated operator r(kA)^{n} approximates exp(−t_{n}A) for t_{n} = nk > 0. We first formulate a simple consequence of (4.3) and (4.4).

Lemma 4.1. Assume that (4.3) and (4.4) hold. Then for any θ_{0} ∈ (0, \frac{π}{2} ) and R > 0 there are positive numbers C, c such that

|r(z)^{n} − e^{−nz}| ≤ Cn|z|^{p+1}e^{−cn|z|}, for |z| ≤ R, | arg z| ≤ θ_{0}, n ≥ 1.

Proof. We first note that we can find C > 0 such that

(4.5)  |r(z) − e^{−z}| ≤ C|z|^{p+1},  for |z| ≤ R, Re z > 0.

By (4.3) this holds for some small R, and in view of (4.4) then also for arbitrary R > 0. We next observe that

(4.6)  |e^{−z}| ≤ e^{−c|z|},  for | arg z| ≤ θ_{0},

if c ≤ cos θ_{0}. We may then select c > 0 such that

(4.7)  |r(z)| ≤ e^{−c|z|},  for |z| ≤ R, | arg z| ≤ θ_{0}.

This follows from (4.5) and (4.6) for some small R, and then for arbitrary R by noting that, by (4.4) and the maximum principle, we have |r(z)| < 1 for Re z > 0. Hence

\begin{aligned}|r(z)^{n} − e^{−nz}| & = |(r(z) − e^{−z}) ∑_{j=0}^{n−1} r(z)^{j}e^{−(n−1−j)z}| \\ & ≤ C|z|^{p+1}ne^{−(n−1)c|z|} ≤ Cn|z|^{p+1}e^{−cn|z|}, \end{aligned}

and the lemma is proved.

We can now show that if v ∈ D(A^{p}), then

∥[r(kA)^{n} − exp(nkA)]v∥ ≤ Ck^{p}∥A^{p}v∥.

This is a consequence of the following theorem.

Theorem 4.2. Under the hypotheses (4.1)–(4.4) on A and r(z), there is a constant C such that

∥[r(kA)^{n} − exp(−nkA)]A^{−j}∥ ≤ C k^{j}, for 1 ≤ j ≤ p, n ≥ 1, 0 < k < 1.

The constant is the same for all A satisfying (4.1) and (4.2).

Proof. Let θ_{0} be as in (4.1). Since the rational function r(kz)^{n}z^{−j} vanishes at infinity, the operator r(kA)^{n}A^{−j} may be represented by Cauchy’s integral formula, cf. [11, p. 69]. Using also the contour integral representation of exp(−t_{n}A), we thus have

[r(kA)^{n} − exp(−nkA)]A^{−j} = \frac{1}{2πi} ∫_{Γ} F_{n}(kz)z^{−j}(zI − A)^{−1} dz,

where F_{n}(z) = r(z)^{n} − e^{−nz} and Γ = Γ_{+} ∪ Γ_{−}, with Γ_{±} = {ρe^{±iθ_{0}} : 0 ≤ ρ < ∞}, is the boundary of Σ traversed in the positive sense.

Using (4.2) we get

\begin{aligned}∥ ∫_{Γ_{+}} F_{n}(kz)z^{−j}(zI − A)^{−1} dz∥ & ≤ M ∫_{0}^{∞} |F_{n}(kρe^{iθ_{0}})| \frac{dρ}{ρ^{j+1}} \\ & ≤ Mk^{j} ∫_{0}^{∞} |F_{n}(ρe^{iθ_{0}})| \frac{dρ}{ρ^{j+1}} . \end{aligned}

Here, by Lemma 4.1, we have

∫_{0}^{1} |F_{n}(ρe^{iθ_{0}})| \frac{dρ}{ρ^{j+1}} ≤ Cn ∫_{0}^{1} ρ^{p−j}e^{−cnρ} dρ ≤ C,

and, in view of (4.4),

∫_{1}^{∞} |F_{n}(ρe^{iθ_{0}})| \frac{dρ}{ρ^{j+1}} ≤ 2 ∫_{1}^{∞} \frac{dρ}{ρ^{j+1}} = C.

The integral along Γ_{−} is estimated in the same way. This completes the proof.

We next show that under the additional assumption that |r(∞)| < 1, we have, for any “nonsmooth” initial data v ∈ X,

∥[r(kA)^{n} − exp(nkA)]v∥ ≤ Ck^{p}t_{n}^{−p}∥v∥.

We will need the following result about the behavior of r(z)^{n} for large z.

Lemma 4.3. Let the rational function r be such that(4.4) holds and assume that |r(∞)| < 1. Then for any R > 0 and θ_{0} ∈ (0, \frac{π}{2} ) there are positive numbers C, c such that, with γ = r(∞),

|r(z)^{n} − γ^{n}| ≤ C|z|^{−1}e^{−cn}, for |z| ≥ R, | arg z| ≤ θ_{0}, n ≥ 1.

Proof. By the maximum principle we have |r(z)| < 1 for Re z > 0. Using also the fact that the rational function r(z) − γ is zero at infinity, we obtain

|r(z) − γ| ≤ C|z|^{−1},
|r(z)| ≤ e^{−c},

for |z| ≥ R, | arg z| ≤ θ_{0}. Hence

|r(z)^{n} − γ^{n}| = |(r(z) − γ) ∑_{j=0}^{n−1} r(z)^{j}γ^{n−1−j}| ≤ C|z|^{−1}ne^{−(n−1)c} ≤ C|z|^{−1}e^{−cn}.

We are now ready to state and prove the following result.

Theorem 4.4. In addition to the assumptions of Theorem 4.2 we assume that |r(∞)| < 1. Then there is a constant C such that

∥r(kA)^{n} − exp(−nkA)∥ ≤ Ck^{p}t_{n}^{−p}, for k > 0, t_{n} = nk > 0.

The constant is the same for all A satisfying (4.1) and (4.2).

Proof. Let F_{n}(z) = r(z)^{n} − e^{−nz}, γ = r(∞) and define G_{n}(z) = F_{n}(z) − γ^{n}z/(1 + z). Since |γ| < 1 and kA(I + kA)^{−1} is bounded, we have

∥γ^{n}kA(I + kA)^{−1}∥ ≤ C|γ|^{n} ≤ \frac{C}{n^{p}} = Ck^{p}t_{n}^{−p},

and it remains to estimate G_{n}(kA). The rational function r(kz)^{n} − γ^{n}kz/(1 + kz) is zero at infinity, so that

G_{n}(kA) = \frac{1}{2πi} ∫_{Γ} G_{n}(kz)(zI − A)^{−1} dz.

In view of (4.2) we have

∥ ∫_{Γ_{+}} G_{n}(kz)(zI − A)^{−1} dz∥ ≤ M ∫_{0}^{∞} |G_{n}(kρe^{iθ_{0}})|\frac{dρ}{ρ} = M ∫_{0}^{∞} |G_{n}(ρe^{iθ_{0}})|\frac{dρ}{ρ}.

By Lemma 4.1 and since |z/(1 + z)| ≤ |z| for Re z > 0, we have

∫_{0}^{1} |G_{n}(ρe^{iθ_{0}})|\frac{dρ}{ρ} ≤ ∫_{0}^{1} |F_{n}(ρe^{iθ_{0}})|\frac{dρ}{ρ} + |γ|^{n} ∫_{0}^{1} dρ

≤ Cn ∫_{0}^{∞} ρ^{p}e^{−cnρ} dρ + |γ|^{n}

≤ \frac{C}{n^{p}} ∫_{0}^{∞} ρ^{p}e^{−ρ} dρ + |γ|^{n} ≤ \frac{C}{n^{p}} .

Next we write G_{n}(z) = (r(z)^{n} − γ^{n}) + γ^{n}/(1 + z) − e^{−nz} and use Lemma 4.3 to get

∫_{1}^{∞} |G_{n}(ρe^{iθ_{0}})|\frac{dρ}{ρ} ≤ Ce^{−nc} ∫_{1}^{∞} \frac{dρ}{ρ^{2}} + |γ|^{n} ∫_{1}^{∞} \frac{dρ}{ρ^{2}} + ∫_{1}^{∞} e^{−cnρ} \frac{dρ}{ρ} ≤ \frac{C}{n^{p}} .

The proof is now complete.

We remark that assumption (4.4) is A-stability in the sense of Dahlquist. Clearly, this can be replaced in the above discussion by A(θ_{0})-stability, i.e., |r(z)| ≤ 1 needs to hold only in the sector | arg z| ≤ θ_{0}.

5. Completely discrete schemes. As indicated in the introduction we shall study a class of completely discrete schemes for (1.1), where the approximate solution W_{n} = (U_{n}, V_{n})^{T} ∈ S_{h} × S_{h} at discrete time t_{n} = nk is defined by the recursion

(5.1)  \begin{aligned}W_{n} & = r(kB_{h})W_{n−1},  n ≥ 1, \\ W_{0} & = w_{0h} = (ϕ_{h}, ψ_{h})^{T} , \end{aligned}

or, equivalently, W_{n} = r(kB_{h})^{n}w_{0h}. Here k > 0 is a time step and r(z) is a rational function approximating e^{−z}, and

B_{h} = [ \begin{matrix}0 & −I \\ A_{h} & αA_{h} \end{matrix} ]

is the discrete analogue of the operator B introduced in Section 2. Lemma 2.2 shows that B_{h} satisfies the hypotheses of Section 4 independently of h.

We shall first show the following smooth data result.

Theorem 5.1. Assume that r(z) satisfies conditions (4.3) and (4.4). Let ϕ, ψ ∈ Ḣ^{s}, s = max(r, 2p), and assume that

∥ϕ_{h} − ϕ∥ ≤ Ch^{r}|ϕ|_{r}, ∥ψ_{h} − ψ∥ ≤ Ch^{r}|ψ|_{r}.

Then for the solutions of (1.1) and of (5.1) we have

\begin{aligned}∥U_{n} − u(t_{n})∥ + ∥V_{n} − u_{t}(t_{n})∥ & ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r}) \\ & + Ck^{p}(|ϕ|_{2p} + |ψ|_{2p}), for t_{n} ≥ 0. \end{aligned}

Equivalently the result of the theorem can be stated as

(5.2)  ∥W_{n} − w(t_{n})∥ ≤ Ch^{r}|w_{0}|_{r} + Ck^{p}|w_{0}|_{2p},  for t_{n} ≥ 0,

where we now let | · |_{s} denote also the norm in Ḣ^{s} × Ḣ^{s}. In order to prove the latter bound we recall that Theorem 3.4 shows

(5.3)  ∥w_{h}(t) − w(t)∥ ≤ Ch^{r}|w_{0}|_{r},  for t ≥ 0,

Applying Theorem 4.2 with A replaced by B_{h} and X = S_{h} × S_{h}, we have

∥W_{n} − w_{h}(t_{n})∥ ≤ Ck^{p}∥B_{h}^{p}w_{0h}∥, for t_{n} ≥ 0.

The latter estimate is unsatisfactory in that it places a discrete norm on the discrete initial data. By means of a technique used for a parabolic problem in Baker, Bramble and Thomée [1] (see also Thomée [13, Chapter 7]) we shall obtain an estimate in terms of norms of the continuous initial values as stated in (5.2).

We begin by showing that the discrete solution operator r(kB_{h})^{n} is stable in L_{2} × L_{2}.

Lemma 5.2. There is a constant C such that

∥r(kB_{h})^{n}∥ ≤ C, for n ≥ 0, 0 < k < 1.

Proof. In the notation of the proof of Lemma 2.2 we must show the uniform boundedness of the family of 2 × 2 matrices {r(kB(λ))^{n} : n ≥ 1, 0 < k < 1, λ ≥ c_{0}}, where B(λ) is defined in (2.10). By Schur’s lemma B(λ) is orthogonally equivalent to

S(λ) = [ \begin{matrix}τ_{−}(λ) & β(λ) \\ 0 & τ_{+}(λ) \end{matrix} ] ,

where |β(λ)| is bounded by the sum of the moduli of the elements of B(λ), so that |β(λ)| ≤ Cλ for λ ≥ c_{0}. An elementary calculation shows that for an arbitrary rational function f we have

f(S(λ)) = [ \begin{matrix}f(τ_{−}(λ)) & β(λ) \frac{f(τ_{+}(λ)) − f(τ_{−}(λ))}{τ_{+}(λ) − τ_{−}(λ)} \\ 0 & f(τ_{+}(λ)) \end{matrix} ] ,

when τ_{+}(λ) ≠ τ_{−}(λ). In particular, when f(z) = r(kz)^{n}, we find by (4.4) that the diagonal elements are bounded by 1 in modulus, and hence it now suffices to show (5.4)

|β_{k,n}(λ)| = |β(λ) \frac{r(kτ_{+}(λ))^{n} − r(kτ_{−}(λ))^{n}}{τ_{+}(λ) − τ_{−}(λ)} | ≤ C, λ ≥ c_{0}, n ≥ 1, 0 < k < 1.

To see this, we first consider λ large, λ ≥ 8/α^{2}, say. Then

|τ_{+}(λ) − τ_{−}(λ)| ≥ \frac{α}{\sqrt{2}}λ ≥ c|β(λ)|, with c > 0,

and since |r(z)| ≤ 1, (5.4) follows. For the remaining case, c_{0} ≤ λ ≤ 8/α^{2}, we have c_{1} ≤ |τ_{±}(λ)| ≤ c_{2} for some c_{1}, c_{2} > 0, so that by (4.7)

|r(kτ_{±}(λ))^{n}| ≤ e^{−cnk}, with c > 0.

Hence, since now β(λ) is bounded, we conclude that

|β_{k,n}(λ)| ≤ Ck| ∑_{j=0}^{n−1} r(kτ_{−}(λ))^{n−1−j}r(kτ_{+}(λ))^{j}| ≤ Cnke^{−c(n−1)k} ≤ C,

showing the uniform boundedness when τ_{+}(λ) ≠ τ_{−}(λ), i.e., when λ ≠ 4/α^{2}, and thus, by continuity, also for λ = 4/α^{2}. The proof is now complete.

Proof of Theorem 5.1. In view of (5.3) we only have to estimate

W_{n} − w_{h}(t_{n}) = [r(kB_{h})^{n} − exp(−t_{n}B_{h})]w_{0h}.

By stability of the discrete solution operator r(kB_{h})^{n} in L_{2} × L_{2} and (2.4) there is no restriction of generality in assuming that w_{0h} = P_{h}w_{0}, where we extend the notation P_{h} to mean also the orthogonal projection of L_{2} × L_{2} onto S_{h} × S_{h}. Thus, we shall bound F_{n}P_{h}w_{0}, where F_{n} = F_{n}(kB_{h}) = r(kB_{h})^{n} − exp(−nkB_{h}).

We set

w_{0}^{k} = ∑_{kλ_{j}≤1} ŵ_{0,j}ω_{j},

where ŵ_{0,j} ∈ C^{2} are the Fourier coefficients of w_{0} with respect to the eigensystem of A (see the notation of the proof of Lemma 2.2). It is easy to verify that

(5.5)  ∥w_{0} − w_{0}^{k}∥ ≤ Ck^{p}|w_{0}|_{2p},
(5.6)  |w_{0}^{k}|_{2p} ≤ |w_{0}|_{2p},
(5.7)  |w_{0}^{k}|_{r+2j} ≤ Ck^{−j}|w_{0}|_{r},  for 0 ≤ j ≤ p.

Using the identity

P_{h} = ∑_{j=0}^{p−1} B_{h}^{−j}(P_{h}B^{−1} − B_{h}^{−1}P_{h})B^{j+1} + B_{h}^{−p}P_{h}B^{p},

we have

\begin{aligned}F_{n}P_{h}w_{0} & = ∑_{j=0}^{p−1} F_{n}B_{h}^{−j}(P_{h}B^{−1} − B_{h}^{−1}P_{h})B^{j+1}w_{0}^{k} \\ & + F_{n}B_{h}^{−p}P_{h}B^{p}w_{0}^{k} + F_{n}P_{h}(w_{0} − w_{0}^{k}). \end{aligned}

In order to bound the terms on the right we combine Theorem 4.2 (for 1 ≤ j ≤ p) and Lemma 5.2 (for j = 0) to obtain

∥F_{n}B_{h}^{−j}∥ ≤ Ck^{j}, for 0 ≤ j ≤ p.

Further, we have

B^{−1} = [ \begin{matrix}αI & T \\ −I & 0 \end{matrix} ] ,

and similarly for B_{h}^{−1}. Therefore

P_{h}B^{−1} − B_{h}^{−1}P_{h} = [ \begin{matrix}0 & P_{h}T − T_{h} \\ 0 & 0 \end{matrix} ] ,

which, in view of (2.3), implies, for f = (f_{1}, f_{2})^{T},

∥(P_{h}B^{−1} − B_{h}^{−1}P_{h})f∥ = ∥(P_{h}T − T_{h})f_{2}∥ ≤ Ch^{r}∥f∥_{r−2}.

Hence, using also (5.7), and the fact that B is a second order partial differential operator, we find

∥F_{n}B_{h}^{−j}(P_{h}B^{−1} − B_{h}^{−1}P_{h})B^{j+1}w_{0}^{k}∥ ≤ Ck^{j}∥(P_{h}B^{−1} − B_{h}^{−1}P_{h})B^{j+1}w_{0}^{k}∥
≤ Ck^{j}h^{r}∥B^{j+1}w_{0}^{k}∥_{r−2} ≤ Ch^{r}k^{j}|w_{0}^{k}|_{r+2j}
≤ Ch^{r}|w_{0}|_{r}, for 0 ≤ j ≤ p − 1.

Similarly, using (5.6), we have

∥F_{n}B_{h}^{−p}P_{h}B^{p}w_{0}^{k}∥ ≤ Ck^{p}∥B^{p}w_{0}^{k}∥ ≤ Ck^{p}|w_{0}^{k}|_{2p} ≤ Ck^{p}|w_{0}|_{2p}.

Finally, by stability and (5.5), we conclude

∥F_{n}P_{h}(w_{0} − w_{0}^{k})∥ ≤ C∥w_{0} − w_{0}^{k}∥ ≤ Ck^{p}|w_{0}|_{2p}.

Together with (5.3) these estimates show (5.2) and the theorem is proved.

Similarly to the situation for the semidiscrete problem it is possible to obtain an estimate for the error in the function values u alone under slightly reduced regularity demands on ψ. More precisely we have the following.

Theorem 5.3. Assume that r(z) satisfies (4.3) and (4.4). Let ϕ ∈ Ḣ^{s}, ψ ∈ Ḣ^{s−2} with s = max(r, 2p), and assume that

∥ϕ_{h} − ϕ∥ ≤ Ch^{r}|ϕ|_{r}, ψ_{h} = P_{h}ψ.

Then for the solutions of (1.1) and of (5.1) we have

\begin{aligned}∥U_{n} − u(t_{n})∥ & ≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r−2}) \\ & + Ck^{p}(|ϕ|_{2p} + |ψ|_{2p−2}), for t_{n} ≥ 0. \end{aligned} (5.8)

Proof. The proof will parallel that of Theorem 5.1 and we shall merely give the essential changes. The error in the semidiscrete solution is now taken care of by Theorem 3.1. By the same argument as in the proof of Theorem 5.1 we may assume without loss of generality that ϕ_{h} = P_{h}ϕ. Letting as before W_{n} − w_{h}(t_{n}) = F_{n}P_{h}w_{0}, we shall estimate

M_{h}F_{n}P_{h}w_{0} = M_{h}F_{n}M_{h}^{−1}M_{h}P_{h}w_{0} ≡ F̃_{n}M_{h}P_{h}w_{0}, with M_{h} = [ \begin{matrix}I & 0 \\ 0 & T_{h} \end{matrix} ] ,

in L_{2} × L_{2}, thus in effect estimating the second component of the error in a discrete negative norm. One easily checks that

F̃_{n} = r(kB̃_{h})^{n} − exp(−t_{n}B̃_{h}),

where

B̃_{h} = B̃(A_{h}) = M_{h}B_{h}M_{h}^{−1} = [ \begin{matrix}0 & −A_{h} \\ I & αA_{h} \end{matrix} ] .

Further, since B̃(λ) has the same eigenvalues as B(λ), the estimate for the resolvent in Lemma 2.2 holds for B̃_{h}, so that the results of Section 4 apply. Also, the stability statement of Lemma 5.2 remains valid for B̃_{h} by the same argument as for B_{h}.

We now write

(5.9) F̃_{n}M_{h}P_{h}w_{0} = F̃_{n}P_{h}Mw_{0} + F̃_{n} [ \begin{matrix}0 \\ (T_{h} − P_{h}T)ψ \end{matrix} ] ,  with M = [ \begin{matrix}I & 0 \\ 0 & T \end{matrix} ] .

By the stability in L_{2} × L_{2}, the latter term is bounded by

(5.10)  C∥(T_{h} − P_{h}T)ψ∥ ≤ Ch^{r}|ψ|_{r−2}.

For the first term on the right of (5.9) we write as in the proof of Theorem 5.1,

\begin{aligned}F̃_{n}P_{h}Mw_{0} & = ∑_{j=0}^{p−1} F̃_{n}B̃_{h}^{−j}(P_{h}B̃^{−1} − B̃_{h}^{−1}P_{h})B̃^{j+1}Mw_{0}^{k} \\ & + F̃_{n}B̃_{h}^{−p}P_{h}B̃^{p}Mw_{0}^{k} + F̃_{n}P_{h}M(w_{0} − w_{0}^{k}), \end{aligned} (5.11)

where now

B̃ = MBM^{−1} = [ \begin{matrix}0 & −A \\ I & αA \end{matrix} ] .

For the terms in the sum we have by Theorem 4.2 and Lemma 5.2, and since

P_{h}B̃^{−1} − B̃_{h}^{−1}P_{h} = [ \begin{matrix}0 & 0 \\ T_{h} − P_{h}T & 0 \end{matrix} ] ,

and by (5.7), that

∥F̃_{n}B̃_{h}^{−j}(P_{h}B̃^{−1} − B̃_{h}^{−1}P_{h})B̃^{j+1}Mw_{0}^{k}∥ ≤ Ck^{j}∥(P_{h}B̃^{−1} − B̃_{h}^{−1}P_{h})B̃^{j+1}Mw_{0}^{k}∥
≤ Ck^{j}h^{r}∥B̃^{j+1}Mw_{0}^{k}∥_{r−2}
≤ Ch^{r}k^{j}|Mw_{0}^{k}|_{r+2j}
≤ Ch^{r}(|ϕ|_{r} + |ψ|_{r−2}).

Finally, as in the proof of Theorem 5.1, the last two terms on the right of (5.11) are easily bounded by

Ck^{p}(|ϕ|_{2p} + |ψ|_{2p−2}).

This completes the proof of the theorem.

We note that in the proof of Theorem 5.3 we have, in fact, estimated also the discrete negative norm

|u_{h,t}(t_{n}) − V_{n}|_{−2,h} = ∥T_{h}(u_{h,t}(t_{n}) − V_{n})∥.

This is somewhat analogous to the case σ = 2 in Lemma 2.3. A corresponding estimate for the semidiscrete solution may be easily obtained directly by solving (3.3) for T_{h}e_{t}, and we may thus bound |V_{n} − u_{t}(t_{n})|_{−2,h} by the right-hand side of (5.8).

We also remark that the assumption ψ_{h} = P_{h}ψ in Theorem 5.3 may be weakened to the condition

|ψ_{h} − ψ|_{−2,h} ≤ Ch^{r}|ψ|_{r−2}.

For, the error in M_{h}W_{n} caused by such a perturbation is bounded by

∥M_{h} r(kB_{h})^{n} [ \begin{matrix}0 \\ ψ_{h} − ψ \end{matrix} ] ∥ = ∥r(kB̃_{h})^{n}M_{h} [ \begin{matrix}0 \\ ψ_{h} − ψ \end{matrix} ] ∥ ≤ C|ψ_{h} − ψ|_{−2,h}.

This also applies to the error in the time derivative just mentioned. We shall not insist on the details.

We conclude by an observation relating to initial data of low regularity. Recall that for time discretization schemes with |r(∞)| < 1, Theorem 4.4 yields the nonsmooth data error estimate

∥U_{n} − u_{h}(t_{n})∥ + ∥V_{n} − u_{h,t}(t_{n})∥ ≤ Ck^{p}t_{n}^{−p}(∥ϕ_{h}∥ + ∥ψ_{h}∥), for t_{n} > 0.

Unfortunately, no nonsmooth data estimate for the error in the spatially semidiscrete solution is available that matches this exactly. However, in view of Theorem 3.1, we may state the following result in which the regularity assumptions are weaker than in Theorem 5.1.

Theorem 5.4. Assume that r(z) satisfies (4.3) and (4.4), and in addition that |r(∞)| < 1. Let ϕ ∈ H^{r} ∩ Ḣ^{max(r−2,1)}, ψ ∈ Ḣ^{r−2}, and assume that

∥ϕ_{h} − ϕ∥ ≤ Ch^{r}∥ϕ∥_{r}, ψ_{h} = P_{h}ψ.

Then for the solutions of (1.1) and of (5.1) we have

∥U_{n} − u(t_{n})∥ + ∥V_{n} − u_{t}(t_{n})∥ ≤ C(t_{n})(h^{r} + k^{p})(∥ϕ∥_{r} + |ψ|_{r−2}), for t_{n} > 0.

In particular, for r ≤ 4, we thus have an optimal order error estimate for positive time as long as ϕ ∈ H^{r} ∩ H_{0}^{1} and ψ ∈ H^{r−2} ∩ H_{0}^{1} if r = 3, 4 (ψ ∈ L_{2} if r = 2). For r = 5 the corresponding error estimate follows for the function values by use of (3.14).

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Stig Larsson and Vidar Thomée, Department of Mathematics, Chalmers University of Technology, S–412 96 Göteborg Sweden

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

Lars B. Wahlbin, Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14853, USA

E-mail address: wahlbin@math.cornell.edu