Asymptotic behaviour of the semidiscrete FE approximations to weakly damped wave equations with minimal smoothness on initial data

TL;DR

This paper analyzes the asymptotic behavior of semidiscrete finite element approximations to weakly damped wave equations, establishing decay estimates and error bounds with minimal smoothness assumptions, supported by numerical validation.

Contribution

It provides uniform decay estimates and optimal error bounds for semidiscrete FE methods applied to weakly damped wave equations with minimal initial data smoothness.

Findings

Decay rates match continuous case

Optimal error estimates under minimal smoothness

Numerical experiments confirm theoretical results

Abstract

Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable continuous, a semidiscrete system is analysed, and uniform decay estimates are derived with precisely the same decay rate as in the continuous case. Optimal error estimates with minimal smoothness assumptions on the initial data are established, which preserve exponential decay rate, and for a 2D problem, the maximum error bound is also proved. The present analysis is then generalized to include the problems with non-homogeneous forcing function, space-dependent damping, and problems with compensator. It is observed that decay rates are improved with large viscous damping and compensator. Finally, some numerical experiments are performed to validate the theoretical results established in this paper.

Tables

Table 1. Table 1 : Example 1: Errors and rate of convergences in ‖ u − u h ‖ , ‖ u − u h ‖ ∞ norm 𝑢 subscript 𝑢 ℎ subscript norm 𝑢 subscript 𝑢 ℎ \|u-u_{h}\|,\;\|u-u_{h}\|_{\infty} and ‖ u − u h ‖ 1 subscript norm 𝑢 subscript 𝑢 ℎ 1 \|u-u_{h}\|_{1} .

Table 2. Table 2 : Example 2: Errors and rate of convergences in ‖ u − u h ‖ , ‖ u − u h ‖ ∞ norm 𝑢 subscript 𝑢 ℎ subscript norm 𝑢 subscript 𝑢 ℎ \|u-u_{h}\|,\;\|u-u_{h}\|_{\infty} and ‖ u − u h ‖ 1 subscript norm 𝑢 subscript 𝑢 ℎ 1 \|u-u_{h}\|_{1}

Table 3. Table 3 : Example 3: Errors and rate of convergences in ‖ u − u h ‖ , ‖ u − u h ‖ ∞ norm 𝑢 subscript 𝑢 ℎ subscript norm 𝑢 subscript 𝑢 ℎ \|u-u_{h}\|,\;\|u-u_{h}\|_{\infty} and ‖ u − u h ‖ 1 subscript norm 𝑢 subscript 𝑢 ℎ 1 \|u-u_{h}\|_{1}

Table 4. Table 4 : Example 4: Errors and rate of convergences in ‖ u − u h ‖ , ‖ u − u h ‖ ∞ norm 𝑢 subscript 𝑢 ℎ subscript norm 𝑢 subscript 𝑢 ℎ \|u-u_{h}\|,\;\|u-u_{h}\|_{\infty} and ‖ u − u h ‖ 1 subscript norm 𝑢 subscript 𝑢 ℎ 1 \|u-u_{h}\|_{1}

Table 5. Table 5 : Example 5: Errors and rate of convergences in ‖ u − u h ‖ , ‖ u − u h ‖ ∞ norm 𝑢 subscript 𝑢 ℎ subscript norm 𝑢 subscript 𝑢 ℎ \|u-u_{h}\|,\;\|u-u_{h}\|_{\infty} and ‖ u − u h ‖ 1 subscript norm 𝑢 subscript 𝑢 ℎ 1 \|u-u_{h}\|_{1}

Equations

$$u^{\prime \prime }+\alpha u^{\prime }+Au=0,x\in \Omega ,t>0$$

$$u(x,0)=u_0(x),u_t(x,0)=u_1(x),x\in \Omega$$

$$u=0,(x,t)\in \partial \Omega \times (0,\infty ),$$

$$A\phi=-\sum_{i,j=1}^{d}\frac{\partial}{\partial x_{i}}\big{(}a_{ij}(x)\frac{\partial\phi}{\partial x_{j}}\big{)}+a_{0}(x)\phi,$$

$$H_0^1(\Omega )=\{\varphi \in H^1(\Omega ):\varphi =0\text{mbox}on\partial \Omega \}.$$

$$u^{\prime \prime }+\alpha u^{\prime }+Au=0,t>0,u(0)=u_0,u_t(0)=u_1.$$

$$\Vert v\Vert \le \frac{1}{\sqrt{{\lambda }_1}}\Vert A^{1/2}v\Vert ,$$

$$\Vert A^{1/2}v\Vert \le \frac{1}{\sqrt{{\lambda }_1}}\Vert Av\Vert ,$$

$$u\in L^{\infty }\left(0,T;D(A)\right),u^{\prime }\in L^{\infty }\left(0,T;D(A^{1/2})\right),u^{\prime \prime }\in L^{\infty }\left(0,T;L^2(\Omega )\right),$$

$$u^{\prime \prime }+\alpha u^{\prime }+Au=0,\text{mbox}a.e.t>0$$

$$u(0)=u_0,u_t(0)=u_1.$$

$$a(v,w)=:(A^{1/2}v,A^{1/2}w):=\sum _{i,j=1}^d\left(a_{ij}\frac{\partial v}{\partial x_j},\frac{\partial w}{\partial x_i}\right)+\left(a_{0}v,w\right).$$

$$(u^{\prime \prime },\chi )+\alpha (u^{\prime },\chi )+a(u,\chi )=0,\chi \in D(A^{1/2}),$$

$$u(0)=u_0,\text{mbox}and{u}^{\prime }(0)=u_1.$$

$${\mathcal{E}}^{(1)}(t)=\frac{1}{2}\left(\Vert u^{\prime }{\Vert }^2+\Vert A^{1/2}u{\Vert }^2\right).$$

$${\mathcal{E}}^{(1)}(t)\le 3e^{-2\delta t}{\mathcal{E}}^{(1)}(0),t\ge 0.$$

$$(u^{\prime \prime },u^{\prime }+ϵu)+\alpha (u^{\prime },u^{\prime }+ϵu)+a\left(u,u^{\prime }+ϵu\right)=0,$$

$$\frac{1}{2}\frac{d}{dt}\left(\Vert u^{\prime }{\Vert }^2+\Vert A^{1/2}u{\Vert }^2+ϵ\alpha \Vert u{\Vert }^2\right)+ϵ(u^{\prime \prime },u)+\alpha \Vert u^{\prime }{\Vert }^2+ϵ\Vert A^{1/2}u{\Vert }^2=0.$$

$$\frac{d}{dt}{\mathcal{E}}^{(1)}(t)+\frac{ϵ\alpha }{2}\frac{d}{dt}\Vert u{\Vert }^2+ϵ(u^{\prime \prime },u)+\alpha \Vert u^{\prime }{\Vert }^2+ϵ\Vert A^{1/2}u{\Vert }^2=0.$$

$$ϵ(u^{\prime \prime },u)=ϵ\frac{d}{dt}(u^{\prime },u)-ϵ\Vert u^{\prime }{\Vert }^2.$$

$$\frac{d}{dt}\left({\mathcal{E}}^{(1)}(t)+ϵ(u^{\prime },u)+\frac{\alpha ϵ}{2}\Vert u{\Vert }^2\right)+\left((\alpha -ϵ)\Vert u^{\prime }{\Vert }^2+ϵ\Vert A^{1/2}u{\Vert }^2\right)=0.$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)={\mathcal{E}}^{(1)}(t)+ϵ(u^{\prime },u)+\frac{\alpha ϵ}{2}\Vert u{\Vert }^2,$$

$$F(t)=(\alpha -ϵ)\Vert u^{\prime }{\Vert }^2+ϵ\Vert A^{1/2}u{\Vert }^2,$$

$$\frac{d}{dt}{\mathcal{E}}_{ϵ}^{(1)}(t)+F(t)=0.$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)\ge {\mathcal{E}}^{(1)}(t)-ϵ\left(\frac{\Vert u^{\prime }{\Vert }^2}{2\alpha }+\frac{\alpha }{2}\Vert u{\Vert }^2\right)+\frac{\alpha ϵ}{2}\Vert u{\Vert }^2={\mathcal{E}}^{(1)}(t)-\frac{ϵ}{2\alpha }\Vert u^{\prime }{\Vert }^2,$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)\ge \left(1-\frac{ϵ}{\alpha }\right){\mathcal{E}}^{(1)}(t).$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)\ge \frac{1}{2}{\mathcal{E}}^{(1)}(t).$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)\le \frac{3}{2}{\mathcal{E}}^{(1)}(t)+\left(\frac{ϵ}{2\alpha }-\frac{1}{4}\right)\Vert u^{\prime }{\Vert }^2-\frac{1}{4}\Vert A^{1/2}u{\Vert }^2+ϵ\alpha \Vert u{\Vert }^2.$$

$${\mathcal{E}}_{ϵ}^{(1)}(t)\le \frac{3}{2}{\mathcal{E}}^{(1)}(t)+\left(\frac{ϵ}{2\alpha }-\frac{1}{4}\right)\Vert u^{\prime }{\Vert }^2+\left(\frac{ϵ\alpha }{{\lambda }_1}-\frac{1}{4}\right)\Vert A^{1/2}u{\Vert }^2.$$

$$\frac{ϵ}{2\alpha }-\frac{1}{4}\le 0,\text{mbox}and\frac{ϵ\alpha }{{\lambda }_1}-\frac{1}{4}\le 0,$$

Full text

Keywords. Weakly damped wave equation, uniform decay estimates, Galerkin finite elements, optimal error estimates, numerical experiments.

Mathematics Subject Classification. 65M60, 65M15, 35L20.

1 Introduction

This paper deals with uniform exponential decay rates for the semidiscrete finite element solution of the following weakly damped wave equation:

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

where $u^{\prime}=\frac{\partial u}{\partial t}$, $\Omega$ is a convex polygonal or polyhedral domain in $\mathbb{R}^{d}$ with boundary $\partial\Omega$, and $\alpha$ is a fixed positive constant. Here, $A$ is a second order linear elliptic operator given by

[TABLE]

where the coefficients $a_{ij}$ and $a_{0}$ are smooth with $a_{0}(x)\geq 0$ and uniformly positive definite matrix $\{a_{ij}\}_{1\leq i,j\leq 2}.$

The equation (1.1) is known as the damped wave or telegraphers equation [11, 12], which arises in many applications such as acoustics, linear elasticity, electro-magnetics, heat transfer, or particle transport, etc. Due to many applications, the damped wave equation has attracted significant interest in the literature. For the existence of a weak solution with regularity results using the Bubnov-Galerkin method and weak compactness arguments, we may refer to [26, Theorems 4.1-4.2 of Chapter II] and [1, Section 1.8].

In the literature, explicit nonuniform decay rates have been established using a control-theoretic method for weakly damped linear systems in Hilbert space, see, [23]. The decay rate for the problem (1.1) is given in terms of the first positive eigenvalue of the operator $A$ and the weak damping coefficient $\alpha>0$ in [26, Proposition 1.2 of Chapter 4]. In this paper, we have proved a better decay rate not only for the first energy, but also for higher order energy. When the damped coefficient $\alpha=\alpha(x)>0,$ the decay rate involves the minimum and maximum of this coefficient and the first positive eigenvalue of $A$ in [20]. For related papers, see, also [5]-[7] and references, therein. In all these papers mentioned above, a large damping coefficient does not necessarily give rise to a large decay rate as it also depends on the minimum positive eigenvalue of the associated elliptic eigenvalue problem. Subsequently, Chen [8] has developed and analysed improved decay rates by a new stabilization scheme that combines viscous damping and compensation. We shall discuss it in section 4.3 under generalizations.

We use the standard notation for Sobolev spaces and their norms. In particular, let $L^{2}(\Omega)$ denote the space of square integrable functions on $\Omega$ with natural inner product $(\cdot,\cdot)$ and induced norm $\|\cdot\|$. For a nonnegative integer $k$, let $H^{k}$ denote the Hilbert Sobolev space $H^{k}(\Omega)$ with norm $\|\cdot\|_{k}$. Let

[TABLE]

With $A$ in (1.4) as a linear self-adjoint and positive definite operator on $L^{2}=:L^{2}(\Omega)$ with dense domain $D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$ we define $\dot{H}^{r}=\dot{H}^{r}(\Omega)=:D(A^{r/2})$ as a subspace of $H^{r}$ with norm $|v|_{r}=\|A^{r/2}\|.$ Essentially, for $r\geq 0$ and $r/2-1/4$ is not an integer, the space $\dot{H}^{r}=\{v\in H^{r}:A^{j}v=0\;\mbox{on}\;\partial\Omega,\;\mbox{for}\;j<r/2-1/4\}$ and its norm is equivalent to $H^{r}$-norm. Now, $H^{1}_{0}=D(A^{1/2})=\dot{H}^{1}_{0}$ and $\dot{H}^{2}=H^{2}\cap H^{1}_{0}.$

In general, uniform decay property of the continuous problem (1.1) may not be preserved by the approximate solution when standard numerical schemes are applied. This may be due to existence of high frequency modes which are only weakly damped. Therefore, several stabilized methods have been developed and analysed, which give rise to uniform decay property of the semidiscrete-in-space schemes, keeping time variable continuous, see, [19] and [28] and references, therein. It is to be noted that mixed finite element methods are also employed to preserve uniform exponential decay property, see, [12]. This paper follows a different strategy to discuss uniform decay property of the semidiscrete scheme, when $C^{0}$-conforming finite element method is applied in the spatial direction. The key to the success of the present scheme is based on the energy arguments with the bound on the Poincaré type inequality, which provides a decay estimate in a range and similar to the decay rate as predicted by the continuous problem. However, the decay rate given by the present analysis may not be optimal and this is due to non-conservative bounds in our estimates. The main contributions of this paper are as follows.

• (i)

The first part of this paper focusses on the problem (1.1)-(1.3) and higher order in time regularity results are derived along with exponential decay properties using energy arguments of [26, Theorems 4.1-4.2 of Chapter II] and [1, Section 1.8]. It is observed that the decay rate is calculated in a range involving the damping parameter $\alpha$ and the first positive eigenvalue of the operator $A.$ In case, $u_{0}\in D(A^{(k)})$ and $u_{1}\in D(A^{(k-1/2)})$, the corresponding energy $2{\cal{E}}_{A^{(k)}}(u)=\|A^{(k-1/2)}u^{\prime}\|^{2}+\|A^{(k)}u\|^{2}$ decays exponentially with the same decay rate. In fact, it is observed that for large damping parameter, the decay rates may not be higher.

• (ii)

Based on $C^{0}$-conforming finite element (FE) discretization in spatial variables keeping time variable continuous, a semidiscrete scheme is proposed, and uniform exponential decay estimates, which are uniform with respect to the discretizing parameter, are derived.

• (iii)

Optimal error estimates are established with minimal smoothness assumption on the initial data, that is, when $u_{0}\in H^{3}\cap H^{1}_{0}$ and $u_{1}\in H^{2}\cap H^{1}_{0},$ which have the same decay rate as observed for the semidiscrete solution. When $d=2$, the maximum norm estimate is also obtained with exactly same decay rate.

• (iv)

The analysis is then extended to include the nonhomogeneous problem and the problem with the space dependent viscous damping.

• (v)

Decay rates are improved by the new stabilization method of combined with viscous damping and compensator for the semidiscrete solution. Compared to (i)-(iii), the decay rates can be made larger by choosing large damping parameter and large compensator.

• (vi)

Finally, several numerical experiments are conducted to confirm our theoretical findings.

We now emphasise here that Rauch [21] has earlier initiated the discussion on optimal order of convergence using $C^{0}$-conforming linear FE method to a second linear order wave equation with minimal initial data, that is, $u_{0}\in H^{3}\cap H^{1}_{0}$ and $u_{1}=0,$ see also [24], [25] and [17] and references, therein.

An outline of this paper is as follows. In Section 2, we discuss weak formulation, regularity, and decay estimates for the continuous problem. Section 3 deals with the semidiscrete scheme. We establish decay estimates and optimal error estimates for the semidiscrete scheme. Section 4 is devoted to some generalizations involving inhomogeneous problems, space dependent damping problems, and problems with damping and compensator. Section 5 discuses a completely discrete scheme along with its energy conservation properties. Finally, several numerical experiments are conducted, whose results confirm our theoretical findings.

2 Weak formulation, Regularity results and Decay properties

This section deals with the weak formulation, some regularity results, and also the decay properties for the continuous problem.

Note that the operator $A$ in (1.4) is, infact, a linear self-adjoint and positive definite operator on $L^{2}=:L^{2}(\Omega)$ with dense domain $D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$ Then, the problem (1.1)-(1.3) in its abstract form in $L^{2}$ is to seek $u(t)\in D(A)$ for $t>0$ satisfying

[TABLE]

Let us recall here the following Poincaré inequalities for our subsequent use. For $v\in D(A^{1/2})$

[TABLE]

and for $v\in D(A)=:H^{2}\cap H_{0}^{1}$,

[TABLE]

where $\lambda_{1}$ is the minimum positive eigenvalue of $A.$

We now state the following theorem without proof on the existence of a unique weak solution, whose proof can be found in [18, Theorem 1.1], [26, Theorem 4.1 of Chapter II].

Theorem 2.1**.**

Assume that $u_{0}\in D(A)$ and $u_{1}\in D(A^{1/2}).$ Then, the problem (1.1)-(1.3) admits a unique strong solution $u$ satisfying

[TABLE]

and

[TABLE]

with

[TABLE]

Now, the bilinear form $a(\cdot,\cdot)$ on $V=D(A^{1/2})$ associated with $A$ is defined for $v,w\in D(A^{1/2})$ by

[TABLE]

Then, rewrite (2.1) as

[TABLE]

2.1 Decay Property

This subsection focuses on the decay properties for the continuous problem (1.1)-(1.3). Now, define the energy functional

[TABLE]

Theorem 2.2**.**

The solution $u$ of (1.1)-(1.3) satisfies for $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$

[TABLE]

Proof. Set $\chi=u^{\prime}+\epsilon u$ in (2.5) to obtain

[TABLE]

and hence, rewrite it as

[TABLE]

A use of the energy (2.7) shows

[TABLE]

Note that

[TABLE]

Using (2.12) in (2.11), we obtain

[TABLE]

Define

[TABLE]

and

[TABLE]

to rewrite (2.13) as

[TABLE]

Observe that

[TABLE]

Using the Young’s inequality, we obtain

[TABLE]

and hence,

[TABLE]

Now, choose $\epsilon$ in a suitable manner with $\frac{\epsilon}{\alpha}\leq\frac{1}{2}$, i.e., $0<\epsilon\leq\frac{\alpha}{2}$ to arrive at

[TABLE]

Again, recall the definition of ${\cal E}_{\epsilon}^{(1)}(t)$ and use the Cauchy-Schwarz and Young’s inequalities to find that

[TABLE]

Using Poincaré inequality (2.2), the inequality (2.18) becomes

[TABLE]

In order to derive an estimate of the form ${\cal E}_{\epsilon}^{(1)}(t)\leq\frac{3}{2}{\cal E}^{(1)}(t)$, we must have

[TABLE]

that is, choose

[TABLE]

Therefore, combining (2.17) and (2.20), we obtain

[TABLE]

provided $0<\epsilon\leq\frac{1}{2}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$. A use of the definition of $F$ and ${\cal E}^{(1)}$ shows

[TABLE]

Note that for $0<\epsilon=\min\left(\frac{\alpha}{2},\frac{\lambda_{1}}{4\alpha}\right)$, $\alpha-2\epsilon\geq 0$, there holds

[TABLE]

Thus, from (2.13), we obtain

[TABLE]

which implies that

[TABLE]

An integration with respect to $t$ implies

[TABLE]

and a use of (2.21) yields

[TABLE]

where $\delta\in\left(0,\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)\right)$. This completes the rest of the proof. $\blacksquare$

The next theorem is on time derivatives.

Theorem 2.3**.**

The solution $u$ of (1.1)-(1.3) satisfies for $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$

[TABLE]

where

[TABLE]

and $u^{(j)}$ stands for $j^{\mbox{th}}$ time derivative of $u$.

Proof. For proving the result for ${\cal E}^{(j)}$, we apply the induction hypothesis. Assume that the result is true for $j-1$, that is

[TABLE]

we shall show the result (2.23) holds for $j$

[TABLE]

Choose $w=u^{(j-1)}$ in (2.24) and set $v=w^{\prime}+\epsilon\,w$ the resulting equation to arrive at

[TABLE]

where

[TABLE]

Since equation (2.25) is similar to the equation (2.14) when $u$ is replaced by $w$, on repeating the same arguments as earlier, we obtain

[TABLE]

Now replacing $w=u^{(j-1)}$ in (2.26), we arrive at

[TABLE]

which can be written as

[TABLE]

This completes the induction. Therefore, (2.23) holds true and this completes the rest of the proof of the theorem. $\blacksquare$

Theorem 2.4**.**

The solution $u$ of (1.1)-(1.3) satisfies for $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$

[TABLE]

where

[TABLE]

Proof. The analysis closely follow the proof technique of Theorem 2.2. Forming an inner product between (1.1) and $Au^{\prime}$, we arrive at

[TABLE]

and hence,

[TABLE]

Next take an inner product between (1.1) and $-\epsilon Au$ to obtain

[TABLE]

Adding (2.29) with (2.30), we find that

[TABLE]

where

[TABLE]

We now proceed exactly in the proof technique of Theorem 2.2 by replacing ${\cal E}^{(1)}$ by ${\cal E}_{A}^{(1)},\;{\cal E}_{\epsilon}^{(1)}$ by ${\cal E}_{A,\epsilon}^{(1)}$ and using Poincaré inequality (2.3) to arrive at

[TABLE]

whenever $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$. This completes the rest of the proof. $\blacksquare$

Remark 2.1**.**

Since

[TABLE]

A use of elliptic regularity yields $\|\Delta u(t)\|\geq C_{R}\|u(t)\|_{2}$. Hence, using the Sobolev embedding result, we obtain

[TABLE]

Remark 2.2**.**

Following the proof technique of Theorem 2.4, the following result

[TABLE]

where

[TABLE]

can be proved by using induction hypothesis.

Remark 2.3**.**

Assume that $u_{0}\in D(A^{(k)})$ and $u_{1}\in D(A^{(k-1/2)})$ for $k>1.$ Then, following the arguments in Theorem 2.2-2.4 and using induction, there holds:

[TABLE]

where ${\cal E}_{A^{(k)}}^{(j)}=\frac{1}{2}\left(\|A^{(k-1/2)}u^{(j)}(t)\|^{2}+\|A^{(k)}u^{(j-1)}(t)\|^{2}\right).$

3 Semidiscrete scheme

This section analyses the semidiscrete method for the problem (1.1)-(1.3), see, [2], [10]; and discuss decay estimates along with the optimal error estimates.

Let $\{S_{h}^{0}\}_{h>0}$ be a family of subspaces of $H^{1}_{0}$ with the following approximation property:

[TABLE]

The semidiscrete formulation is to find $u_{h}:[0,\infty)\rightarrow S_{h}^{0}$ such that

[TABLE]

where $u_{0,h}$ and $u_{1,h}$ are appropriate approximations of $u_{0}$ and $u_{1}$ respectively, in $S_{h}^{0}$ to be defined later. Since $S_{h}^{0}$ is finite dimensional, (3.2) gives rise to a system of linear ODEs. An application of the Picard’s theorem yields the existence of a unique discrete solution $u_{h}(t)\in S^{0}_{h},\;t\in(0,\infty)$.

Let us first define a discrete counterpart $A_{h}:S_{h}^{0}\mapsto S_{h}^{0}$ of the operator $A$ as

[TABLE]

Then, we rewrite (3.2) as

[TABLE]

3.1 Decay Property

This subsection discusses the decay estimates for the solution of semidiscrete equation. Now, define the energy functional as

[TABLE]

where $\|A_{h}^{1/2}u_{h}\|^{2}:=a(u_{h},u_{h}).$

Theorem 3.1**.**

For $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$, the solution $u_{h}$ of (3.2)-(3.3) satisfies the following decay property

[TABLE]

Proof. A use of $\chi=u_{h}^{\prime}(t)+\epsilon\,u_{h}(t)$ in (3.2) yields

[TABLE]

Since $u_{h}(t)\in S_{h}^{0}\subset H_{0}^{1},$ then by Poincaré inequality (2.2)

[TABLE]

We then proceed exactly like the proof of the Theorem 2.2 replacing $u$ by $u_{h}$ to obtain

[TABLE]

This completes the rest of the proof. $\blacksquare$

Theorem 3.2**.**

For $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$, the solution $u_{h}$ of (3.2)-(3.3) satisfies

[TABLE]

where

[TABLE]

Proof. We prove the result ${\cal E}^{(j)}$ by using the induction hypothesis. Assume that the result is true for $j-1,$ that is,

[TABLE]

We now consider

[TABLE]

Choose $w_{h}=u_{h}^{(j-1)}$ and $\chi=w_{h}^{\prime}+\epsilon\,w_{h}$ and follow similar steps like proof of Theorem 2.3 replacing ${\cal E}$ by ${\cal E}_{h}$ to obtain

[TABLE]

This completes the rest of the proof. $\blacksquare$

Remark 3.1**.**

If $\|A_{h}^{1/2}u_{0h}\|\leq C\|u_{0}\|_{1}$ and $\|u_{1h}\|\leq C\|u_{1}\|$ then

[TABLE]

Observe that using coercivity property of the bilinear form $\|A_{h}^{1/2}u_{h}\|^{2}=a(u_{h},u_{h})\geq\alpha_{0}\|\nabla u_{h}\|^{2},$ we arrive at

[TABLE]

As a consequence of the Sobolev embedding for $d=2$, see, [27]

[TABLE]

Theorem 3.3**.**

For the energy

[TABLE]

where $\hat{u}_{h}(t)=\int_{0}^{t}u_{h}(s)\,ds,$ the following decay estimate holds

[TABLE]

Proof. Integrate the equation (3.2) from [math] to $t$ on both sides to obtain

[TABLE]

Choose $\chi=u_{h}+\epsilon\hat{u}_{h}$ in (3.9) to obtain

[TABLE]

Using the fact $\epsilon(u_{h}^{\prime},\hat{u}_{h})=\epsilon\frac{d}{dt}(u_{h},\hat{u}_{h})-\epsilon\|u_{h}\|^{2}$, we arrive at

[TABLE]

We rewrite the equation (3.1) as

[TABLE]

Define

[TABLE]

and then proceed in a similar manner exactly like the proof of Theorem 2.2 to obtain the required result. This concludes the rest of the proof. $\blacksquare$

Theorem 3.4**.**

For $0<\delta\leq\frac{1}{3}\min\left(\alpha,\frac{\lambda_{1}}{2\alpha}\right)$, and a positive constant $C,$ the solution $u_{h}$ of (3.2)-(3.3) satisfies

[TABLE]

where

[TABLE]

Proof. Forming inner product between equation (3.5) and $A_{h}u^{\prime}_{h}+\epsilon u_{h}$ to obtain

[TABLE]

We then proceed in a similar manner exactly like the proof of Theorem 2.2 and using for $v_{h}\in S_{h}^{0}$, Poincaré inequality (3.7),

[TABLE]

that is, $\|A_{h}^{1/2}v_{h}\|\leq\frac{1}{\sqrt{\lambda_{1}}}\|A_{h}v_{h}\|$ and obtain

[TABLE]

This completes the rest of the proof. $\blacksquare$

3.2 Error estimates.

This subsection deals with optimal error estimates for the semidiscrete scheme. Throughout this subsection, we shall use $r=2$, that is, $S_{h}^{0}$ consisting of $C^{0}$-conforming piecewise linear elements; and for general $r>2$, all the ensuing results hold under assumptions of higher regularity on the exact solution.

Let $R_{h}u$ be the elliptic projection of $u$ defined by

[TABLE]

We split the error as

[TABLE]

Note that $a(\cdot,\cdot)$ satisfies the boundedness and coercivity properties. Setting $\eta=u-R_{h}u$, then the following estimates are easy to obtain

[TABLE]

For details, see, [4].

We subtract the equation (3.2) from (2.5), and using the elliptic projection (3.12), we obtain the error equation in $\theta$ as

[TABLE]

Lemma 3.1**.**

Let $\theta$ satisfies (3.15). Then, there holds

[TABLE]

Proof. Setting $\chi=\theta^{\prime}$ in (3.2), we obtain

[TABLE]

A use of the Cauchy-Schwarz inequality with the Young’s inequality in (3.16) shows

[TABLE]

and hence,

[TABLE]

Substituting $\chi=\epsilon\,\theta$ in (3.2), we obtain

[TABLE]

Apply again the Cauchy-Schwarz inequality in (3.19) to arrive at

[TABLE]

A use of $\epsilon(\theta^{\prime\prime},\theta)=\epsilon\frac{d}{dt}(\theta^{\prime},\theta)-\epsilon\|\theta^{\prime}\|^{2}$ with the Poincaré inequality and the Young’s inequality yields

[TABLE]

Set

[TABLE]

to arrive at

[TABLE]

Denote $F(t)=\left(\frac{\alpha}{2}-\epsilon\right)\|\theta^{\prime}(t)\|^{2}+\frac{\epsilon}{2}\|A^{1/2}\theta\|^{2}$, and obtain

[TABLE]

With

[TABLE]

it follows that

[TABLE]

On substitution, we arrive at

[TABLE]

We rewrite the above equation (3.23) as

[TABLE]

On integration from [math] to $t$, it follows that

[TABLE]

With $2\delta=\frac{4}{3}\epsilon,\;\delta=\frac{2}{3}\epsilon$ and using ${\cal E}^{1}_{\epsilon}$ in terms of ${\cal E}^{1}$, we complete the rest of the proof. $\blacksquare$

Remark 3.2**.**

When $u_{0h}=R_{h}u_{0},$ then $\theta(0)=0$ and therefore,

[TABLE]

With $u_{1h}$ either $L^{2}$ projection or interpolant of $u_{1}$ in $S_{h}^{0}$, we obtain

[TABLE]

Therefore, we arrive at the following superconvergent result for $\|A^{1/2}_{h}\theta(t)\|$

[TABLE]

Since from Remark 2.3 with $k=1$ and $j=3,$ there holds

[TABLE]

and

[TABLE]

A use of $u^{\prime\prime}(0)=-\alpha u_{1}-Au_{0}$ with $u^{(3)}(0)=-\alpha u^{\prime\prime}(0)-Au_{1}=(\alpha^{2}-A)u_{1}+Au_{0}$ implies

[TABLE]

and

[TABLE]

*Hence, we obtain the superconvergence result *

[TABLE]

As a by-product and using triangle inequality with (3.14), there hods

[TABLE]

When $u_{0h}$ is chosen as $L^{2}$-projection or an interpolant, then

[TABLE]

and hence, using the coercivity of the bilinear form $a(\cdot,\cdot),$ we find that

[TABLE]

A use of

[TABLE]

shows the following optimality error estimate.

Theorem 3.5**.**

There holds for $u_{0h}$ and $u_{1h}$ such that

[TABLE]

As a consequence of superconvergent result of $\|\nabla\theta(t)\|$ in (3.30), we apply the Sobolev embedding lemma for $d=2$, (see, [27]) to obtain

[TABLE]

Since

[TABLE]

then, when $d=2$

[TABLE]

provided $\|u(t)\|_{W^{2,\infty}}=O\left(e^{-\delta t}\right)$.

From the superconvergence result for $\|\nabla\theta(t)\|$ in (3.30), one obtains estimate of $\|\theta(t)\|$, but with the assumption of higher regularity, that is, $u_{0}\in H^{4}\cap H^{1}_{0}$ and $u_{1}\in H^{3}\cap H^{1}_{0}$ and only with $u_{0h}=R_{h}u_{0}$.

Below, we directly deduce using a modified version of Baker’s arguments [2], an optimal error estimate of $\|u(t)-u_{h}(t)\|$ requiring $\|u^{\prime}(t)\|=O\left(e^{-2\delta t}\right)$ and $u_{0h}$ as $L^{2}$-projection or interpolant of $u_{0}$ onto $S_{h}^{0}$.

Theorem 3.6**.**

Let $u$ and $u_{h}$ be a solution of (2.5) and (3.2), respectively. Then, there exists a positive constant $C$ independent of $h$ such that

[TABLE]

Proof. Integrate (3.15) with respect to $t$ and obtain

[TABLE]

With a choice of $u_{0h}$ and $u_{1h}$ as $L^{2}$-projection of $u_{0}$ and $u_{1}$, respectively, i.e.,

[TABLE]

Set $\chi=\theta$ in (3.33) to arrive at

[TABLE]

A use of the Cauchy-Schwarz inequality with the Young’s inequality in (3.34), we obtain

[TABLE]

Integrating on both sides of the equation (3.35) from [math] to $t$ and substituting the estimates of $\|\eta\|$ and $\|\eta^{\prime}\|$, we obtain the estimate for $\theta(t)$. Substituting (3.14) in (3.13), and a use of triangle inequality completes the rest of the proof. $\blacksquare$

4 Some Generalizations

In the previous sections, we have discussed exponential decay estimates of a general linear weakly damped wave equation. We now present below the generalizations of our results to weakly damped wave equation with nonhomogeneous forcing function, space dependent damping coefficient, viscous damping and compensation and weakly damped beam equations.

4.1 Inhomogeneous equations

This subsection is on the weakly damped wave equation with nonhomogeneous forcing function:

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

Here, $f=f(x,y)$ is a given function of $(x,y)$ only.

Theorem 4.1**.**

If $u_{\infty}$ is the solution of

[TABLE]

then with $w(t)=u(t)-u_{\infty},$ there holds

[TABLE]

Here, for $j=1,$ there holds $w^{(1)}=w(0)$, and for $j>1$, it follows that $w^{(j)}(0)=u^{(j)}(0).$

Proof. Now $w(t)$ in its abstract form satisfies

[TABLE]

On following the decay properties in Theorem 2.2 and Theorem 2.3, we complete the rest of the proof. $\blacksquare$

Remark 4.1**.**

Following Theorem 2.4 and Theorem 2.3, we again arrive at

[TABLE]

Thus, as in Remark 2.1, we find for $d=2$

[TABLE]

This implies $u(t)\rightarrow u_{\infty}$ in $L^{\infty}(\Omega)$ as $t\rightarrow\infty$.

For semidiscrete scheme, similar results holds good for the semidiscrete solution $w_{h}(t)=u_{h}(t)-u_{\infty,h}$ and we derive $\|u_{h}(t)-u_{\infty,h}\|_{\infty}=O\left(e^{-\delta t}\right)$.

Remark 4.2**.**

In case $f(t)=O\left(e^{-\delta_{0}t}\right)$, then also the solution decay exponentially with decay rate $\delta^{*}=\min\;(\delta_{0},\delta)$.

4.2 On space dependent damping term

In this subsection, we briefly discuss weakly damped wave equation with space dependent damping coefficient of the form, (see, [5], [20] and [9])

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

Here, the space dependent damping coefficient $\alpha(x)$, $x\in\bar{\Omega}$ satisfies

[TABLE]

To indicate the decay property, for simplicity, assume that $\alpha_{2}\alpha_{1}\leq\lambda_{1}$, where $\lambda_{1}$ is first positive eigenvalue of the operator $A.$ Now, an appropriate modification of the analysis of Rauch [20] shows that the continuous energy

[TABLE]

decays like

[TABLE]

Similarly, by differentiating $j$ times in the temporal variable, it is easily follows that

[TABLE]

For the corresponding semidiscrete system: Find $u_{h}(t)\in S_{h}^{0}$ such that

[TABLE]

Setting $w_{h}=e^{\frac{\alpha_{1}}{2}t}u_{h}(t),$ we now rewrite (4.14) in terms of $w_{h}$ as

[TABLE]

Now choose $\chi_{h}=w_{h}^{\prime}$ in (4.15) and define

[TABLE]

Then, as $(\alpha-\alpha_{1})\geq 0$, there holds

[TABLE]

and an integration with respect to time shows

[TABLE]

Note that $\frac{\alpha_{1}^{2}}{4}-\frac{\alpha\alpha_{1}}{2}\leq-\frac{\alpha_{1}^{2}}{4}<0.$ Since $e^{\frac{\alpha_{1}}{2}t}u_{h}^{\prime}=w_{h}^{\prime}-\frac{\alpha_{1}}{2}w_{h}(t),$ it follows using $(a-b)^{2}\leq 2(a^{2}+b^{2})$ that

[TABLE]

Since $\alpha_{2}\alpha_{1}+2\left(\frac{\alpha_{1}^{2}}{4}-\frac{\alpha\alpha_{1}}{2}\right)\geq\frac{\alpha_{1}^{2}}{2}$, we obtain

[TABLE]

A use of the Poincaré inequality $\|w_{h}(t)\|^{2}\leq\frac{1}{\lambda_{1}}\|A^{1/2}_{h}w_{h}(t)\|^{2}$ shows

[TABLE]

If $\alpha_{1}^{2}<\alpha_{1}\alpha_{2}\leq\frac{\lambda_{1}}{2}$, then $\alpha_{2}\alpha_{1}-\lambda_{1}\leq 0$. Thus, a use of (4.17) shows

[TABLE]

Since $\left(\frac{\alpha_{1}^{2}}{4}-\frac{\alpha\alpha_{1}}{2}\right)\leq\frac{\alpha_{1}^{2}}{4},$ we note with $e^{-\frac{\alpha_{1}}{2}t}w_{h}^{\prime}(t)=u_{h}^{\prime}(t)+\frac{\alpha_{1}}{2}u_{h}(t)$ and Poincaré inequality that

[TABLE]

On substitution of (4.19) in (4.18), we arrive at

[TABLE]

Similarly,

[TABLE]

Moreover, we derive all the error estimates as in Section 3. In particular, when $d=2$

[TABLE]

Remark 4.3**.**

In section 3, since $\alpha$ is a constant, the decay rate is $O\left(e^{-\frac{\alpha_{1}}{2}t}\right)$, provided $\alpha_{1}\alpha_{2}=\alpha^{2}\leq\lambda_{1}.$ In fact, the analysis of this subsection improves the decay rate compared to the decay rate in the Section 3.

4.3 On viscous damping and compensation

This subsection focuses on improved decay rates due to both viscous damping and compensation, which is influenced by the paper of Chen [8].

Now, consider the wave equation with viscous damping and compensation: Given positive constants $\alpha$ and $\beta$, find $u$ in $\Omega\times[0,\infty)$ such that

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

Here, $\alpha$ and $\beta$ are called the viscous damping and compensation coefficient, respectively. When $A=-\Delta$, this problem is discussed in [5], and improved exponential decay rates are proved. For general second order linear self-adjoint positive elliptic operator, appropriate modifications will provide the following improved decay estimates for the energy in terms of the theorem.

Theorem 4.2**.**

For any $\delta>0$ with

[TABLE]

the energy

[TABLE]

decays exponentially, that is,

[TABLE]

where the the positive constant $C=O(\delta^{3}).$

Note that for large $\alpha$ and $\beta$, it is possible to derive decay rate $\delta>0$, which remains large. Moreover, for a given $\delta>0$ with (4.23) and $u_{0}\in D(A^{(k)})$ and $u_{1}\in D(A^{(k-1/4)})$ for $k>1,$ there holds using the arguments to arrive at (4.24) and using induction

[TABLE]

where ${\cal E}_{A^{(k)}}^{(j)}=\frac{1}{2}\left(\|A^{(k-1/4)}u^{(j)}(t)\|^{2}+\|A^{(k)}u^{(j-1)}(t)\|^{2}\right).$

Now, the corresponding semidiscrete system is to seek $u_{h}(t)\in S_{h}^{0}$ such that

[TABLE]

With a choice of $\chi_{h}=u_{h}^{\prime}+\delta u_{h}$ in (4.25), it follows using definition $A_{h}$ as in (3.4) with the energy

[TABLE]

and extended energy

[TABLE]

that

[TABLE]

where

[TABLE]

Since from (4.23), the condition

[TABLE]

shows using -\delta^{2}(u_{h}^{\prime},u_{h})\geq-\big{(}(\delta^{2}/2)\|u_{h}^{\prime}\|^{2}+(\delta^{2}/2)\|u_{h}\|^{2}\big{)} that

[TABLE]

On substitution of (4.28) in (4.26), we arrive at

[TABLE]

and hence, an integration with respect to time yields

[TABLE]

Again a use of (4.23) shows

[TABLE]

For obtaining an upper bound, we note using (4.23), $\delta(u^{\prime}_{h},u_{h})\leq(1/2)(\|u_{h}^{\prime}\|^{2}+\delta^{2}\|u_{h}\|^{2})$ and Poincaré inequality (3.7)

[TABLE]

With \frac{1}{2}C(\lambda_{1},\delta)=\frac{1}{2\lambda_{1}}\big{(}2\lambda_{1}+(2\delta+7\delta^{2}+3\delta^{3})\big{)}=O(\delta^{3}), we arrive from (4.30)-(4.31) at

[TABLE]

On substitution in (4.29), we obtain

[TABLE]

Moreover, following the similar line of arguments, there holds for $j\geq 1$

[TABLE]

Further, a use of definition of $A_{h}$ in (3.4) yields

[TABLE]

Following the argument that leads to (4.33) and also the error analysis in section 3, we easily derive optimal error estimates for $\delta>0$ as

[TABLE]

and for $d=2$

[TABLE]

4.4 On weakly damped beam equations

This subsection deals with beam equation with a weakly damping [13].

For a convex polygonal or polyhedral domain $\Omega$ in $\mathbb{R}^{d}$ with boundary $\partial\Omega$ and fixed positive constant $\alpha$, the problem is to find $u(x,t)$ for $(x,t)\in\Omega\times(0,\infty)$ satisfying

[TABLE]

with initial conditions

[TABLE]

and either homogeneous clamped boundary conditions

[TABLE]

where $\nu$ is the outward unit normal to the boundary $\partial\Omega$ or hinged boundary conditions or simply supported boundary conditions.

With $A=\Delta^{2}$ and $D(A)=H^{4}(\Omega)\cap H^{2}_{0}(\Omega)$, results of the previous sections remain valid in the present case with appropriate changes. For semidiscrete FEM, choose $S_{h}^{0}$ be a finite element subspace of $H_{0}^{2}(\Omega)$ satisfying the following approximation property:

[TABLE]

Then, the rest of the decay property holds similarly. Moreover, the following estimates are easy to prove

[TABLE]

5 Numerical Experiments

In this section, we perform some numerical experiments and validate the theoretical results established in the previous sections. We shall carry our numerical experiments for the completely discrete scheme.

5.1 Completely Discrete Scheme

Let $k>0$ be the time step and let $t_{n}=nk,\;n\geq 0$. Set $\varphi^{n}=\varphi(t_{n})$,

[TABLE]

with ${\bar{\partial}}^{0}_{t}\varphi^{n}=\varphi^{n}$.

We define

[TABLE]

Let $\varphi^{n+\frac{1}{2}}=\frac{\varphi^{n+1}+\varphi^{n}}{2}$. We define

[TABLE]

The discrete time finite element approximations $U^{n}$ of $u(t_{n})$ is defined as solution of

[TABLE]

with $U^{0}=u_{0,h}$ and $U^{1}=u_{1,h},$ where, $u_{0,h},u_{1,h}\in S_{h}^{0}$ are appropriate approximations to be defined later.

We now define the discrete energy

[TABLE]

Setting $\chi=\delta_{t}U^{n}$ in (5.1) to obtain

[TABLE]

We note that

[TABLE]

and

[TABLE]

Substituting (5.4)-(5.5) in (5.3), we obtain

[TABLE]

Taking summation for $n=1$ to $m$, we arrtive at

[TABLE]

Therefore, the discrete energy satisfies

[TABLE]

For numerical experiments, examples 1, 2, and 6 are related to the homogeneous weakly damped wave equation with various damping parameter values. Examples 3, 4, 5, and 7 are related to the weakly damped wave equation with a nonhomogeneous forcing function. In all the examples 1-7, the equations are solved up to the final time $T=1.0$ and the value of time step $k=h^{2}$. The numerical experiments are performed using FreeFem++ with piecewise linear elements [14].

In each case, the experimental convergence rate of the error is computed using

[TABLE]

where $E_{h_{i}}$ denotes the norm of the error using $h_{i}$ as the spatial discretizing parameter at $i^{\small{\mbox{th}}}$ stage.

Example 1. [22] For weakly damped wave equation:

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

the exact solution is given by

[TABLE]

In the Table 1, the errors and rate of converges in $L^{2},\;L^{\infty}$ and $H^{1}$- norms are presented with convergence rates confirm with the theoretical results.

In the Figure 1, the decay estimates of errors in Example 1 are shown.

Observe that from Figure 1, we can see the exponentially decay phenomenon for all the three norms $L^{\infty},\,L^{2}$ and $H^{1}$.

Example 2. [12] For the weakly damped wave equation:

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

the exact solution is given by

[TABLE]

Table 2, shows the errors and rate of converges in $L^{2},\;L^{\infty}$ and $H^{1}$-norms, confirming our theoretical findings.

From Figure 2, we observe that $\alpha=8.9$ exponentially decay faster than $\alpha=9.5$ and $\alpha=10$. This confirms that exponentially decay phenomenon for all the three norms $L^{\infty},\,L^{2}$ and $H^{1}$.

Example 3. Consider the following inhomogeneous weakly damped wave equation

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

We compute the unknowns $f,\;u_{0}$ and $u_{1}$ with the help of the exact solution

[TABLE]

In Table 3, we show the numerical results for $\alpha=0.1$, the errors and rate of converges in $L^{2},\;L^{\infty}$ and $H^{1}$-norms.

We observe that errors decay exponentially in Figure 3.

Example 4. [20] Consider the following weakly damped wave equation with space dependent damping coefficient of the form

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

where $\alpha(x)=\alpha_{0}|x|^{-\gamma}$ with some $\alpha_{0}>0$ and $\gamma=[0,1)$. We compute the unknowns $f,\;u_{0}$ and $u_{1}$ with the help of the exact solution

[TABLE]

Below, in Table 4, the errors and rate of converges in $L^{2},\;L^{\infty}$ and $H^{1}$-norms are shown.

In Figure 4, we observe that errors decay exponentially.

Example 5. Consider the following semilinear weakly damped wave equation, see [3] and [16]

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

where $f(u)=u^{3}-u$.

We compute the unknowns $g,\;u_{0}$ and $u_{1}$ with the help of the exact solution

[TABLE]

The errors and rate of converges in $L^{2},\;L^{\infty}$ and $H^{1}$-norms are shown in the Table 5.

In Figure 5, we observe that errors decay exponentially.

Example 6. Consider the following semilinear weakly damped wave equation, see [15]

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

where $f(u)=u^{3}-u$.

We observe that errors decay exponentially in Figure 6.

In Figure 7, we show the decay plots for different values of damping coefficient $\alpha$. We observe that $\alpha=7$ decay exponentially faster than $\alpha=3$ and $\alpha=5$. This confirms that exponentially decay phenomenon for the norms $L^{2}$ and $L^{\infty}$.

Example 7. Consider the following wave equation with viscous damping and compensation

[TABLE]

with initial conditions

[TABLE]

and the boundary condition

[TABLE]

We calculate the values of damping coefficient $\alpha$ and compensation coefficient $\beta$ from (4.23). If we choose $\delta=2$ and $\delta=5$, that is, decay rate $1$ and $5/2,$ respectively, then we obtain $\alpha=10,\;\beta=32$ and $\alpha=40,\;\beta=335$, respectively. Now the decay plots for different values of damping coefficient $\alpha$ and compensation coefficient $\beta$ are shown in Figure 8.

From Figure 8(a),(c), we observe that when $\alpha=40$ and $\beta=335$, the errors decay exponentially faster than $\alpha=10$ and $\beta=32$. This confirms that for any arbitrary $\delta,$ one may choose damping coefficient $\alpha$ and compensation parameter $\beta$ appropriately so that errors in $L^{2},H^{1}$ and $L^{\infty}$-norms decay exponentially with decay rate $\delta/2.$ Now, we examine the decay estimates by setting compensation coefficient $\beta=0$. We observe that the decay estimates of Figure 8(a),(c) and Figure 8(b),(d). Comparing both the decay estimates, we see that the errors in Figure 8(a),(c) decay exponentially faster than the errors in Figure 8(b),(d). This confirms that the compensation term $\beta$ is helping in the weakly damping equation to get the errors decay exponentially faster.

It is further observe through numerical experiments for the wave equation with different viscous damping coefficients and compensation coefficients in Figure 9 that for large decay rates one may choose the compensation term and damping coefficient large as given in the subsection 4.3 which is better than the decay rate predicted in the Sections 2 and 3. Say, for example with decay rates $1$, the damping coefficients $\alpha=10$ and the compensation parameter $32,$ the predicted decay rate as in Sections 2 and 3 for the Example 7 is less than equal to $11/20$ with $\lambda_{1}=1$ and $\frac{1}{3}\min(10,33/20)=11/20,$ which confirms our results in subsection 4.3.

We now calculate decay rate $\delta/2$ numerically. When $\alpha=10,\;\beta=32$, from Figure 9, it is noticed that $\delta$ is converging close to $2,$ that is, decay rate $1,$ which confirms theoretical result in subsection 4.3. Further with $\alpha=10,\;\beta=32,$ $\delta$ is converging close to $6.6,$ that is, the decay rate in this case is roughly $3.2$, which seems to be better than the decay rate $2.5$ as predicted by the Theorem 4.2. This suggests that the choice of $\alpha$ and $\beta$ in terms of $\delta$ may not be conservative.

5.2 Conclusions

In this article, the uniform exponential decay estimates for the linear weakly damped wave equation are developed and analyzed for continuous and semidiscrete problem. Semidiscrete approximations are obtained by applying FEM to discretize in space directions keeping the time variable continuous. Compared to the existing literature, improved decay rates with rates lying in a range are derived. It is further observed that optimal error estimates, which depict the decay behaviour are proved with minimal smoothness assumptions on the initial data. The present analysis is extended to problems with inhomogeneous forcing function, space dependent damping coefficient, viscous damping and compensation. As a consequence of our abstract analysis, the proof technique is also generalized to a weakly damped beam equation. Several numerical experiments are performed to validate the theoretical results established in this article. The optimal rate of convergence is achieved in Table 1-5 and uniform exponential decay behaviour is observed in Figure 1-8. In examples 5-6, it is shown numerically that the semidiscrete solution of the semilinear weakly damped equation decays exponentially, and in future, we shall develop similar results as in linear case. Moreover, our future investigation will include the uniform exponential decay estimates for the complete discrete schemes.