Virtual element methods for weakly damped wave equations on polygonal meshes

Abstract

We develop a virtual element method for weakly damped wave equations on polygonal meshes. Very general polygonal meshes are used for the spatial discretization. In both $L^{2}$ norm and $H^{1}$ semi-norm, optimal order of convergence is obtained for the spatially discrete approximation. We employ the Crank–Nicolson temporal discretization scheme for the fully discrete problem and derive the convergence analysis. Numerical experiments are illustrated to confirm our theoretical findings.

Appendix

Proof of Lemma 3.1

We take $t\rightarrow 0^{+}$ in (3.1) and then using definition of the elliptic projection operator $R_{h},$ we get

$$\begin{aligned} m_{h}(u_{htt}(0),w_{h})&=-s_{h}(u_{ht}(0),w_{h})-a_{h}(u_{h}(0),w_{h})+(f_{h}(0),w_{h}) \nonumber \\&=-s_{h}(R_{h}v_{0},w_{h})-a_{h}(R_{h}u_{0},w_{h})+(f_{h}(0),w_{h})\nonumber \\&=-s_{h}(R_{h}v_{0},w_{h})-a(u_{0},w_{h})+(f_{h}(0),w_{h}). \end{aligned}$$

(5.1)

From the continuity of the bilinear form $s_{h}$ and the $H^{1}$-stability of $R_{h},$ we have

$$\begin{aligned} s_{h}(R_{h}v_{0},w_{h}) \le \gamma _{2}\Vert R_{h}v_{0}\Vert \Vert w_{h}\Vert \le C\Vert v_{0}\Vert \Vert w_{h}\Vert . \end{aligned}$$

(5.2)

By using Green’s formula and boundary condition the second term can be bounded as

$$\begin{aligned} -a(u_{0},w_{h})=(\nabla \cdot (\sigma (x)\nabla u_{0}),w_{h})\le C\Vert u_{0}\Vert _{2}\Vert w_{h}\Vert . \end{aligned}$$

(6.1)

Again, from Lemma 2.1 and the fact that, for any Banach space ${\mathbb {B}},$ (cf. Robinson 2001, Proposition 7.1)

$$\begin{aligned} \sup _{0\le t \le T}\Vert v(t)\Vert _{{\mathbb {B}}}\le C \Vert v\Vert _{H^{1}({\mathcal {J}};{\mathbb {B}})},\quad v\in H^{1}({\mathcal {J}};{\mathbb {B}}), \end{aligned}$$

(6.2)

we obtain

$$\begin{aligned} (f_{h}(0),w_{h}) \le C\Vert f\Vert _{H^{1}(L^{2})}\Vert w_{h}\Vert . \end{aligned}$$

(6.3)

Now, substituting $w_{h}=u_{htt}(0),$ using the stability of $m_{h}$ and the estimates in (6.2), (6.3) and (4.10) in (6.1) yields

$$\begin{aligned} \Vert u_{htt}(0)\Vert \le C(\Vert u_{0}\Vert _{2}+\Vert v_{0}\Vert +\Vert f\Vert _{H^{1}(L^{2})}). \end{aligned}$$

(6.4)

Again, from the $H^{1}$-stability of $R_{h},$ we have

$$\begin{aligned} \Vert u_{ht}(0)\Vert _{1}=\Vert R_{h}v_{0}\Vert _{1} \le C\Vert v_{0}\Vert _{1}. \end{aligned}$$

This proves the result for $k=2.$ For $k=3,$ from (3.1) and (2.1) for $t\rightarrow 0^+,$ we have

$$\begin{aligned} m_{h}(u_{htt}(0),w_{h})-(u_{tt}(0),w_{h})=s(v_{0},w_{h}) -s_{h}(R_{h}v_{0},w_{h})+(f_{h}(0)-f(0),w_{h}). \end{aligned}$$

Here, we have used the definition of $R_{h}$ operator. Now, from the definition of $L^{2}$-projection operators $L_{i,h},$ $i=1,2,$ we obtain

$$\begin{aligned} m_{h}(u_{htt}(0)-L_{1,h}u_{tt}(0),w_{h})=s_{h}(L_{2,h}v_{0} -R_{h}v_{0},w_{h})+(f_{h}(0)-f(0),w_{h}). \end{aligned}$$

Setting $w_{h}=u_{htt}(0)-L_{1,h}u_{tt}(0),$ using stability of $m_{h},$ continuity of $s_{h},$ approximation properties of operators $R_{h}$ and $L_{2,h},$ and definition of $f_{h},$ we obtain

$$\begin{aligned} \Vert u_{htt}(0)-L_{1,h}u_{tt}(0)\Vert&\le C(\Vert L_{2,h}v_{0}-R_{h}v_{0}\Vert + \Vert f_{h}(0)-f(0)\Vert ) \end{aligned}$$

(6.5)

$$\begin{aligned}&\le Ch(\Vert v_{0}\Vert _{1}+\Vert f\Vert _{H^{1}(H^{1})}). \end{aligned}$$

(6.6)

Estimate (4.12) along with inverse inequality (2.5) and $H^{1}$-stability of $L_{1,h}$ yields

$$\begin{aligned} \Vert u_{htt}(0)\Vert _{1}&\le Ch^{-1}\Vert u_{htt}(0)-L_{1,h}u_{tt}(0)\Vert + \Vert L_{1,h}u_{tt}(0)\Vert _{1}\nonumber \\&\le C(\Vert v_{0}\Vert _{1}+\Vert f\Vert _{H^{1}(H^{1})}+\Vert u_{tt}(0)\Vert _{1}). \end{aligned}$$

(6.7)

From (1.1), we have

$$\begin{aligned} u_{tt}(0)=-\alpha (x)v_{0}+\nabla \cdot (\sigma (x)\nabla u_{0})+f(0). \end{aligned}$$

(6.8)

Now, using (6.4), we get

$$\begin{aligned} \Vert u_{tt}(0)\Vert _{1}\le C(\Vert u_{0}\Vert _{3}+\Vert v_{0}\Vert _{1}+\Vert f\Vert _{H^{1}(H^{1})}). \end{aligned}$$

(6.9)

Hence, using (6.11), (6.9) reduces to

$$\begin{aligned} \Vert u_{htt}(0)\Vert _{1}\le C(\Vert u_{0}\Vert _{3}+\Vert v_{0}\Vert _{1}+\Vert f\Vert _{H^{1}(H^{1})}). \end{aligned}$$

(6.10)

Next, differentiating (3.1) partially with respect to variable t and then taking $t \rightarrow 0^{+},$ we obtain

$$\begin{aligned} m_{h}(u_{httt}(0),w_{h})&=-s_{h}(u_{htt}(0),w_{h}) -a_{h}(R_{h}v_{0},w_{h})+(f_{ht}(0),w_{h})\\&=-s_{h}(u_{htt}(0),w_{h})-a(v_{0},w_{h})+(f_{ht}(0),w_{h}). \end{aligned}$$

Following the arguments as in (6.6), we can easily derive

$$\begin{aligned} \Vert u_{httt}(0)\Vert \le C(\Vert u_{0}\Vert _{2}+\Vert v_{0}\Vert _{2}+\Vert f\Vert _{H^{2}(L^{2})}). \end{aligned}$$

(6.11)

This proves the result for $k=3.$ For $k=4,$ we first observe that

$$\begin{aligned} m_{h}(u_{htttt}(0),w_{h})&=-s_{h}(u_{httt}(0),w_{h}) -a_{h}(u_{htt}(0),w_{h})+(f_{htt}(0),w_{h}) \nonumber \\&=-s_{h}(u_{httt}(0),w_{h})-a_{h}(u_{htt}(0)-R_{h}u_{tt}(0),w_{h})\nonumber \\&\quad -a(u_{tt}(0),w_{h})+(f_{htt}(0),w_{h}) \nonumber \\&=-s_{h}(u_{httt}(0),w_{h})-a_{h}(u_{htt}(0)-R_{h}u_{tt}(0),w_{h})\nonumber \\&\quad +(\nabla \cdot (\sigma (x)\nabla u_{tt}(0)),w_{h})+(f_{htt}(0),w_{h}) \nonumber \\&\le C\left( \Vert u_{httt}(0)\Vert +h^{-1}\Vert u_{htt}(0)-R_{h}u_{tt}(0)\Vert _{1}\right. \nonumber \\&\quad \left. +\Vert u_{tt}(0)\Vert _{2}+ \Vert f\Vert _{H^{3}(L^{2})}\right) \Vert w_{h}\Vert . \end{aligned}$$

(6.12)

Now using inverse inequality (2.5) and (6.7), we have

$$\begin{aligned} \Vert u_{htt}(0)-R_{h}u_{tt}(0)\Vert _{1}&\le Ch^{-1} \Vert u_{htt}(0) -L_{1,h}u_{tt}(0)\Vert +\Vert L_{1,h}u_{tt}(0)-R_{h}u_{tt}(0)\Vert _{1}\nonumber \\&\le Ch(\Vert v_{0}\Vert _{2}+\Vert f\Vert _{H^{1}(H^{2})}+\Vert u_{tt}(0)\Vert _{2}). \end{aligned}$$

(6.13)

Similar to (6.11) from (6.10), we have

$$\begin{aligned} \Vert u_{tt}(0)\Vert _{2}\le C(\Vert u_{0}\Vert _{4}+\Vert v_{0}\Vert _{2}+\Vert f\Vert _{H^{1}(H^{2})}). \end{aligned}$$

(6.14)

Substituting $w_{h}=u_{htttt}(0)$ and then using estimate (6.16) together with (6.13) and (6.15) in (6.14), gives us

$$\begin{aligned} \Vert u_{htttt}(0)\Vert \le C(\Vert u_{0}\Vert _{4}+\Vert v_{0}\Vert _{2}+\Vert f\Vert _{H^{3}(H^{2})}). \end{aligned}$$

(6.15)

Now, using the same argument employed in (6.9), we get

$$\begin{aligned} \Vert u_{httt}(0)\Vert _{1}&\le Ch^{-1}\Vert u_{httt}(0)-L_{1,h}u_{ttt}(0)\Vert +\Vert L_{1,h}u_{ttt}(0)\Vert _{1}\nonumber \\&\le C(\Vert u_{0}\Vert _{3}+\Vert v_{0}\Vert _{3}+\Vert f\Vert _{H^{2}(H^{1})}). \end{aligned}$$

$\square$

Proof of Lemma 3.2

Differentiating (3.1) twice with respect to t and then substituting $w_{h}=u_{httt},$ we obtain

$$\begin{aligned} \dfrac{1}{2}\dfrac{d}{dt}m_{h}(u_{httt},u_{httt})+s_{h}(u_{httt},u_{httt})+ \dfrac{1}{2}\dfrac{d}{dt}a_{h}(u_{htt},u_{htt})=(f_{htt},u_{httt}). \end{aligned}$$

Now integrating from 0 to T,  application of Young’s inequality and stability of $m_{h}$ yields

$$\begin{aligned} \int _{0}^{T}\Vert u_{httt}(t)\Vert ^{2}dt\le C\Big (\Vert u_{httt}(0)\Vert ^{2}+\Vert u_{htt}(0)\Vert _{1}^{2}+\int _{0}^{T} \Vert f_{htt}(t)\Vert ^{2}dt\Big ). \end{aligned}$$

Further, using Lemma 3.1 in the above equation, we have

$$\begin{aligned} \Vert u_{httt}\Vert ^{2}_{L^{2}(L^{2})}\le C(\Vert u_{0}\Vert _{3}^{2}+\Vert v_{0}\Vert _{2}^{2}+\Vert f\Vert ^{2}_{H^{2}(H^{1})}). \end{aligned}$$

This proves for $k=3.$ For the case $k=4,$ proceeding in a similar manner, we obtain

$$\begin{aligned} \Vert u_{htttt}\Vert ^{2}_{L^{2}(L^{2})}\le C(\Vert u_{0}\Vert _{4}^{2}+\Vert v_{0}\Vert _{3}^{2}+\Vert f\Vert ^{2}_{H^{3}(H^{2})}). \end{aligned}$$

$\square$