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.
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Appendix
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
We can restate the above equation as
Now, integrate the above equation with respect to time from 0 to t and apply the Cauchy-Schwarz inequality; we get
We can rearrange the above equation as
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,
and
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
Now, applying the Cauchy-Schwarz inequality together with the estimate (6.3) in the above equation with λ = 2, we obtained
In the previous estimate, we have used the fact that (cf. [43], Proposition 7.1)
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
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
Then, it follows from (6.2) that
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
□
Lemma 6.1
Let w ∈ H1(0,T; H2(Ω)) be the solutions of the (3.19) and wh be its SFWG approximation. Then, there exists a constant C such that
Proof
The following analysis used to derive (3.9), we obtain
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
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 \(\tilde {e}_{h}: = \mathcal {Q}_{h}w-{w}_{h}\)
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
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
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
with wh(τ) = 0.
Setting φh = wh in (6.14) and using the coercive property (2.12), we obtain
Next, integrate the above equation in [0,τ] to obtain
Here, we used the fact that wh(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
When we set \( \varphi _{h}=w_{h}^{\prime }\) in (6.14), we can get
The following estimates are satisfied by wh, which is the SFWG approximation to w (see estimate (6.9))
Now, we combine estimates (6.15) and (6.17) to obtain
As a consequence, we can prove that
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Kumar, N. Supercloseness analysis of a stabilizer-free weak Galerkin finite element method for viscoelastic wave equations with variable coefficients. Adv Comput Math 49, 12 (2023). https://doi.org/10.1007/s10444-023-10010-w
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DOI: https://doi.org/10.1007/s10444-023-10010-w
Keywords
- Viscoelastic wave equations
- Stabilizer-free weak Galerkin method
- Semidiscrete and fully discrete schemes
- Supercloseness