Supercloseness analysis of a stabilizer-free weak Galerkin finite element method for viscoelastic wave equations with variable coefficients

Abstract

In this article, we are concerned about a stabilizer-free weak Galerkin (SFWG) finite element method for approximating a second-order linear viscoelastic wave equation with variable coefficients. For SFWG solutions, both semidiscrete and fully discrete convergence analysis is considered. The second-order Newmark scheme is employed to develop the fully discrete scheme. We obtain supercloseness of order two, which is two orders higher than the optimal convergence rate in $L^{\infty }(L^{2})$ and $L^{\infty }(H^{1})$ norms. In other words, we attain $\mathcal {O}(h^{k+3}+\tau ^{2})$ in $L^{\infty }(L^{2})$ norm and $\mathcal {O}(h^{k+2}+\tau ^{2})$ in $L^{\infty }(H^{1})$ norm. Several numerical experiments in a two-dimensional setting are carried out to validate our theoretical convergence findings. These experiments confirm the robustness and accuracy of the proposed method.

Appendix

Proof of Lemma 3.1.

Differentiating (3.1) twice with respect to time t and substitute $\phi _{h}= u^{\prime \prime \prime }_{h}(t),$ we have

$$ \begin{array}{@{}rcl@{}} (u^{\prime\prime\prime\prime}_{h},u^{\prime\prime\prime}_{h})+\mathcal{A}_{1,w}(u^{\prime\prime}_{h},u^{\prime\prime\prime}_{h})+\mathcal{ A}_{2,w}(u^{\prime\prime\prime}_{h},u^{\prime\prime\prime}_{h}) = (f^{\prime\prime}, u^{\prime\prime\prime}_{h}). \end{array} $$

We can restate the above equation as

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{2}\frac{d}{dt}\Big(\|u^{\prime\prime\prime}_{h}\|^{2}+\mathcal{A}_{1,w}(u^{\prime\prime}_{h}, u^{\prime\prime}_{h})\Big)+\mathcal{A}_{2,w}(u^{\prime\prime\prime}, u^{\prime\prime\prime}) = (f^{\prime\prime}, u^{\prime\prime\prime}). \end{array} $$

Now, integrate the above equation with respect to time from 0 to t and apply the Cauchy-Schwarz inequality; we get

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{2}\|u^{\prime\prime\prime}_{h}\|^{2}+\frac{1}{2}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert u^{\prime\prime}_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}+{{\int}_{0}^{t}}\|u^{\prime\prime\prime}_{h}\|^{2}ds+{{\int}_{0}^{t}}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert u^{\prime\prime\prime}_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}ds \\&& \quad\leq \frac{1}{2}\|u^{\prime\prime\prime}_{h}(0)\|^{2}+\frac{1}{2}\left\vert\left\vert\left\vert{u^{\prime\prime}_{h}(0)}\right\vert\right\vert\right\vert^{2}+\frac{1}{2}\Big({{\int}_{0}^{t}}\|u^{\prime\prime\prime}_{h}(s)\|^{2}ds+{{\int}_{0}^{t}}\|f^{\prime\prime}\|^{2}ds\Big). \end{array} $$

We can rearrange the above equation as

$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}}\|u^{\prime\prime\prime}_{h}(s)\|^{2}ds+{{\int}_{0}^{t}}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert u^{\prime\prime\prime}_{h}(s)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}ds \leq C\Big(\|u^{\prime\prime\prime}_{h}(0)\|^{2}+\left\vert\left\vert\left\vert{u^{\prime\prime\prime}_{h}(0)}\right\vert\right\vert\right\vert^{2}\\&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+{{\int}_{0}^{t}}\|f^{\prime\prime}(s)\|^{2}ds\Big). \end{array} $$

(6.2)

Now, we need to bound $\|u^{\prime \prime \prime }_{h}(0)\|^{2}$ and ${\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert u^{\prime \prime \prime }_{h}(0)\right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }^{2}$ in (6.2). To this end, taking t → 0⁺ in (1.1), it follow that for 0 ≤ λ ≤ k,

$$ \begin{array}{@{}rcl@{}} \|u^{\prime\prime}(0)\|_{\lambda} \leq C \Big(\|u^{0}\|_{\lambda+2}+\|u^{\prime}(0)\|_{\lambda+2}+\|f\|_{H^{1}(H^{\lambda})}\Big) \end{array} $$

(6.3)

and

$$ \begin{array}{@{}rcl@{}} \|u^{\prime\prime\prime}(0)\|_{\lambda} \leq C \Big(\|u^{0}\|_{\lambda+4}+\|u^{\prime}(0)\|_{\lambda+4}+\|f\|_{H^{2}(H^{\lambda})}\Big) \end{array} $$

(6.4)

Next, we differentiate (3.1) with respect to time t and using the definition of $\mathcal {E}_{h}$ operator (3.18). Then, setting t → 0⁺ to have

$$ \begin{array}{@{}rcl@{}} (u^{\prime\prime\prime}_{h}(0), \phi_{0}) &=&-\mathcal{A}_{1,w}(u^{\prime}_{h}(0),\phi_{h})-\mathcal{A}_{2,w}(u_{h}^{\prime\prime}(0),\phi_{h})+(f^{\prime}(0),\phi_{0}) \\ &=&-\mathcal{A}_{1,w}(\mathcal{E}_{h}u^{\prime}(0),\phi_{h})-\mathcal{A}_{2,w}(\mathcal{E}_{h}u^{\prime\prime}(0),\phi_{h})+(f^{\prime}(0),\phi_{0}) \\ &=& (\nabla\cdot(\alpha\nabla u^{\prime}(0),\phi_{h}))+(\nabla\cdot(\beta\nabla u^{\prime\prime}(0)),\phi_{h})+(f^{\prime}(0),\phi_{0}) \end{array} $$

Now, applying the Cauchy-Schwarz inequality together with the estimate (6.3) in the above equation with λ = 2, we obtained

$$ \begin{array}{@{}rcl@{}} \|u^{\prime\prime\prime}_{h}(0)\|\leq C \Big(\|u^{0}\|_{H^{4}({\Omega})}+\|v^{0}\|_{H^{4}({\Omega})}+\|f\|_{H^{2}(J;H^{2}({\Omega}))}\Big). \end{array} $$

(6.5)

In the previous estimate, we have used the fact that (cf. [43], Proposition 7.1)

$$ \sup_{0\le t \le T}\|v(t)\|_{\mathcal{B}}\leq C(T)\|v\|_{H^{1}(J;\mathcal{B})} \forall v\in H^{1}(J;\mathcal{B}), $$

(6.6)

for any Banach space ${\mathscr{B}}.$

As a consequence of estimate (6.5) together with standard inverse inequality, estimate (6.4) with λ = 2, and the fact that $\|u\|_{L^{2}(K)}\leq Ch\|u\|_{2,K},$ we obtain

$$ \begin{array}{@{}rcl@{}} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert u^{\prime\prime\prime}_{h}(0)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert} \!\leq\! C h^{-1}\|{u^{\prime\prime\prime}_{h}(0)}\|\!\leq\! C \Big(\|u^{0}\|_{H^{6}({\Omega})}+\|v^{0}\|_{H^{6}({\Omega})}+\|f\|_{H^{2}(J;H^{2})}\Big).~~~~~~~~~ \end{array} $$

(6.7)

Again, we are differentiating (3.1) thrice with respect to time t and substitute $\phi _{h}= u^{\prime \prime \prime \prime }_{h}(t)$, we get

$$ \begin{array}{@{}rcl@{}} (u^{\prime\prime\prime\prime\prime}_{h},u^{\prime\prime\prime\prime}_{h})+\mathcal{A}_{1,w}(u^{\prime\prime\prime}_{h},u^{\prime\prime\prime\prime}_{h})+\mathcal{ A}_{2,w}(u^{\prime\prime\prime\prime}_{h},u^{\prime\prime\prime\prime}_{h}) = (f^{\prime\prime\prime},u^{\prime\prime\prime\prime}_{h}). \end{array} $$

Then, it follows from (6.2) that

$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}}\|u^{\prime\prime\prime\prime}_{h}(s)\|^{2}ds+{{\int}_{0}^{t}}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert u^{\prime\prime\prime\prime}_{h}(s)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}ds \leq C\Big(\|u^{\prime\prime\prime\prime}_{h}(0)\|^{2}+\left\vert\left\vert\left\vert{u^{\prime\prime\prime}_{h}(0)}\right\vert\right\vert\right\vert^{2}\\&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad +{{\int}_{0}^{t}}\|f^{\prime\prime\prime}(s)\|^{2}ds\Big). \end{array} $$

(6.8)

Here, the term ${\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert u^{\prime \prime \prime }_{h}(0)\right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }$ can be bound using the estimate (6.7). To the bound $\|u^{\prime \prime \prime \prime }_{h}(0)\|$ and in (6.8), we follow the step from (6.3)–(6.7), and we get

$$ \begin{array}{@{}rcl@{}} \|u^{\prime\prime\prime\prime}_{h}(0)\|\leq C \Big(\|u^{0}\|_{H^{6}({\Omega})}+\|v^{0}\|_{H^{6}({\Omega})}+\|f\|_{H^{3}(J;H^{2}({\Omega}))}\Big). \end{array} $$

□

Lemma 6.1

Let w ∈ H¹(0,T; H²(Ω)) be the solutions of the (3.19) and w_h be its SFWG approximation. Then, there exists a constant C such that

$$ \begin{array}{@{}rcl@{}} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w-w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}\leq Ch\|\zeta_{u}\|_{L^{2}(J;L^{2}({\Omega}))}, \end{array} $$

(6.9)

Proof

The following analysis used to derive (3.9), we obtain

$$ \begin{array}{@{}rcl@{}} \mathcal{A}_{1,w}(\mathcal{Q}_{h}w, \phi_{h})&+&\mathcal{A}_{2,w}((\mathcal{Q}_{h}w)_{t}, \phi_{h}) = (f_{w}, \phi_{0})+\ell_{1}(w, \phi_{h})+ \ell_{2}(w, \phi_{h}) \\&+&\ell_{3}(w^{\prime}, \phi_{h})+ \ell_{4}(w^{\prime}, \phi_{h}), \forall \phi_{h}=\{\phi_{0}, \phi_{b}\}\in \mathcal{V}_{h}^{0}. \end{array} $$

(6.10)

Next, we may define $w_{h}\in \mathcal {V}_{h}^{0}$ as the solution to the SFWG approximation of the equation (3.19) that follows

$$ \mathcal{A}_{1,w}(w_{h}, \varphi_{h})+\mathcal{A}_{2,w}(w_{h}^{\prime}, \varphi_{h}) = (f_{w}, \varphi_{0}), \forall \varphi_{h}=\{ \varphi_{0}, \varphi_{b}\}\in \mathcal{V}_{h}^{0}, $$

(6.11)

with $w_{h}(\tau ) = \mathcal {Q}_{h}w^{0}.$

Now, subtracting (6.11) from the equation (6.10), we arrive at the following error relation for $\tilde {e}_{h}: = \mathcal {Q}_{h}w-{w}_{h}$

$$ \begin{array}{@{}rcl@{}} &&\mathcal{A}_{1,w}(\tilde{e}_{h}(t), \phi_{h})+\mathcal{A}_{2,w}(\tilde{e}_{h}^{\prime}(t), \phi_{h}) = \ell_{1}(w, \phi_{h})+ \ell_{2}(w, \phi_{h})+\ell_{3}(w^{\prime}, \phi_{h}) \\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \ell_{4}(w^{\prime},\phi_{h}), \forall \varphi_{h}\in \mathcal{V}_{h}^{0}, t\in (0, T]. \end{array} $$

(6.12)

Finally, putting $\phi _{h}=\tilde {e}_{h}$ in (6.12) and then standard analysis as we did in Theorem 3.2 combined with the estimations (3.32) and (3.36) yields the following estimate

$$ \begin{array}{@{}rcl@{}} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \tilde{e}_{h}(t)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}&\le& C\big({\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \tilde{e}_{h}(0)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}+h^{2}\|w\|_{H^{1}(0, T;H^{2}({\Omega}))}^{2}\big)\\ &\le& Ch^{2}\|w\|_{H^{1}(0, T;H^{2}({\Omega}))}^{2}\\ &\le& Ch^{2}\|\zeta_{u}\|^{2}_{L^{2}(J;L^{2}({\Omega}))}. \end{array} $$

Here, we have used the estimate (3.26) together with the fact that $\tilde {e}_{h}(0)= 0.$ The proof is completed. □

Remark 6.1

We recall a dual problem that seeks a solution w ∈ H¹(J; H²(Ω)) such that

$$ \begin{array}{@{}rcl@{}} -\nabla\cdot\big((\alpha\nabla w)-(\beta\nabla w^{\prime})\big) = \zeta_{u} \text{in} {\Omega}\times J, \end{array} $$

(6.13)

and w(τ) = 0 for some τ ∈ J.

We may define $w_{h}\in \mathcal {V}_{h}^{0}$ as the solution to the discrete problem of the equation (6.13) that follows

$$ \mathcal{A}_{1,w}(w_{h}, \varphi_{h})-\mathcal{A}_{2,w}(w_{h}^{\prime}, \varphi_{h}) = (\zeta_{u}, \varphi_{0}), \forall \varphi_{h}=\{ \varphi_{0}, \varphi_{b}\}\in \mathcal{V}_{h}^{0}, $$

(6.14)

with w_h(τ) = 0.

Setting φ_h = w_h in (6.14) and using the coercive property (2.12), we obtain

$$ \begin{array}{@{}rcl@{}} \sigma_{*}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}-\frac{1}{2}\frac{d}{dt}\big(\mathcal{A}_{2,w}(w_{h}(s), w_{h}(s))\big)\leq \|\zeta_{u}\|\|w_{0}\|. \end{array} $$

Next, integrate the above equation in [0,τ] to obtain

$$ \begin{array}{@{}rcl@{}} \sigma_{*}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2}+\frac{1}{2}\mathcal{A}_{2,w}(w_{h}(0), w_{h}(0))\Big)\leq \|\zeta_{u}\|\|w_{0}\|. \end{array} $$

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

Now, we apply the Poincaŕe-type inequality (2.16) and positive definiteness of $\mathcal {A}_{2,w}(\cdot , \cdot )$ in the above estimate, we get

$$ {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}\le C\|\zeta_{u}\|. $$

(6.15)

When we set $\varphi _{h}=w_{h}^{\prime }$ in (6.14), we can get

$$ {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert w_{h}^{\prime}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}\le C\|\zeta_{u}\|. $$

(6.16)

The following estimates are satisfied by w_h, which is the SFWG approximation to w (see estimate (6.9))

$$ {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w-w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}\leq Ch\|\zeta_{u}\|_{L^{2}(L^{2})}. $$

(6.17)

Now, we combine estimates (6.15) and (6.17) to obtain

$$ \begin{array}{@{}rcl@{}} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}={\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w-w_{h}+w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert} &\leq& {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w-w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert} +{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert w_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}\\ &\leq & C\|\zeta_{u}\|_{L^{2}(L^{2})}. \end{array} $$

(6.18)

As a consequence, we can prove that

$$ \begin{array}{@{}rcl@{}} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w^{\prime}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}={\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{Q}_{h}w^{\prime}-w_{h}^{\prime}+w^{\prime}_{h}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert} \leq C\|\zeta_{u}\|_{L^{2}(L^{2})}. \end{array} $$

(6.19)