A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes

Abstract

In this study, we design and analyze weak Galerkin finite element methods to approximate diffusive viscus wave equations with variable coefficients on polygonal meshes. The proposed method has numerous assets, including supporting a higher order of convergence and general polygonal meshes. We investigated the convergence analysis using a two-step technique that discretizes first in space and then in time. A second-order Newmark scheme is employed to develop the temporal discretization and obtain the optimal order of convergence rate in $L^{\infty }(L^2)$ and $L^{\infty }(H^1)$ norms. In other words, we attain ${{\mathcal {O}}}(h^{k+1}+\tau ^2)$ in $L^{\infty }(L^2)$ norm and ${{\mathcal {O}}}(h^{k}+\tau ^2)$ in $L^{\infty }(H^1)$ norm. We performed several numerical experiments in a two-dimensional setting, illustrating our theoretical convergence findings.

Appendix

Lemma 6.1

Let $w\in H^1(0,T; H^2(\Omega ))$ be the solutions of the equation (3.15) and $w_h$ be its WG approximation. Then, there exists a constant C such that

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_{h}w-w_{h}\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\le Ch\Vert \zeta _u\Vert _{L^2(J;L^2(\Omega ))}, \end{aligned}$$

(6.1)

Proof

Following the analysis used to derive (3.7), we obtain

$$\begin{aligned}{} & {} {{\mathcal {A}}}_{1,w}({\Pi }_{h}w, \phi _h)+{\mathcal A}_{2,w}(({\Pi }_{h}w)_{t}, \phi _{h}) = (f_w, \phi _0)+\ell _1(w, \phi _h)+ \ell _2(w, \phi _{h}) \nonumber \\{} & {} \quad +\ell _3(w^{\prime }, \phi _h)+ \ell _4(w^{\prime }, \phi _{h}),\;\;\forall \;\phi _h=\{\phi _0, \phi _b\}\in {\mathcal M}_h^0. \end{aligned}$$

(6.2)

Next, we may define $w_h\in {{\mathcal {M}}}_h^0$ as the solution to the weak Galerkin approximation of the Eq. (3.15) that follows

$$\begin{aligned} {{\mathcal {A}}}_{1,w}(w_h, \varphi _h)+{{\mathcal {A}}}_{2,w}(w_h^{\prime }, \varphi _h) = (f_{w}, \varphi _0),\;\;\forall \; \varphi _h=\{ \varphi _0, \varphi _b\}\in {{\mathcal {M}}}_h^0, \end{aligned}$$

(6.3)

with $w_h(\tau ) = {\Pi }_{h}w^0.$

Now, subtracting (6.3) from the Eq. (6.2), we arrive at the following error relation for ${\tilde{e}}_h: = {\Pi }_{h}w-{w}_{h}$

$$\begin{aligned}{} & {} {{\mathcal {A}}}_{1,w}({\tilde{e}}_h(t), \phi _h)+{\mathcal A}_{2,w}({\tilde{e}}_{h}^{\prime }(t), \phi _{h}) = \ell _1(w, \phi _h)+ \ell _2(w, \phi _{h})+\ell _3(w^{\prime }, \phi _h) \nonumber \\{} & {} \quad +\ell _4(w^{\prime },\phi _{h}),\;\forall \;\varphi _h\in {\mathcal M}_h^0,\;t\in (0, T]. \end{aligned}$$

(6.4)

Finally, setting $\phi _h={\tilde{e}}_h$ in (6.4) and then standard analysis as we did in Theorem 3.1 combined with the estimations (3.28) and (3.32) yields the following estimate

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\tilde{e}}_h(t)\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }^2\le & {} C\big ({\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\tilde{e}}_h(0)\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }^2+h^2\Vert w\Vert _{H^1(0, T;H^{2}(\Omega ))}^2\big )\\\le & {} Ch^2\Vert w\Vert _{H^1(0, T;H^{2}(\Omega ))}^2\\\le & {} Ch^2\Vert \zeta _u\Vert ^2_{L^2(J;L^2(\Omega ))}. \end{aligned}$$

Here, we have used the estimate (3.22) together with the fact that ${\tilde{e}}_h(0)= 0.$ The proof is completed. $\square$

Remark 6.1

We recall a dual problem that seeks a solution $w\in H^1(J; H^2(\Omega ))$ such that

$$\begin{aligned} -\nabla \cdot \big (\varepsilon \nabla w)-(\beta \nabla w^{\prime })\big ) = \zeta _u\;\;\text{ in }\;\;\Omega \times J, \end{aligned}$$

(6.5)

and $w(\tau ) = 0$ for some $\tau \in J.$

We may define $w_h\in {{\mathcal {M}}}_h^0$ as the solution to the discrete problem of the Eq. (6.5) that follows

$$\begin{aligned} {{\mathcal {A}}}_{1,w}(w_h, \varphi _h)-{{\mathcal {A}}}_{2,w}(w_h^{\prime }, \varphi _h) = (\zeta _{u}, \varphi _0),\;\;\forall \; \varphi _h=\{ \varphi _0, \varphi _b\}\in {{\mathcal {M}}}_h^0, \end{aligned}$$

(6.6)

with $w_h(\tau ) = 0.$

Setting $\varphi _h=w_h$ in (6.6) and using the coercive property (2.11), we obtain

$$\begin{aligned} \sigma _{*}{\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }^2-\frac{1}{2}\frac{d}{dt}\big ({\mathcal A}_{2,w}(w_h(s), w_h(s))\big )\le \Vert \zeta _{u}\Vert \Vert w_0\Vert . \end{aligned}$$

Next, integrate the above equation in $[0, \tau ]$ to obtain

$$\begin{aligned} \sigma _{*}{\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }^2+\frac{1}{2}{{\mathcal {A}}}_{2,w}(w_h(0), w_h(0)))\le \Vert \zeta _{u}\Vert \Vert w_0\Vert . \end{aligned}$$

Here, we used the fact that $w_h(\tau ) =0$ and hence, ${\mathcal A}_{2,w}(w_h(\tau ), w_h(\tau )) = 0.$

Now, we apply the Poincaŕe-type inequality (2.15) and positive definiteness of ${{\mathcal {A}}}_{2,w}(\cdot , \cdot )$ in the above estimate, we get

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\le C\Vert \zeta _{u}\Vert . \end{aligned}$$

(6.7)

When we set $\varphi _h=w_h^{\prime }$ in (6.6), we can get

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| w_h^{\prime }\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\le C\Vert \zeta _{u}\Vert . \end{aligned}$$

(6.8)

The following estimates are satisfied by $w_h$, which is the WG approximation to w (see, estimate (6.1))

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_hw-w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\le Ch\Vert \zeta _u\Vert _{L^2(L^2)}. \end{aligned}$$

(6.9)

Now, we combine estimates (6.7) and (6.9) to obtain

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_{h}w\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }={\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_hw-w_h+w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\le & {} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_hw-w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } +{\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| w_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }\nonumber \\\le & {} C\Vert \zeta _u\Vert _{L^2(L^2)}. \end{aligned}$$

(6.10)

As a consequence, we can prove that

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_{h}w^{\prime }\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }={\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| {\Pi }_hw^{\prime }-w_h^{\prime }+w^{\prime }_h\right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } \le C\Vert \zeta _u\Vert _{L^2(L^2)}. \end{aligned}$$

(6.11)