A Piezoelectric Euler-Bernoulli Beam with Dynamic Boundary Control: Stability and Dissipative FEM

Abstract

We present a mathematical and numerical analysis on a control model for the time evolution of a multi-layered piezoelectric cantilever with tip mass and moment of inertia, as developed by Kugi and Thull (Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, pp. 351–368, 2005). This closed-loop control system consists of the inhomogeneous Euler-Bernoulli beam equation coupled to an ODE system that is designed to track both the position and angle of the tip mass for a given reference trajectory. This dynamic controller only employs first order spatial derivatives, in order to make the system technically realizable with piezoelectric sensors. From the literature it is known that it is asymptotically stable (Kugi and Thull in Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, pp. 351–368, 2005). But in a refined analysis we first prove that this system is not exponentially stable.

In the second part of this paper, we construct a dissipative finite element method, based on piecewise cubic Hermitian shape functions and a Crank-Nicolson time discretization. For both the spatial semi-discretization and the full x−t-discretization we prove that the numerical method is structure preserving, i.e. it dissipates energy, analogous to the continuous case. Finally, we derive error bounds for both cases and illustrate the predicted convergence rates in a simulation example.

Appendices

Appendix A

The following proof is an adaption of the proof of Theorem 8.1 in [24], for the system studied here. It is included for the sake of completeness.

Proof of Theorem 4

(a)–existence: Let $\{ \hat{w}_{k} \}_{k=1}^{\infty}$ be a sequence of functions that is an orthonormal basis for H, and an orthogonal basis for V. We introduce $W_{m} := \mathit{span} \{ \hat {w}_{1}, \dots, \hat{w}_{m}\}, \forall m \in\mathbb{N}$. Furthermore, let sequences $\hat{u}_{m0}, \hat{v}_{m0} \in W_{m}$ be given so that

$$ \begin{aligned} \hat{u}_{m0} &\rightarrow \hat{u}_0 \quad \text{in } V, \\ \hat{v}_{m0} &\rightarrow\hat{v}_0 \quad \text{in } H. \end{aligned} $$

(A.1)

For a fixed $m \in\mathbb{N}$ we consider the Galerkin approximation

$$\hat{u}_m(t)= \bigl((u_m)_x(L),u_m(L), u_m\bigr) = \sum_{k=1}^{m}{d_m^k(t) \hat{w}_k}, $$

with $d_{m}^{k}(t) \in\mathbb{R}$, which solves the formulation (3.3) for all $\hat{w} \in W_{m}$:

$$ \bigl((\hat{u}_m)_{tt}, \hat{w} \bigr)_H + a(\hat{u}_m, \hat{w}) + b\bigl(( \hat{u}_m)_t, \hat{w}\bigr) + e_1( \zeta_{1,m}, \hat{w}) + e_2(\zeta_{2,m}, \hat{w})= 0, $$

(A.2)

and ζ _1,m ,ζ _2,m solve the ODE system

$$\begin{aligned} \begin{aligned} (\zeta_{1,m})_t(t) & = A_{1} \zeta_{1,m}(t) + b_{1} \,^1(\hat{u}_m)_{t} (t), \\ (\zeta_{2,m})_t(t) & = A_{2} \zeta_{2,m}(t) + b_{2} \,^2(\hat {u}_m)_{t} (t), \end{aligned} \end{aligned}$$

(A.3)

with the initial conditions

$$\begin{aligned} \hat{u}_m(0) = & \hat{u}_{m0} , \\ (\hat{u}_m)_t(0) = & \hat{v}_{m0} , \\ \zeta_{1,m}(0) = & \zeta_{0,1} , \\ \zeta_{2,m}(0) = & \zeta_{0,2}. \end{aligned}$$

This problem is a linear system of second order differential equations, with a unique solution satisfying $\hat{u}_{m} \in C^{2}([0, T]; V)$ and $\zeta_{1,m}, \zeta_{2,m} \in C^{1}([0,T];\mathbb{R}^{n})$. Next, we define an energy functional, analogous to (3.11), for the trajectory $(\hat{u}, \zeta_{1}, \zeta_{2})$:

$$\begin{aligned} \hat{E}(t; \hat{u}, \zeta_1, \zeta_2) := & \frac{1}{2} \bigl\| \hat{u}(t) \bigr\| ^2_V + \frac{k_1}{2} \bigl(^1\hat{u}(t)\bigr)^2 + \frac{k_2}{2} \bigl(^2\hat{u}(t)\bigr)^2 + \frac{1}{2} \bigl\| \hat{u}_t(t) \bigr\| ^2_H \\ &{} + \frac{1}{2} \zeta_{1}^{\top}(t) P_1 \zeta_{1}(t) + \frac{1}{2} \zeta_{2}^{\top}(t) P_2 \zeta_{2}(t) \\ = & \bigl\| \bigl(u, u_t, \zeta_1, \zeta_2, J u_{tx}(J), M u_t(L)\bigr) \bigr\| _{\mathcal{H}}. \end{aligned}$$

Taking $\hat{w} = (\hat{u}_{m})_{t}$ in (A.2) and using the smoothness of $\hat{u}_{m}, \zeta_{1,m}, \zeta_{2,m}$, a straightforward calculation yields

$$\begin{aligned} \frac{d}{dt} \hat{E}(t;\hat{u}_m, \zeta_{1,m}, \zeta_{2,m}) = & - \delta_1 \bigl(^1(\hat{u}_m)_{t}\bigr)^2 - \frac{1}{2} \bigl(\zeta_{1,m} \cdot q_1 + \tilde{\delta}_1 \bigl(^1(\hat{u}_m)_{t} \bigr) \bigr)^2 \\ &{} - \delta_2 \bigl(^2(\hat{u}_m)_{t} \bigr)^2 - \frac{1}{2} \bigl(\zeta_{2,m} \cdot q_2 + \tilde{\delta}_2\bigl(^2( \hat{u}_m)_{t}\bigr) \bigr)^2 \\ &{} - \frac{\epsilon_1}{2} (\zeta_{1,m})^{\top} P_1 \zeta_{1,m}- \frac{\epsilon_2}{2} (\zeta_{2,m})^{\top} P_2 \zeta_{2,m} \\ =:& F(t;\hat{u}_m, \zeta_{1,m}, \zeta_{2,m}) \le0, \end{aligned}$$

(A.4)

which is analogous to (3.1) for the continuous solution. Hence

$$ \hat{E}(t;\hat{u}_m,\zeta_{1,m}, \zeta_{2,m}) \le\hat{E}(0;\hat{u}_m,\zeta_{0,1}, \zeta_{0,2}), \quad t \ge0, $$

which implies

$$ \begin{aligned} &\{\hat{u}_m \}_{m \in\mathbb{N}} \quad \text{is bounded in } C\bigl([0,T]; V\bigr), \\ &\bigl\{ (\hat{u}_m)_t \bigr\} _{m \in\mathbb{N}} \quad \text{is bounded in } C\bigl([0,T]; H\bigr), \\ &\{\zeta_{1,m} \}_{m \in\mathbb{N}}, \qquad \{\zeta_{2,m} \}_{m \in\mathbb{N}} \quad \text{are bounded in } C\bigl([0,T]; \mathbb{R}^n \bigr). \end{aligned} $$

(A.5)

Due to these boundedness results, it holds $\forall\hat{w} \in V$:

$$\begin{aligned} \bigl|a\bigl(\hat{u}_m(t), \hat{w}\bigr) + b\bigl((\hat{u}_m)_t(t), \hat{w}\bigr) + e_1\bigl(\zeta_{1,m}(t), \hat{w}\bigr) + e_2\bigl(\zeta_{2,m}(t), \hat{w}\bigr)\bigr| \le& D_1 \| \hat{w} \|_{V}, \end{aligned}$$

a.e. on (0,T), with some constant D ₁>0 which does not depend on m. Now, let $m \in\mathbb{N}$ be fixed. Furthermore, let $\hat{w} \in V$, and $\hat{w} = \hat{w}_{1} + \hat{w}_{2}$, such that $\hat{w}_{1} \in W_{m}$ and $\hat{w}_{2}$ orthogonal to W _m in H. Then we obtain from (A.2):

$$\begin{aligned} \bigl((\hat{u}_m)_{tt}, \hat{w}\bigr)_H = & \bigl((\hat{u}_m)_{tt}, \hat{w}_1 \bigr)_H \\ = & -a(\hat{u}_m, \hat{w}_1) - b\bigl(( \hat{u}_m)_t, \hat{w}_1\bigr) - e_1(\zeta_{1,m}, \hat{w}_1) - e_2( \zeta_{2,m}, \hat{w}_1) \\ \le& D_1 \|\hat{w}_1 \|_V \le D_1 \|\hat{w} \|_V. \end{aligned}$$

This implies that also $(\hat{u}_{m})_{tt}$ is bounded in L ²(0,T;V′). Furthermore, from (A.3) it trivially follows that $\{(\zeta_{1,m})_{t} \}_{m \in\mathbb{N}}$ and $(\{\zeta_{2,m})_{t} \}_{m \in\mathbb{N}}$ are also bounded in $L^{2}(0,T; \mathbb{R}^{n})$.

According to the Eberlein-S̆muljan Theorem, there exist subsequences $\{ \hat{u}_{m_{l}}\}_{l \in\mathbb{N}}$, $\{ \zeta_{1,m_{l}}\}_{l \in \mathbb{N}}$, $\{ \zeta_{2,m_{l}}\}_{l \in\mathbb{N}}$, and $\hat{u} \in L^{2}(0,T ; V)$, with $\hat{u}_{t} \in L^{2}(0,T ; H)$, $\hat{u}_{tt} \in L^{2}(0,T ; V')$, and $\zeta_{1}, \zeta_{2} \in H^{1}(0,T;\mathbb{R}^{n})$ such that

$$\begin{aligned} \{\hat{u}_{m_l} \} \rightharpoonup& u \quad \text{in }L^{2}(0,T; V), \\ \bigl\{ (\hat{u}_{m_l})_t \bigr\} \rightharpoonup& u_t \quad \text{in }L^{2}(0,T; H), \\ \bigl\{ (\hat{u}_{m_l})_{tt} \bigr\} \rightharpoonup& u_{tt} \quad \text{in } L^{2}\bigl(0,T; V'\bigr), \\ \{\zeta_{1,m_l} \} \rightharpoonup& \zeta_1 \quad \text{in } L^{2}\bigl(0,T; \mathbb{R}^n\bigr), \\ \{\zeta_{2,m_l} \} \rightharpoonup& \zeta_2 \quad \text{in } L^{2}\bigl(0,T;\mathbb{R}^n\bigr), \\ \bigl\{ (\zeta_{1,m_l})_t \bigr\} \rightharpoonup& ( \zeta_1)_t \quad \text{in } L^{2}\bigl(0,T; \mathbb{R}^n\bigr), \\ \bigl\{ (\zeta_{2,m_l})_t \bigr\} \rightharpoonup& ( \zeta_2)_t \quad \text{in } L^{2}\bigl(0,T; \mathbb{R}^n\bigr). \end{aligned}$$

(A.6)

Therefore, passing to the limit in (A.2) and (A.3), we see that $\hat{u}$ and ζ ₁,ζ ₂ solve (3.4) and (3.5).

(b)–additional regularity: From $\zeta_{1}, \zeta_{2} \in H^{1}(0, T; \mathbb{R}^{n})$ follows the continuity of the controller functions, i.e. (3.7b). It is easily seen from the construction of the weak solution and (A.5) that $\hat{u}$ satisfies (3.7a). Equation (3.7c) follows immediately due to Lemma 2, after, possibly, a modification on a set of measure zero. Equation (3.7d) follows from Lemma 2 and the ‘Duality Theorem’ (see [24], Chap. 6.2, p. 29) which states: for all θ∈(0,1), it holds

$$[X, Y]_{\theta}' = \bigl[Y', X'\bigr]_{1 - \theta}. $$

(a)–initial conditions, uniqueness: It remains to show that $\hat{u}$, ζ ₁, and ζ ₂ satisfy the initial conditions. For this purpose, we integrate by parts (in time) in (3.4), with $\hat{w} \in C^{2}([0, T]; V)$ such that $\hat{w}(T) = 0$ and $\hat{w}_{t}(T) = 0$:

$$\begin{aligned} & \int_{0}^{T}{ \bigl[ (\hat{u}, \hat{w}_{tt})_H + a(\hat{u}, \hat{w}) + b( \hat{u}_t, \hat{w}) + e_1(\zeta_1, \hat{w}) + e_2(\zeta_2, \hat{w}) \bigr] \, d \tau} \\ &\quad = -\bigl(\hat{u}(0), \hat{w}_t(0)\bigr)_H + {_{V'}\bigl\langle \hat{u}_t(0), \hat{w}(0)\bigr\rangle _{V}}. \end{aligned}$$

(A.7)

Similarly, for a fixed m it follows from (A.2):

$$\begin{aligned} & \int_{0}^{T}{ \bigl[ (\hat{u}_m, \hat{w}_{tt})_H + a( \hat{u}_m, \hat{w}) + b\bigl((\hat{u}_m)_t, \hat{w}\bigr) + e_1(\zeta_{1m}, \hat{w}) + e_2(\zeta_{2m}, \hat{w}) \bigr] \, d \tau} \\ &\quad = -\bigl(\hat{u}_{m0}, \hat{w}_t(0)\bigr)_H + \bigl(\hat{v}_{m0}, \hat{w}(0)\bigr)_H. \end{aligned}$$

(A.8)

Due to (A.1) and (A.6), passing to the limit in (A.8) along the convergent subsequence $\{\hat{u}_{m_{l}}\}$ gives

$$\begin{aligned} & \int_{0}^{T}{ \bigl[ (\hat{u}, \hat{w}_{tt})_H + a(\hat{u}, \hat{w}) + b( \hat{u}_t, \hat{w}) + e_1(\zeta_1, \hat{w}) + e_2(\zeta_2, \hat{w}) \bigr] \, d \tau} \\ &\quad = -\bigl(\hat{u}_0, \hat{w}_t(0)\bigr)_H + \bigl(\hat{v}_0, \hat{w}(0)\bigr)_H. \end{aligned}$$

(A.9)

Comparing (A.7) with (A.9), implies $\hat {u}(0) = \hat{u}_{0}$ and $\hat{u}_{t}(0) = \hat{v}_{0}$. Analogously we obtain ζ ₁(0)=ζ _0,1 and ζ ₂(0)=ζ _0,2.

In order to show uniqueness, let $(\hat{u}, \zeta_{1}, \zeta_{2})$ be a solution to (3.4) and (3.5) with zero initial conditions. Let s∈(0,T) be fixed, and set

$$ \hat{U}(t):= \left \{ \begin{array}{l@{\quad}l} \int_{t}^{s}{\hat{u}(\tau) \,d\tau}, & t < s, \\ 0, & t \ge s, \end{array} \right . $$

and

$$ Z_i(t):= \int_{0}^{t}{ \zeta_i(\tau) \,d\tau}, $$

for i=1,2. Integrating (3.5) over (0,t) yields with (1.7)

$$\begin{aligned} \frac{1}{2} \frac{d}{dt}\bigl(Z_i^{\top} P_i Z_i\bigr) (t) = & - \frac{1}{2} \epsilon_i Z_i^{\top}(t) P_i Z_i(t) - \frac{1}{2} \bigl(q_i \cdot Z_i(t) + \tilde{\delta}_i \bigl(^i \hat{u}(t)\bigr)\bigr)^2 \\ &{} + (d_i - \delta_i) \bigl(^i \hat{u}(t)\bigr)^2 + Z_i(t) \cdot c_i \bigl(^i \hat{u}(t)\bigr), \end{aligned}$$

(A.10)

for 0≤t≤T, i=1,2. Integrating (3.4) with $\hat{w} = \hat{U}$ over [0,T], and performing partial integration in time, yields

$$\begin{aligned} & \int_{0}^{s}{\bigl( \hat{u}_{t}(\tau), \hat{u}(\tau)\bigr)_{H} - a\bigl( \hat{U}_t(\tau), \hat{U}(\tau)\bigr) + b\bigl(\hat{u}(\tau), \hat{u}( \tau)\bigr) \,d\tau} \\ &\quad {} + \sum_{i = 1}^{2} \int _{0}^{s}{Z_i(\tau) \cdot c_i \bigl(^i\hat{u}(\tau)\bigr) \,d\tau}= 0. \end{aligned}$$

(A.11)

From (A.10) and (A.11) follows

$$\begin{aligned} & \int_{0}^{s}{\frac{d}{dt} \Biggl( \frac{1}{2} \bigl\| \hat{u}(\tau)\bigr\| ^2_{H} - \frac{1}{2} a\bigl(\hat{U}(\tau), \hat{U}(\tau)\bigr) + \frac{1}{2} \sum_{i=1}^2{Z_i^{\top}( \tau) P_i Z_i(\tau)} \Biggr) \,d\tau} \\ &\quad = - \sum_{i = 1}^{2} \int _{0}^{s}{ \biggl( \delta_i \bigl(^i\hat{u}(\tau)\bigr)^2 + \frac{\epsilon_i}{2} Z_i^{\top}(\tau)P_i Z_i(\tau) + \frac{1}{2} \bigl(q_i \cdot Z_i(\tau) + \tilde{\delta}_i \bigl(^i \hat{u}\bigr) (\tau) \bigr)^2 \biggr) \,d\tau}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{2} \bigl\| \hat{u}(s)\bigr\| ^2_{H} + \frac{1}{2} a\bigl(\hat{U}(0), \hat{U}(0)\bigr) + \sum _{i=1}^2{\frac{1}{2} Z_i^{\top}(s) P_i Z_i(s)} \le& 0. \end{aligned}$$

The matrices P _j ,j=1,2 are positive definite, and the bilinear form a(.,.) is coercive. Hence $\hat{u}(s) = 0$, $\hat{U}(0) = 0$, and Z _i (s)=0. Since s∈(0,T) was arbitrary, $\hat{u} \equiv0$, ζ _i ≡0, i=1,2 follows. □

Before the proof of the continuity in time of the weak solution, a definition and a lemma will be stated.

Definition 1

Let Y be a Banach space. Then

$$\begin{aligned} C_w\bigl([0, T]; Y\bigr) := & \bigl\{ w \in L^{\infty}(0,T;Y) \colon\forall f \in Y' \\ & t \mapsto\bigl(f,w(t)\bigr) \text{ is continuous on } [0, T]\bigr\} , \end{aligned}$$

denotes the space of weakly continuous functions with values in Y.

The following lemma was stated and proven in [24] (Chap. 8.4, p. 275).

Lemma 3

Let X, Y be Banach spaces, X⊂Y with continuous injection, X reflexive. Then

$$L^{\infty}(0, T; X) \cap C_w(0, T; Y) = C_w(0, T; X). $$

Proof of Theorem 5

This proof is an adaption of standard strategies to the situation at hand (cf. Sect. 8.4 in [24] and Sect. 2.4 in [36]). Using Lemma 3 with X=V, Y=H, we conclude from (3.7a), (3.7c) that $\hat{u} \in C_{w}([0, T]; V)$. Similarly, (3.7a) and (3.7d) imply $\hat{u}_{t} \in C_{w}([0, T]; H)$.

Next, we take the scalar cut-off function $O_{I} \in C^{\infty}(\mathbb {R})$ such that it equals one on some interval I⋐[0,T], and zero on $\mathbb{R} \setminus[0, T]$. Then the functions $O_{I} \hat{u} : \mathbb{R} \rightarrow V$ and $O_{I} \zeta_{1},O_{I} \zeta_{2} : \mathbb{R} \rightarrow\mathbb{R}^{n}$ are compactly supported. Let $\eta^{\epsilon} : \mathbb{R} \rightarrow\mathbb{R}$ be a standard mollifier in time. Then we define

$$\begin{aligned} \hat{u}^{\epsilon} :=& \eta^{\epsilon} \ast O_{I} \hat{u} \in C^{\infty}_c(\mathbb{R}, V), \\ \zeta_1^{\epsilon} :=& \eta^{\epsilon} \ast O_{I} \zeta_1 \in C^{\infty}_c\bigl( \mathbb{R}, \mathbb{R}^n\bigr), \\ \zeta_2^{\epsilon} :=& \eta^{\epsilon} \ast O_{I} \zeta_2 \in C^{\infty}_c\bigl( \mathbb{R}, \mathbb{R}^n\bigr). \end{aligned}$$

Now $\zeta_{1}^{\epsilon}$ and $\zeta_{2}^{\epsilon}$ converge uniformly on I to ζ ₁ and ζ ₂, respectively. Moreover, $\hat{u}^{\epsilon}$ converges to $\hat{u}$ in V, and $\hat {u}^{\epsilon}_{t}$ to $\hat{u}_{t}$ in H a.e. on I. Then, $\hat{E}(t;\hat{u}^{\epsilon},\zeta_{1}^{\epsilon}, \zeta _{2}^{\epsilon})$ converges to $\hat{E}(t;\hat{u},\zeta_{1}, \zeta_{2})$ a.e. on I as well. Since $\hat{u}^{\epsilon}, \zeta_{1}^{\epsilon}, \zeta_{2}^{\epsilon}$ are smooth, a straightforward calculation on I yields

$$\begin{aligned} \frac{d}{dt} \hat{E}\bigl(t;\hat{u}^{\epsilon}, \zeta_1^{\epsilon}, \zeta_2^{\epsilon}\bigr) = & F\bigl(t;\hat{u}^{\epsilon},\zeta_1^{\epsilon}, \zeta_2^{\epsilon}\bigr), \end{aligned}$$

(A.12)

with F defined in (A.4). Passing to the limit in (A.12) as ϵ→0

$$\begin{aligned} \frac{d}{dt} \hat{E}(t; \hat{u}, \zeta_1, \zeta_2) = & F(t;\hat{u},\zeta_1, \zeta_2) \end{aligned}$$

(A.13)

holds in the sense of distributions on I. Since I was arbitrary, (A.13) holds on all compact subintervals of (0,T). Now $t \mapsto\hat{E}(t; \hat{u}, \zeta_{1}, \zeta_{2})$ is an integral of an L ¹-function (note that the input functions of F satisfy: $^{1}\hat{u}_{t}, ^{2}\hat{u}_{t} \in L^{2}(0, T)$), so it is absolutely continuous.

For a fixed t, let lim _n→+∞ t _n =t and let the sequence χ _n be defined by

$$\begin{aligned} \chi_n := & \frac{1}{2} \bigl\| \hat{u}(t) - \hat{u}(t_n)\bigr\| ^2_V + \frac{1}{2} \bigl\| \hat{u}_t(t) - \hat{u}_t(t_n) \bigr\| ^2_H \\ &{} + \frac{k_1}{2} \bigl(^1\hat{u}(t)-\,^1 \hat{u}(t_n)\bigr)^2 + \frac{k_2}{2} \bigl(^2\hat{u}(t)-\,^2\hat{u}(t_n) \bigr)^2 \\ &{} + \frac{1}{2} \bigl(\zeta_{1}(t)-\zeta_{1}(t_n) \bigr)^{\top} P_1 \bigl(\zeta_{1}(t) - \zeta_{1}(t_n)\bigr) \\ &{} + \frac{1}{2} \bigl(\zeta_{2}(t)-\zeta_{2}(t_n) \bigr)^{\top} P_2 \bigl(\zeta_{2}(t) - \zeta_{2}(t_n)\bigr). \end{aligned}$$

Then

$$\begin{aligned} \chi_n = & \hat{E}(t;\hat{u}, \zeta_1, \zeta_2) + \hat{E}(t_n;\hat{u}, \zeta_1, \zeta_2) - \bigl(\hat{u}(t), \hat{u}(t_n) \bigr)_V - \bigl(\hat{u}_t(t), \hat{u}_t(t_n) \bigr)_H \\ &{} - k_1 \,^1\hat{u}(t) ^1 \hat{u}(t_n) - k_2 \,^2\hat{u}(t) ^2\hat{u}(t_n) - \zeta_{1}(t)^{\top} P_1 \zeta_{1}(t_n) - \zeta_{2}(t) ^{\top} P_2 \zeta_{2}(t_n). \end{aligned}$$

Due to the t-continuity of the energy function, weak continuity of $\hat{u}, \hat{u}_{t}$, and continuity of ζ ₁,ζ ₂, it follows

$$\lim_{n \rightarrow+\infty}{\chi_n} = 0. $$

Finally, it follows that

$$\begin{aligned} \lim_{n \rightarrow\infty}{ \bigl\| \hat{u}_t(t) - \hat{u}_t(t_n) \bigr\| ^2_H} = &0 , \\ \lim_{n \rightarrow\infty}{ \bigl\| \hat{u}(t) - \hat{u}(t_n) \bigr\| ^2_V} = &0, \end{aligned}$$

which proves the theorem. □

Appendix B

Proof of Theorem 8

First we obtain from (3.21) and (3.22) (written in the style of (3.3)):

$$\begin{aligned} & \frac{u^{n+1} - u^{n}}{\Delta t} = \frac{v^{n+1} + v^{n}}{2}, \end{aligned}$$

(B.1)

$$\begin{aligned} & \int^{L}_{0} {\mu\frac{v^{n+1} - v^{n}}{\Delta t} w_h \,dx} + \int^{L}_{0}{\varLambda \frac{u^{n+1}_{xx} + u^{n}_{xx}}{2} (w_{h})_{xx} \,dx} \\ &\quad {} + M \frac{v^{n+1}(L) - v^{n}(L)}{\Delta t} w_h(L) + J \frac {v^{n+1}_{x} (L) - v^{n}_{x} (L)}{\Delta t} (w_h)_{x}(L) \\ &\quad {} + k_1 \frac{u^{n+1}_x(L) + u^{n}_x(L) }{2} (w_h)_x(L) + k_2 \frac{u^{n+1}(L) + u^{n}(L) }{2} w_h(L) \\ &\quad {} + d_1 \frac{v^{n+1}_x(L) + v^{n}_x(L) }{2} (w_h)_x(L) + d_2 \frac{v^{n+1}(L) + v^{n}(L) }{2} w_h(L) \\ &\quad {} + c_1 \cdot\frac{\zeta_1^{n+1} + \zeta_1^{n}}{2} (w_h)_x(L) + c_2 \cdot\frac{\zeta_2^{n+1} + \zeta_2^{n}}{2} w_h(L) = 0, \quad \forall w_h \in W_h. \end{aligned}$$

(B.2)

Next we multiply (B.1) by μ(v ⁿ⁺¹−v ⁿ), and integrate over [0,L] to obtain

$$\frac{1}{2} \int_{0}^{L}{\mu \bigl[\bigl(v^{n+1}\bigr)^2 - \bigl(v^{n}\bigr)^2 \bigr]\,dx} = \int_{0}^{L}{\mu\frac{u^{n+1} - u^{n}}{\Delta t} \bigl(v^{n+1} - v^{n}\bigr) \,dx}, $$

and w _h =u ⁿ⁺¹ in (B.2):

$$\begin{aligned} \frac{1}{2} \int_{0}^{L}{\varLambda \bigl(u^{n+1}_{xx}\bigr)^2 \,dx} =& - \frac{1}{2} \int_{0}^{L}{\varLambda u^{n+1}_{xx} u^{n}_{xx} \,dx} - \int _{0}^{L}{\mu\frac{v^{n+1} - v^{n}}{\Delta t} u^{n+1} \,dx} \\ &{}- M \frac{v^{n+1}(L) - v^{n}(L)}{\Delta t} u^{n+1}(L) - J \frac{v^{n+1}_x(L) - v^{n}_x(L)}{\Delta t} u^{n+1}_x(L) \\ &{}- k_1 \frac{u^{n+1}_x(L) + u^{n}_x(L)}{2} u^{n+1}_x(L) - k_2 \frac{u^{n+1}(L) + u^{n}(L)}{2} u^{n+1}(L) \\ &{}-d_1 \frac{v^{n+1}_x(L) + v^{n}_x(L)}{2} u^{n+1}_x(L) - d_2 \frac{v^{n+1}(L) + v^{n}(L)}{2} u^{n+1}(L) \\ &{}- c_1 \cdot\frac{\zeta_1^{n+1} + \zeta_1^{n}}{2} u^{n+1}_x(L) - c_2 \cdot\frac{\zeta_2^{n+1} + \zeta_2^{n}}{2} u^{n+1}(L). \end{aligned}$$

We next set w _h =u ⁿ in (B.2):

$$\begin{aligned} \frac{1}{2} \int_{0}^{L}{\varLambda \bigl(u^{n}_{xx}\bigr)^2 \,dx} =& - \frac{1}{2} \int_{0}^{L}{\varLambda u^{n+1}_{xx} u^{n}_{xx} \,dx} - \int _{0}^{L}{\mu\frac{v^{n+1} - v^{n}}{\Delta t} u^{n} \,dx} \\ &{}- M \frac{v^{n+1}(L) - v^{n}(L)}{\Delta t} u^{n}(L) -J \frac{v^{n+1}_x(L) - v^{n}_x(L)}{\Delta t} u^{n}_x(L) \\ &{}- k_1 \frac{u^{n+1}_x(L) + u^{n}_x(L)}{2} u^{n}_x(L) - k_2 \frac{u^{n+1}(L) + u^{n}(L)}{2} u^{n}(L) \\ &{}- d_1 \frac{v^{n+1}_x(L) + v^{n}_x(L)}{2} u^{n}_x(L) - d_2 \frac{v^{n+1}(L) + v^{n}(L)}{2} u^{n}(L) \\ &{}- c_1 \cdot\frac{\zeta_1^{n+1} + \zeta_1^{n}}{2} u^{n}_x(L) - c_2 \cdot\frac{\zeta_2^{n+1} + \zeta_2^{n}}{2} u^{n}(L) . \end{aligned}$$

This yields for the norm of the time-discrete solution, as defined in (3.20):

$$\begin{aligned} & \bigl\| z^{n+1} \bigr\| ^2 - \bigl\| z^{n} \bigr\| ^2 \\ &\quad = M \biggl( - \frac{v^{n+1}(L) - v^{n}(L)}{\Delta t} \bigl(u^{n+1}(L) - u^{n}(L)\bigr) + \frac{v^{n+1}(L)^2 - v^{n}(L)^2}{2} \biggr) \\ &\qquad{} + J \biggl(- \frac{v^{n+1}_x(L) - v^{n}_x(L)}{\Delta t} \bigl(u^{n+1}_x(L) - u^{n}_x(L)\bigr) + \frac{v^{n+1}_x(L)^2 - v^{n}_x(L)^2}{2} \biggr) \\ &\qquad{} + \frac{k_1}{2} \bigl(- \bigl(u^{n+1}_x(L) + u^{n}_x(L) \bigr) \bigl(u^{n+1}_x(L) - u^{n}_x(L)\bigr) + u^{n+1}_x(L)^2 - u^{n}_x(L)^2 \bigr) \\ &\qquad{} + \frac{k_2}{2} \bigl( - \bigl(u^{n+1}(L) + u^{n}(L) \bigr) \bigl(u^{n+1}(L) - u^{n}(L)\bigr) + u^{n+1}(L)^2 - u^{n}(L)^2 \bigr) \\ &\qquad{} - \frac{d_1}{2} \bigl(v^{n+1}_x(L) + v^{n}_x(L)\bigr) \bigl(u^{n+1}_x(L)-u^{n}_x(L) \bigr) \\ &\qquad{} - \frac{d_2}{2} \bigl(v^{n+1}(L) + v^{n}(L)\bigr) \bigl(u^{n+1}(L) - u^{n}(L)\bigr) \\ &\qquad{} - \frac{1}{2} c_1 \cdot\bigl(\zeta_1^{n+1} + \zeta_1^{n}\bigr) \bigl(u^{n+1}_x(L) - u^{n}_x(L)\bigr) + \frac{1}{2}\bigl( \zeta_1^{n+1}\bigr)^{\top} P_1 \zeta_1^{n+1} - \frac{1}{2}\bigl(\zeta_1^{n} \bigr)^{\top} P_1 \zeta_1^{n} \\ &\qquad{} - \frac{1}{2} c_2 \cdot\bigl(\zeta_2^{n+1} + \zeta_2^{n}\bigr) \bigl(u^{n+1}(L) - u^{n}(L)\bigr) + \frac{1}{2}\bigl(\zeta_2^{n+1} \bigr)^{\top} P_2 \zeta_2^{n+1} - \frac{1}{2}\bigl(\zeta_2^{n}\bigr) ^{\top} P_2 \zeta_2^{n} . \end{aligned}$$

For the first six lines we use (3.21), and for the rest $c_{j}=P_{j}b_{j}+q_{j}\tilde{\delta}_{j}$ (cf. (1.7)) to obtain:

$$\begin{aligned} \bigl\| z^{n+1} \bigr\| ^2 & = \bigl\| z^{n} \bigr\| ^2 - \frac{d_1}{\Delta t} \bigl(u^{n+1}_x(L)-u^{n}_x(L) \bigr)^2 - \frac{d_2}{\Delta t}\bigl(u^{n+1}(L) - u^{n}(L)\bigr)^2 \\ &\quad {} - \frac{ ( \zeta_1^{n+1} + \zeta_1^{n} )^{\top}}{2} (P_1 b_1 + q_1 \tilde{\delta}_1) \bigl(u^{n+1}_x(L) - u^{n}_x(L)\bigr) \\ &\quad {} - \frac{ ( \zeta_2^{n+1} + \zeta_2^{n} )^{\top}}{2} (P_2 b_2 + q_2 \tilde{\delta}_2) \bigl(u^{n+1}(L) - u^{n}(L)\bigr) \\ &\quad {} + \frac{1}{2} \bigl(\zeta_1^{n+1} \bigr)^{\top} P_1 \zeta_1^{n+1} - \frac{1}{2} \bigl(\zeta_1^{n}\bigr)^{\top} P_1 \zeta_1^{n} + \frac{1}{2} \bigl( \zeta_2^{n+1}\bigr)^{\top} P_2 \zeta_2^{n+1} - \frac{1}{2} \bigl(\zeta_2^{n} \bigr)^{\top} P_2 \zeta_2^{n} . \end{aligned}$$

(B.3)

For the second and the third line of (B.3) we now use (3.21), (3.23), and (3.24) from the Crank-Nicholson scheme:

$$\begin{aligned} \bigl\| z^{n+1} \bigr\| ^2 = & \bigl\| z^{n} \bigr\| ^2 - \frac{d_1}{\Delta t} \bigl(u^{n+1}_x(L)-u^{n}_x(L) \bigr)^2 - \frac{d_2}{\Delta t}\bigl(u^{n+1}(L) - u^{n}(L)\bigr)^2 \\ &{} - \frac{ ( \zeta_1^{n+1} + \zeta_1^{n} )^{\top}}{2} P_1 \biggl(\zeta_1^{n+1} - \zeta_1^{n} - \Delta t \; A_1 \frac{\zeta_1^n + \zeta_1^{n+1}}{2} \biggr) \\ &{} - \frac{ ( \zeta_1^{n+1} + \zeta_1^{n} )}{2} \cdot q_1 \tilde{\delta}_1 \bigl(u^{n+1}_x(L) - u^{n}_x(L) \bigr) \\ &{} - \frac{ ( \zeta_2^{n+1} + \zeta_2^{n} ) ^{\top}}{2} P_2 \biggl(\zeta_2^{n+1} - \zeta_2^{n} - \Delta t \; A_2 \frac{\zeta_2^{n+1} + \zeta_2^{n}}{2} \biggr) \\ &{} - \frac{ ( \zeta_2^{n+1} + \zeta_2^{n} )}{2} \cdot q_2 \tilde{\delta}_2 \bigl(u^{n+1}(L) - u^{n}(L)\bigr) \\ &{} + \frac{1}{2} \bigl(\zeta_1^{n+1} \bigr)^ {\top} P_1 \zeta_1^{n+1} - \frac{1}{2} \bigl(\zeta_1^{n}\bigr)^{\top} P_1 \zeta_1^{n} + \frac{1}{2} \bigl( \zeta_2^{n+1}\bigr)^{\top} P_2 \zeta_2^{n+1} - \frac{1}{2} \bigl(\zeta_2^{n} \bigr)^{\top} P_2 \zeta_2^{n}. \end{aligned}$$

Since P _j , j=1,2 are symmetric matrices, this yields

$$\begin{aligned} \bigl\| z^{n+1} \bigr\| ^2 = & \bigl\| z^{n} \bigr\| ^2 - \frac{d_1}{\Delta t} \bigl(u^{n+1}_x(L)-u^{n}_x(L) \bigr)^2 - \frac{d_2}{\Delta t}\bigl(u^{n+1}(L) - u^{n}(L)\bigr)^2 \\ &{} + \Delta t\frac{ ( \zeta_1^{n+1} + \zeta_1^{n} )^{\top}}{2} P_1 A_1 \frac{\zeta_1^n + \zeta_1^{n+1}}{2} \\ &{} - \frac{ ( \zeta_1^{n+1} + \zeta_1^{n} )}{2} \cdot q_1 \tilde{\delta}_1 \bigl(u^{n+1}_x(L) - u^{n}_x(L) \bigr) \\ &{} + \Delta t \frac{ ( \zeta_2^{n+1} + \zeta_2^{n} )^{\top}}{2} P_2 A_2 \frac{\zeta_2^{n+1} + \zeta_2^{n}}{2} \\ &{} - \frac{ ( \zeta_2^{n+1} + \zeta_2^{n} )}{2} \cdot q_2 \tilde{\delta}_2 \bigl(u^{n+1}(L) - u^{n}(L)\bigr) , \end{aligned}$$

which is the claimed result (by using (1.7)). □

Proof of Theorem 9

Let k∈{0,1,…,S} be arbitrary. Taylor’s Theorem yields ∀x∈[0,L]:

$$\begin{aligned} \frac{\breve{u}(t_{k+1},x)-\breve {u}(t_{k},x)}{\Delta t} = & \frac{\breve{u}_{t}(t_{k+1},x) + \breve{u}_{t}(t_{k},x)}{2} + \Delta t \, T^k_1(x), \end{aligned}$$

(B.4)

where

$$\begin{aligned} T^k_1 (x) = & \int_{t_{k + \frac{1}{2}}}^{t_{k+1}}{ \frac{\breve{u}_{ttt}(t, x)}{2 ( \Delta t )^2} (t_{k+1} - t)^2 \;dt} + \int _{t_{k}}^{t_{k+\frac{1}{2}}}{\frac{\breve{u}_{ttt}(t, x)}{2 ( \Delta t )^2} (t_{k} - t)^2 \;dt} \\ &{} - \int_{t_{k + \frac{1}{2}}}^{t_{k+1}}{\frac{\breve{u}_{ttt}(t, x)}{2 \Delta t} (t_{k+1} - t) \;dt} + \int_{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{\breve{u}_{ttt}(t, x)}{2 \Delta t} (t_{k} - t)\;dt}. \end{aligned}$$

From (B.4), we obtain

$$\begin{aligned} \frac{\epsilon^{k+1} - \epsilon^{k}}{\Delta t} + \Delta t \, T_1^k = \frac{\varPhi^{k+1} + \varPhi^{k}}{2}. \end{aligned}$$

(B.5)

Multiplying (B.5) by μ(Φ ^k+1−Φ ^k) and integrating over [0,L] yields:

$$\begin{aligned} & \int_0^L{ \mu \frac{\epsilon^{k+1} - \epsilon^{k}}{\Delta t} \bigl(\varPhi^{k+1} - \varPhi^{k} \bigr) \;dx} \\ &\quad = \frac{1}{2} \int_0^L{ \mu\bigl( \varPhi^{k+1} \bigr)^2 \, dx} - \frac{1}{2} \int _0^L{ \mu\bigl(\varPhi^{k} \bigr)^2 \, dx}- \Delta t \int_0^L{ \mu T_1^k \bigl(\varPhi^{k+1} - \varPhi^{k} \bigr)\;dx }. \end{aligned}$$

(B.6)

Furthermore, from (3.3) with $t = t_{k + \frac{1}{2}}$ and Taylor’s Theorem, we get $\forall w \in\tilde{H}^{2}_{0}(0, L)$:

$$\begin{aligned} &\int^{L}_{0} {\mu \frac{u_t(t_{k+1},x) - u_t(t_{k},x)}{\Delta t} w \,dx} + \int^{L}_{0}{ \varLambda\frac{u_{xx}(t_{k+1},x) + u_{xx}(t_{k},x)}{2} w_{xx} \,dx} \\ &\quad {}+ M \frac{u_t(t_{k+1}, L) - u_t(t_{k}, L)}{\Delta t} w(L) + J \frac {u_{tx}(t_{k+1}, L) - u_{tx}(t_{k}, L)}{\Delta t} w_{x}(L) \\ &\quad {}+ k_1 \frac{u_{x}(t_{k+1}, L) + u_{x}(t_{k}, L) }{2} w_x(L) + k_2 \frac{u(t_{k+1}, L) + u(t_{k}, L) }{2} w(L) \\ &\quad {} + d_1 \frac{u_{tx}(t_{k+1}, L) + u_{tx}(t_{k}, L)}{2} w_x(L) + d_2 \frac{u_{t}(t_{k+1}, L) + u_{t}(t_{k}, L) }{2} w(L) \\ &\quad {} + c_1 \cdot\frac{\zeta_1(t_{k+1}) + \zeta_1(t_k)}{2} w_x(L) + c_2 \cdot\frac{\zeta_2(t_{k+1}) + \zeta_2(t_k)}{2} w(L) = \Delta t \, T^k_2 (w), \end{aligned}$$

(B.7)

with the functional $T^{k}_{2} \colon\tilde{H}^{2}_{0}(0, L) \rightarrow \mathbb{R}$ defined as

$$\begin{aligned} T^k_2(w) &=\int^{L}_{0} {\mu\biggl( \int _{t_{k+\frac{1}{2}}}^{t_{k+1}}{\frac{u_{tttt}(t,x)}{2 (\Delta t)^2} ( t_{k+1} - t )^2\,dt } + \int_{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{tttt}(t,x)}{2 (\Delta t)^2} ( t_{k} - t )^2\,dt } \biggr) w \, dx} \\ &\quad {} + \int^{L}_{0}{\varLambda\biggl( \int _{t_{k+\frac{1}{2}}}^{t_{k+1}}{\frac{u_{ttxx}(t,x)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int_{t_{k}}^{t_{k+\frac{1}{2}}}{\frac{u_{ttxx}(t,x)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w_{xx} \,dx} \\ &\quad {} + M \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{\frac{u_{tttt}(t,L)}{2 (\Delta t)^2} ( t_{k+1} - t )^2\,dt } + \int_{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{tttt}(t,L)}{2 (\Delta t)^2} ( t_{k} - t )^2 \,dt } \biggr) w(L) \\ &\quad {} + J \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{\frac{u_{ttttx}(t,L)}{2 (\Delta t)^2} ( t_{k+1} - t )^2 \,dt } + \int_{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{ttttx}(t,L)}{2 (\Delta t)^2} ( t_{k} - t )^2\,dt } \biggr) w_x(L) \\ &\quad {} + k_1 \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{u_{ttx}(t,L)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{ttx}(t,L)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w_x(L) \\ &\quad {} + k_2 \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{u_{tt}(t,L)}{2 \Delta t} ( t_{k+1} - t ) \,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{tt}(t,L)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w(L) \\ &\quad {} + d_1 \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{u_{tttx}(t,L)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{tttx}(t,L)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w_x(L) \\ &\quad {} + d_2 \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{u_{ttt}(t,L)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{ttt}(t,L)}{2 \Delta t} ( t_{k} - t ) \,dt } \biggr) w(L) \\ &\quad {} + c_1 \cdot\biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{(\zeta_1)_{tt}(t)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{(\zeta_1)_{tt}(t)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w_x(L) \\ &\quad {} + c_2 \cdot\biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{(\zeta_2)_{tt}(t)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{(\zeta_2)_{tt}(t)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) w(L). \end{aligned}$$

(B.8)

Now, from (3.22) and (B.7) follows ∀w _h ∈W _h :

$$\begin{aligned} & \int^{L}_{0} {\mu\frac{\varPhi^{k+1}- \varPhi^{k}}{\Delta t} w_h \,dx} + \int^{L}_{0}{\varLambda \frac{\epsilon_{xx}^{k+1} + \epsilon_{xx}^k}{2} (w_h)_{xx} \,dx} \\ &\qquad {}+ M \frac{\varPhi^{k+1}(L) - \varPhi^{k}(L)}{\Delta t} (w_h) (L) + J \frac{\varPhi^{k+1}_x(L) - \varPhi^{k}_x(L)}{\Delta t} (w_h)_{x}(L) \\ &\qquad {}+ k_1 \frac{\epsilon^{k+1}_x(L) + \epsilon^{k}_x(L)}{2} (w_h)_x(L) + k_2 \frac{\epsilon^{k+1}(L) + \epsilon^{k}(L) }{2} w_h(L) \\ &\qquad {}+ d_1 \frac{\varPhi^{k+1}_x(L) + \varPhi^{k}_x(L)}{2} (w_h)_x(L) + d_2 \frac{\varPhi^{k+1}(L) + \varPhi^{k}(L)}{2} w_h(L) \\ &\qquad {}+ c_1 \cdot\frac{\zeta_{e,1}^{k+1} + \zeta_{e,1}^k}{2} (w_h)_x(L) + c_2 \cdot\frac{\zeta_{e,2}^{k+1} + \zeta_{e,2}^k}{2} w_h(L) \\ &\quad = -\Delta t \, T^k_2 (w_h) + G^{k}_1(w_h), \end{aligned}$$

(B.9)

where the functional $G^{k}_{1}(w_{h})$ is given by

$$\begin{aligned} G^{k}_1(w_h) &:= \int^{L}_{0} {\mu \frac{u^e_t(t_{k+1},x) - u^e_t(t_{k},x)}{\Delta t} w_h \,dx} \\ &\hphantom{:=\,}{}+ M \frac{u^e_t(t_{k+1}, L) - u^e_t(t_{k}, L)}{\Delta t} w_h(L) + J \frac{u^e_{tx}(t_{k+1}, L) - u^e_{tx}(t_{k}, L)}{\Delta t} (w_h)_{x}(L) \\ &\hphantom{:=\,}{}+ d_1 \frac{u^e_{tx}(t_{k+1}, L) + u^e_{tx}(t_{k}, L)}{2} (w_h)_x(L) + d_2 \frac{u^e_{t}(t_{k+1}, L) + u^e_{t}(t_{k}, L) }{2} w_h(L). \end{aligned}$$

(B.10)

A Taylor expansion of ζ _j about $t_{k+\frac{1}{2}}$ yields with (3.5):

$$ \begin{aligned} & \frac{\zeta_1(t_{k+1}) - \zeta_1(t_{k})}{\Delta t} - A_1 \frac{\zeta _1(t_{k+1}) + \zeta_1(t_{k})}{2} - b_1 \frac{u_{tx}(t_{k+1}, L) + u_{tx}(t_{k}, L)}{2} = \Delta t \, T_3^k, \\ & \frac{\zeta_2(t_{k+1}) - \zeta_2(t_{k})}{\Delta t} - A_2 \frac{\zeta _2(t_{k+1}) + \zeta_2(t_{k})}{2} - b_2 \frac{u_{t}(t_{k+1}, L) + u_{t}(t_{k}, L)}{2} = \Delta t \, T_4^k , \end{aligned} $$

(B.11)

with

$$\begin{aligned} T_3^k = & \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{(\zeta_1)_{ttt}(t)}{2 (\Delta t)^2} ( t_{k+1} - t )^2\,dt } + \int _{t_{k}}^{t_{k+\frac{1}{2}}}{\frac{(\zeta_1)_{ttt}(t)}{2 (\Delta t)^2} ( t_{k} - t )^2\,dt } \\ &{} - A_1 \biggl( \int_{t_{k + \frac{1}{2}}}^{t_{k +1 }}{ \frac{(\zeta_1)_{tt}(t)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{(\zeta_1)_{tt}(t)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) \\ &{} - b_1 \biggl( \int_{t_{k + \frac{1}{2}}}^{t_{k+1}}{ \frac{u_{tttx}(t,L)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{tttx}(t, L)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr), \\ T_4^k = & \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{(\zeta_2)_{ttt}(t)}{2 (\Delta t)^2} ( t_{k+1} - t )^2\,dt } + \int _{t_{k}}^{t_{k+\frac{1}{2}}}{\frac{(\zeta_2)_{ttt}(t)}{2 (\Delta t)^2} ( t_{k} - t )^2\,dt } \\ &{} - A_2 \biggl( \int_{t_{k+ \frac{1}{2}}}^{t_{k+1 }}{ \frac{(\zeta_2)_{tt}(t)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{(\zeta_2)_{tt}(t)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) \\ &{} - b_2 \biggl( \int_{t_{k + \frac{1}{2}}}^{t_{k+1}}{ \frac{u_{ttt}(t,L)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{ttt}(t, L)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr). \end{aligned}$$

Using (3.23), (3.24), and (B.11), we get

$$ \begin{aligned} & \frac{\zeta^{k+1}_{e,1} - \zeta^{k}_{e,1}}{\Delta t} - A_1 \frac{\zeta ^{k+1}_{e,1} + \zeta^{k}_{e,1}}{2} - b_1 \frac{\varPhi^{k+1}_x(L) + \varPhi^{k}_x(L)}{2} = -\Delta t \, T_3^k - G^{k}_2, \\ & \frac{\zeta^{k+1}_{e,2} - \zeta^{k}_{e,2}}{\Delta t} - A_2 \frac{\zeta ^{k+1}_{e,2} + \zeta^{k}_{e,2}}{2} - b_2 \frac{\varPhi^{k+1}(L) + \varPhi^{k}(L)}{2} = -\Delta t \, T_4^k - G^{k}_3, \end{aligned} $$

(B.12)

with

$$\begin{aligned} G_2^k = & b_1 \frac{u^e_{tx}(t_{k+1}, L) + u^e_{tx}(t_{k}, L)}{2}, \\ G_3^k = & b_2 \frac{u^e_{t}(t_{k+1}, L) + u^e_{t}(t_{k}, L)}{2}. \end{aligned}$$

In (B.9) we now take $w_{h} := \Delta t \frac{\varPhi ^{k+1} + \varPhi^{k}}{2} \in W_{h}$, due to (B.5). Using (B.6) and (B.12), yields:

$$\begin{aligned} \bigl\| z^{k+1}_e \bigr\| ^2 - \bigl\| z^{k}_e \bigr\| ^2 = & - (\Delta t)^2 \frac{1}{2} \int_{0}^{L}{ \varLambda\bigl( \epsilon^{k+1}_{xx} + \epsilon^{k}_{xx} \bigr) \bigl(T^k_1\bigr)_{xx} \;dx} + \frac{\Delta t}{2} G^k_1\bigl(\varPhi^{k+1} + \varPhi^{k}\bigr) \\ &{}- (\Delta t)^2 \biggl( k_1 \frac{\epsilon_x^{k+1}(L) + \epsilon_x^{k}(L)}{2} \bigl(T_1^k\bigr)_x(L) + k_2 \frac{\epsilon^{k+1}(L) + \epsilon^{k}(L)}{2} T_1^k(L) \biggr) \\ &{} - \frac{\Delta t}{2} \biggl( q_1 \frac{\zeta^{k+1}_{e,1} + \zeta ^{k}_{e,1}}{2} + \tilde{ \delta}_1 \frac{\varPhi^{k+1}_x(L) + \varPhi^{k}_x(L)}{2} \biggr)^2 \\ &{} - \Delta t \delta_1 \biggl( \frac{\varPhi^{k+1}_x(L) + \varPhi^{k}_x(L)}{2} \biggr)^2 - \Delta t \frac{\epsilon_1}{2} \frac{\zeta_{e,1}^{k+1} + \zeta_{e,1}^{k}}{2} \cdot P_1 \frac{\zeta_{e,1}^{k+1} + \zeta_{e,1}^{k}}{2} \\ &{} - P_1 \frac{\zeta_{e,1}^{k+1} + \zeta_{e,1}^{k}}{2} \cdot\bigl( (\Delta t)^2 T^k_3 + \Delta t \, G^k_2 \bigr) \\ &{} - \frac{\Delta t}{2} \biggl( q_2 \frac{\zeta^{k+1}_{e,2} + \zeta ^{k}_{e,2}}{2} + \tilde{ \delta}_2 \frac{\varPhi^{k+1}(L) + \varPhi^{k}(L)}{2} \biggr)^2 \\ &{} - \Delta t \delta_2 \biggl( \frac{\varPhi^{k+1}(L) + \varPhi^{k}(L)}{2} \biggr)^2 - \Delta t \frac{\epsilon_2}{2} \frac{\zeta_{e,2}^{k+1} + \zeta_{e,2}^{k}}{2} \cdot P_2 \frac{\zeta_{e,2}^{k+1} + \zeta_{e,2}^{k}}{2} \\ &{} - P_2 \frac{\zeta_{e,2}^{k+1} + \zeta_{e,2}^{k}}{2} \cdot\bigl( (\Delta t)^2 T^k_4 + \Delta t \, G^k_3 \bigr) \\ &{}- \frac{1}{2} (\Delta t)^2 T^k_2 \bigl(\varPhi^{k+1} + \varPhi^{k}\bigr). \end{aligned}$$

Therefore,

$$\begin{aligned} \bigl\| z^{k+1}_e \bigr\| ^2 - \bigl\| z^{k}_e \bigr\| ^2 \le& - (\Delta t)^2 \frac{1}{2} \int_{0}^{L}{ \varLambda\bigl( \epsilon^{k+1}_{xx} + \epsilon^{k}_{xx} \bigr) \bigl(T^k_1\bigr)_{xx} \;dx} + \frac{\Delta t}{2} G^k_1\bigl(\varPhi^{k+1} + \varPhi^{k}\bigr) \\ &{}- (\Delta t)^2 \biggl( k_1 \frac{\epsilon_x^{k+1}(L) + \epsilon_x^{k}(L)}{2} \bigl(T_1^k\bigr)_x(L) + k_2 \frac{\epsilon^{k+1}(L) + \epsilon^{k}(L)}{2} T_1^k(L) \biggr) \\ &{} - P_1 \frac{\zeta_{e,1}^{k+1} + \zeta_{e,1}^{k}}{2} \cdot\bigl( (\Delta t)^2 T^k_3 + \Delta t \, G^k_2 \bigr) \\ &{} - P_2 \frac{\zeta_{e,2}^{k+1} + \zeta_{e,2}^{k}}{2} \cdot\bigl( (\Delta t)^2 T^k_4 + \Delta t \, G^k_3 \bigr) \\ &{}- \frac{1}{2} (\Delta t)^2 T^k_2 \bigl(\varPhi^{k+1} + \varPhi^{k}\bigr). \end{aligned}$$

(B.13)

Next, from (B.10) follows:

$$\begin{aligned} \bigl|G^{k}_1\bigl(\varPhi^{k+1} + \varPhi^k\bigr)\bigr| \le& C \biggl( \biggl\| \frac{u^e_t(t_{k+1},x) - u^e_t(t_{k},x)}{\Delta t} \biggr\| _{L^2}^2 + \bigl\| \varPhi^{k+1} + \varPhi^k \bigr\| _{L^2}^2 \\ &{} + \biggl| \frac{u^e_t(t_{k+1}, L) - u^e_t(t_{k}, L)}{\Delta t}\biggr|^2 + \biggl| \frac {u^e_{tx}(t_{k+1}, L) - u^e_{tx}(t_{k}, L)}{\Delta t}\biggr|^2 \\ &{}+ \biggl|\frac{u^e_{tx}(t_{k+1}, L) + u^e_{tx}(t_{k}, L)}{2}\biggr|^2 + \biggl|\frac {u^e_{t}(t_{k+1}, L) + u^e_{t}(t_{k}, L) }{2}\biggr|^2 \\ &{}+ \bigl| \varPhi^{k+1}(L) + \varPhi^k(L)\bigr|^2 + \bigl| \varPhi_x^{k+1}(L) + \varPhi_x^{k}(L)\bigr|^2 \biggr) \end{aligned}$$

(B.14)

$$\begin{aligned} \le& C \biggl( \bigl\| \varPhi^{k+1} + \varPhi^k \bigr\| _{L^2}^2 + \bigl| \varPhi^{k+1}(L) + \varPhi^k(L)\bigr|^2 + \bigl| \varPhi_x^{k+1}(L) + \varPhi_x^{k}(L)\bigr|^2 \\ &{} + \frac{1}{\Delta t} \int_{t_k}^{t_k+1}\bigl\| u^e_{tt}(t) \bigr\| ^2_{L^2} + \bigl|u^e_{tt}(t,L)\bigr|^2 \\ &{}+\bigl|u^e_{ttx}(t,L)\bigr|^2 \, dt + \bigl\| u^e_t \bigr\| _{C([t_{k},t_{k+1}]; H^2)}^2 \biggr). \end{aligned}$$

(B.15)

It can easily be seen that

$$\begin{aligned} \bigl\| T^k_1\bigr\| ^2_{H^2} \le&\Delta t \int_{t_k}^{t_{k+1}}{\bigl\| \breve{u}_{ttt}(t) \bigr\| ^2_{H^2} \, dt} \le C \Delta t \int _{t_k}^{t_{k+1}}{\bigl\| u_{ttt}(t) \bigr\| ^2_{H^2} \, dt}, \end{aligned}$$

(B.16)

$$\begin{aligned} \bigl\| T^k_3\bigr\| ^2 \le& C \Delta t \int_{t_k}^{t_{k+1}}{\bigl\| u_{ttt}(t) \bigr\| ^2_{H^2} + \bigl\| (\zeta_1)_{tt} \bigr\| ^2 + \bigl\| (\zeta_1)_{ttt}\bigr\| ^2 \, dt}, \end{aligned}$$

(B.17)

$$\begin{aligned} \bigl\| T^k_4\bigr\| ^2 \le& C \Delta t \int_{t_k}^{t_{k+1}}{\bigl\| u_{ttt}(t) \bigr\| ^2_{H^1} + \bigl\| (\zeta_2)_{tt} \bigr\| ^2 + \bigl\| (\zeta_2)_{ttt}\bigr\| ^2 \, dt}, \end{aligned}$$

(B.18)

and

$$\begin{aligned} T^k_2\bigl(\varPhi^k\bigr) \le& C \biggl( \bigl\| \varPhi^k \bigr\| ^2_{L^2} + \bigl| \varPhi^k(L) \bigr|^2 + \bigl| \varPhi^k_x(L) \bigr|^2 \\ &{} + \Delta t \int_{t_k}^{t_{k+1}}{\bigl\| u_{tt} (t)\bigr\| _{H^4}^2 + \bigl\| u_{ttt}(t) \bigr\| _{H^2}^2 + \bigl\| u_{tttt}(t) \bigr\| _{H^2}^2 \, dt} \\ &{} + \Delta t \int_{t_k}^{t_{k+1}}{\bigl\| ( \zeta_1)_{tt} (t)\bigr\| ^2 + \bigl\| (\zeta_2)_{tt} (t)\bigr\| ^2 \, dt} \biggr). \end{aligned}$$

(B.19)

For the above estimate, we rewrote the second term of $T^{k}_{2}(\varPhi ^{k})$ in (B.8) as:

$$\begin{aligned} &\int^{L}_{0}{ \biggl( \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{ \frac{u_{ttxx}(t,x)}{2 \Delta t} ( t_{k+1} - t )\,dt } - \int _{t_{k}}^{t_{k+\frac{1}{2}}}{ \frac{u_{ttxx}(t,x)}{2 \Delta t} ( t_{k} - t )\,dt } \biggr) \varPhi^k_{xx} \,dx} \\ &\quad = \int_{t_{k+\frac{1}{2}}}^{t_{k+1}}{\frac{t_{k+1} - t}{2 \Delta t} \biggl( u_{ttxx}(t,L) \varPhi^k_x(L) - u_{ttxxx}(t,L) \varPhi^k(L) + \int_{0}^{L}{u_{ttxxxx}(t,x) \varPhi^k \, dx} \biggr) \,dt} \\ &\qquad {}- \int_{t_{k}}^{t_{k+\frac{1}{2}}}{\frac{t_{k} - t}{2 \Delta t} \biggl( u_{ttxx}(t,L) \varPhi^k_x(L) - u_{ttxxx}(t,L) \varPhi^k(L) + \int_{0}^{L}{u_{ttxxxx}(t,x) \varPhi^k \, dx} \biggr) \,dt}, \end{aligned}$$

using $\varPhi^{k}(0) = \varPhi^{k}_{x}(0) = 0$, and then the Sobolev embedding Theorem. From (B.13)–(B.19), now follows:

$$\begin{aligned} \bigl\| z^{k+1}_e \bigr\| ^2 - \bigl\| z^{k}_e \bigr\| ^2 & \le C \Biggl( \Delta t \bigl( \bigl\| z^{k+1}_e \bigr\| ^2 + \bigl\| z^{k}_e \bigr\| ^2\bigr) + \Delta t \bigl\| u^e_t \bigr\| _{C([t_{k},t_{k+1}]; H^2)}^2 \\ &\quad {}+ \int_{t_k}^{t_{k+1}}{\bigl\| u^e_{tt}(t) \bigr\| ^2_{L^2} + \bigl|u^e_{tt}(t,L)\bigr|^2 +\bigl|u^e_{ttx}(t,L)\bigr|^2 \, dt} \\ &\quad {} + (\Delta t)^4 \sum_{i=1}^{2}{ \int_{t_k}^{t_{k+1}}{ \bigl\| (\zeta_i)_{tt} \bigr\| ^2 + \bigl\| (\zeta_i)_{ttt}\bigr\| ^2 \, dt}} \\ &\quad {} + (\Delta t)^4 \int_{t_k}^{t_{k+1}}{ \bigl\| u_{tt} (t)\bigr\| _{H^4}^2 + \bigl\| u_{ttt}(t) \bigr\| _{H^2}^2 + \bigl\| u_{tttt}(t) \bigr\| _{H^2}^2 \, dt} \Biggr). \end{aligned}$$

(B.20)

Let now n∈{1,…,S}. Assuming $\Delta t \le\frac{1}{2 C}$ (with C from (B.20)), and summing (B.20) over k∈{0,…,n}, gives:

$$\begin{aligned} \frac{1}{2} \bigl\| z^{n+1}_e \bigr\| ^2 & \le \frac{3}{2} \bigl\| z^{0}_e \bigr\| ^2 + C \Biggl( \Delta t \sum_{k=1}^{n}{ \bigl\| z^{k}_e \bigr\| ^2} + \bigl\| u^e_t \bigr\| _{C([0, T]; H^2)}^2 + \bigl\| u^e_{tt} \bigr\| ^2_{L^2(0,T; H^2)} \\ &\quad {} + (\Delta t)^4 \Biggl[ \sum_{i = 1}^2{ \bigl\| (\zeta_i)_{tt} (t)\bigr\| _{L^2(0, T; \mathbb{R}^n)}^2 + \bigl\| ( \zeta_i)_{ttt} (t)\bigr\| _{L^2(0, T; \mathbb{R}^n)}^2} \\ &\quad {} + \bigl\| u_{tt} (t) \bigr\| _{L^2(0, T; H^4)}^2 + \bigl\| u_{ttt}(t) \bigr\| _{L^2(0, T; H^2)}^2 + \bigl\| u_{tttt}(t) \bigr\| _{L^2(0, T; H^2)}^2 \Biggr] \Biggr). \end{aligned}$$

(B.21)

Finally, using the discrete-in-time Gronwall inequality and (B.4), we obtain:

$$\begin{aligned} \bigl\| z^{n+1}_e \bigr\| ^2 & \le C \Biggl( \bigl\| z^{0}_e \bigr\| ^2 + h^4 \bigl( \| u_t \|_{C([0, T]; H^4)}^2 +\| u_{tt} \|^2_{L^2(0,T; H^4)} \bigr) \\ &\quad {} + (\Delta t)^4 \Biggl[ \sum_{i = 1}^2{ \bigl\| (\zeta_i)_{tt} (t)\bigr\| _{L^2(0, T; \mathbb{R}^n)}^2 + \bigl\| ( \zeta_i)_{ttt} (t)\bigr\| _{L^2(0, T; \mathbb{R}^n)}^2} \\ &\quad {} + \bigl\| u_{tt} (t) \bigr\| _{L^2(0, T; H^4)}^2 + \bigl\| u_{ttt}(t) \bigr\| _{L^2(0, T; H^2)}^2 + \bigl\| u_{tttt}(t) \bigr\| _{L^2(0, T; H^2)}^2 \Biggr] \Biggr). \end{aligned}$$

(B.22)

The result now follows from (B.22), (3.25), and the triangle inequality. □