Virtual element methods for the three-field formulation of time-dependent linear poroelasticity

Abstract

A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in R. Oyarzúa and R. Ruiz-Baier, (SIAM J. Numer. Anal. 54 2951–2973, 2016) is proposed. The treatment is extended to include also the transient case. Appropriate poroelasticity projector operators are introduced and they assist in deriving energy bounds for the time-dependent discrete problem. Under standard assumptions on the computational domain, optimal a priori error estimates are established. These estimates are valid independently of the values assumed by the dilation modulus and the specific storage coefficient, implying that the formulation is locking-free. Furthermore, the accuracy of the method is verified numerically through a set of computational tests.

Appendix 1: Proof of Theorem 4.3

As in the semi-discrete case, we split the individual errors as

$$ \begin{array}{@{}rcl@{}} &&\boldsymbol{u}(t_{n}) - {\boldsymbol{u}_{h}^{n}} = (\boldsymbol{u}(t_{n}) - I^{h}_{\boldsymbol{u}} \boldsymbol{u}(t_{n})) + (I^{h}_{\boldsymbol{u}} \boldsymbol{u}(t_{n})- {\boldsymbol{u}_{h}^{n}}):= E_{\boldsymbol{u}}^{I,n} + E_{\boldsymbol{u}}^{A,n}, \\ &&\psi(t_{n}) - {\psi_{h}^{n}} = (\psi(t_{n}) - I^{h}_{\psi} \psi(t_{n})) + (I^{h}_{\psi} \psi(t_{n})- {\psi_{h}^{n}}):= E_{\psi}^{I,n} + E_{\psi}^{A,n}, \\ &&p(t_{n}) - {p_{h}^{n}} = (p(t_{n}) - {I^{h}_{p}} p(t_{n})) + ({I^{h}_{p}} p(t_{n})- {p_{h}^{n}}):= E_{p}^{I,n} + E_{p}^{A,n}. \end{array} $$

Then, from estimate (4.2a) and following the steps of the proof of Theorem 4.2, we get the bounds:

$$ \begin{array}{@{}rcl@{}} &&\|E_{\boldsymbol{u}}^{I,n} \|_{1} \leq C h (|\boldsymbol{u}(t_{n})|_{2} + |\psi(t_{n})|_{1}) \le C h (| \boldsymbol{u}(0) |_{2} + | \psi(0) |_{1}\\ &&+ \|\partial_{t}\boldsymbol{u}\|_{\boldsymbol{L}^{1}(0,t_{n}; [H^{2}({\varOmega})]^{2})} + \|\partial_{t} \psi\|_{L^{1}(0,t_{n}; H^{1}({\varOmega}))} ), \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} \|E_{\psi}^{I,n} \|_{0} &\leq& C h (| \boldsymbol{u}(0) |_{2} + | \psi(0) |_{1} + \|\partial_{t}\boldsymbol{u}\|_{\boldsymbol{L}^{1}(0,t_{n}; [H^{2}({\varOmega})]^{2})}\\ &&+ \|\partial_{t}\psi\|_{L^{1}(0,t_{n}; H^{1}({\varOmega}))} ), \end{array} $$

(2)

$$ \begin{array}{@{}rcl@{}} &&\|E_{p}^{I,n} \|_{1} \leq C h (| p(0) |_{2} + \|\partial_{t} p\|_{L^{1}(0,t_{n}; H^{2}({\varOmega}))}). \end{array} $$

(3)

From Eqs. 4.1a, 3.17a, and 2.3a, we readily get:

$$ {a_{1}^{h}}(E_{\boldsymbol{u}}^{A,n},\boldsymbol{v}_{h}) + b_{1}(\boldsymbol{v}_{h}, E_{\psi}^{A,n}) = F^{n}(\boldsymbol{v}_{h}) - F^{h,n}(\boldsymbol{v}_{h}). $$

(A.2)

We then use Eqs. 4.1b and 3.21, and proceed to differentiate (2.3c) with respect to time. This implies

$$ \begin{array}{@{}rcl@{}} &&b_{1}(E_{\boldsymbol{u}}^{A,n} - E_{\boldsymbol{u}}^{A,n-1}, \phi_{h}) + b_{2}(E_{p}^{A,n}-E_{p}^{A,n-1}, \phi_{h})-a_{3}(E_{\psi}^{A,n} - E_{\psi}^{A,n-1}, \phi_{h}) \\ && \quad \quad = b_{1}((\boldsymbol{u}(t_{n}) - \boldsymbol{u}(t_{n-1}))-({\varDelta} t) \partial_{t} \boldsymbol{u}(t_{n}), \phi_{h}) + b_{2}(({I_{p}^{h}} p(t_{n}) - {I_{p}^{h}} p(t_{n-1}))\\ &&\quad\quad\quad -({\varDelta} t) \partial_{t} p(t_{n}), \phi_{h}) \\ &&\quad\quad\quad -a_{3}((I_{\psi}^{h} \psi(t_{n})-I_{\psi}^{h} \psi(t_{n-1}))-({\varDelta} t) \partial_{t} \psi (t_{n}), \phi_{h}). \end{array} $$

(A.3)

After choosing $\boldsymbol {v}_{h} = E_{\boldsymbol {u}}^{A,n} - E_{\boldsymbol {u}}^{A,n-1 }$ in Eq. A.2 and $\phi _{h} = - E_{\psi }^{A,n}$ in Eq. A.3 and adding the outcomes, we readily get:

$$ \begin{array}{@{}rcl@{}} &&{a_{1}^{h}}(E_{\boldsymbol{u}}^{A,n}, E_{\boldsymbol{u}}^{A,n} - E_{\boldsymbol{u}}^{A,n-1}) + a_{3}(E_{\psi}^{A,n} - E_{\psi}^{A,n-1}, E_{\psi}^{A,n}) - b_{2}(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{\psi}^{A,n}) \\ &&= \rho ((t_{n}) - {^{n}_{h}}, E_{\boldsymbol{u}}^{A,n} - E_{\boldsymbol{u}}^{A,n-1 } )_{0, {\varOmega}} - b_{1}((\boldsymbol{u}(t_{n}) - \boldsymbol{u}(t_{n-1})) - ({\varDelta} t) \partial_{t} \boldsymbol{u}(t_{n}), E_{\psi}^{A,n}) \\ &&\quad\quad- b_{2}(({I_{p}^{h}} p(t_{n}) - {I_{p}^{h}} p(t_{n-1})) - ({\varDelta} t) \partial_{t} p(t_{n}), E_{\psi}^{A,n}) \\ &&\quad\quad+ a_{3}((I_{\psi}^{h} \psi(t_{n}) - I_{\psi}^{h} \psi(t_{n-1})) - ({\varDelta} t) \partial_{t} \psi (t_{n}), E_{\psi}^{A,n}). \end{array} $$

(A.4)

Next, and as a consequence of using Eqs. 4.1c, 3.7b, and 2.3b with $q_{h} = E_{p}^{A,n}$, we are left with

$$ \begin{array}{@{}rcl@{}} &&\tilde{a}_{2}^{h}(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{p}^{A,n}) + {\varDelta} t {a_{2}^{h}}(E_{p}^{A,n}, E_{p}^{A,n}) - b_{2}(E_{p}^{A,n}, E_{\psi}^{A,n} - E_{\psi}^{A,n-1}) \\ && = {\varDelta} t (\ell(t_{n})- {\ell^{n}_{h}}, E_{p}^{A,n} )_{0, {\varOmega}} + \tilde{a}_{2}^{h}({I^{h}_{p}} p(t_{n}) - {I^{h}_{p}} p(t_{n-1}), E_{p}^{A,n}) \\ && \quad \quad- \tilde{a}_{2}(({\varDelta} t) \partial_{t} p(t_{n}), E_{p}^{A,n}) + b_{2}(E_{p}^{A,n}, ({\varDelta} t) \partial_{t} \psi - (I^{h}_{\psi} \psi(t_{n})) - I^{h}_{\psi} \psi(t_{n-1})). \end{array} $$

(A.5)

If we then add the resulting (A.4)–(A.5) and repeat the same arguments used in deriving (3.11), we can assert that

$$ \begin{array}{@{}rcl@{}} &&a_{3}(E_{\psi}^{A,n} - E_{\psi}^{A,n-1}, E_{\psi}^{A,n}) - b_{2}(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{\psi}^{A,n})\\ &&\quad\quad- b_{2}(E_{p}^{A,n}, E_{\psi}^{A,n} - E_{\psi}^{A,n-1}) + \tilde{a}_{2}^{h}(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{p}^{A,n}) \\ &&= ({\varDelta} t) \bigg(c_{0}({\varDelta}_{t} E_{p}^{A,n} , E_{p}^{A,n})_{0,{\varOmega}} + \frac{1}{\lambda} \sum\limits_{K} \big(\alpha^{2} ({\varDelta}_{t} (I - {\varPi^{0}_{K}}) E_{p}^{A,n}, (I - {\varPi^{0}_{K}}) E_{p}^{A,n})_{0,K} \\ & &\quad\quad-({\delta}_{t} (\alpha {\varPi^{0}_{K}} E_{p}^{A,n} - E_{\psi}^{A,n}), \alpha {\varPi^{0}_{K}} E_{p}^{A,n} - E_{\psi}^{A,n})_{0,K} \big) \bigg), \end{array} $$

The left-hand side can be bounded by using the inequality (3.24) and then summing over n we get:

$$ \begin{array}{@{}rcl@{}} &&\mu \| \boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,n})\|_{0}^{2} + c_{0} \| E_{p}^{A,n}\|_{0}^{2} + ({\varDelta} t) \frac{\kappa_{\min}}{\eta} \sum\limits_{j=1}^{n} \| \nabla E_{p}^{A,j} \|_{0}^{2} \\ &&\qquad + (1/\lambda) \sum\limits_{K}\bigg(\alpha^{2} \| (I- {\varPi^{0}_{K}}) E_{p}^{A,n} \|_{0,K}^{2} + \| \alpha {\varPi^{0}_{K}} E_{p}^{A,n} - E_{\psi}^{A,n}\|_{0,K}^{2} \bigg) \\ &&\le \mu \| \boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,0})\|_{0}^{2} + c_{0} \| E_{p}^{A,0}\|_{0}^{2} + (1/\lambda) \sum\limits_{K}\\&&\qquad\times\bigg(\alpha^{2} \| (I- {\varPi^{0}_{K}}) E_{p}^{A,0} \|_{0,K}^{2} + \| \alpha {\varPi^{0}_{K}} E_{p}^{A,0} - E_{\psi}^{A,0} \|_{0,K}^{2} \bigg) \\ &&\qquad+ \underbrace{ \sum\limits_{j=1}^{n} \rho (\boldsymbol{b}(t_{j}) - {\boldsymbol{b}^{j}_{h}}, E_{\boldsymbol{u}}^{A,j}- E_{\boldsymbol{u}}^{A,j-1} )_{0,{\varOmega}}}_{=:L_{1}} + \underbrace{ \sum\limits_{j=1}^{n} {\varDelta} t (\ell(t_{j})- {\ell^{j}_{h}}, E_{p}^{A,j} )_{0,{\varOmega}}}_{=:L_{2}} \\ && \qquad- \underbrace{ \sum\limits_{j=1}^{n} b_{1}((\boldsymbol{u}(t_{j}) - \boldsymbol{u}(t_{j-1})) - ({\varDelta} t) \partial_{t} \boldsymbol{u}(t_{j}), E_{\psi}^{A,j})}_{=:L_{3}} \\ &&\qquad- \underbrace{ \sum\limits_{j=1}^{n} b_{2}(({I_{p}^{h}} p(t_{j}) - {I_{p}^{h}} p(t_{j-1})) - ({\varDelta} t) \partial_{t} p(t_{j}), E_{\psi}^{A,j})}_{=:L_{4}} \\ &&\qquad+ \underbrace{ \sum\limits_{j=1}^{n} a_{3}((I_{\psi}^{h} \psi(t_{j}) - I_{\psi}^{h} \psi(t_{j-1})) - ({\varDelta} t) \partial_{t} \psi (t_{j}), E_{\psi}^{A,j}) }_{:=L_{5}} \\ &&\qquad+ \underbrace{ \sum\limits_{j=1}^{n} (\tilde{a}_{2}^{h}({I^{h}_{p}} p(t_{j}) - {I^{h}_{p}} p(t_{j-1}), E_{p}^{A,j}) - \tilde{a}_{2}(({\varDelta} t) \partial_{t} p(t_{j}), E_{p}^{A,j}) )}_{:=L_{6}} \\ && \qquad+ \underbrace{ \sum\limits_{j=1}^{n} b_{2}(E_{p}^{A,j}, ({\varDelta} t) \partial_{t} \psi - (I^{h}_{\psi} \psi(t_{j}) - I^{h}_{\psi} \psi(t_{j-1}))}_{:=L_{7}}. \end{array} $$

We bound the term L₁ with the help of formula (3.25), the estimates of projection $\boldsymbol \varPi ^{0,0}_{K}$, applying Taylor expansion, and using generalised Young’s inequality. This gives

$$ \begin{array}{@{}rcl@{}} L_{1} &=& \rho (((\boldsymbol{b} - \boldsymbol{b}_{h})(t_{n}), E_{\boldsymbol{u}}^{A,n})_{0,{\varOmega}} - ((\boldsymbol{b} - \boldsymbol{b}_{h})(0), E_{\boldsymbol{u}}^{A,0})_{0,{\varOmega}} \\&&- \sum\limits_{j=1}^{n} ({\varDelta} t)({\varDelta}_{t}(\boldsymbol{b} - \boldsymbol{b}_{h})(t_{j}), E_{\boldsymbol{u}}^{A,j-1})_{0,{\varOmega}}) ) \\ &\le& \frac{\mu}{2} \| \boldsymbol{\varepsilon} (E_{\boldsymbol{u}}^{A,n})\|_{0}^{2} + C_{1} \Big(\frac{\rho}{\mu} h|b(0)|_{1}~ \mu \| \boldsymbol{\varepsilon} (E_{\boldsymbol{u}}^{A,0})\|_{0} + \frac{\rho^{2}}{\mu}h^{2} \max\limits_{1 \le j \le n} |\boldsymbol{b}(t_{j}) |_{1}^{2} \\ &&+ ({\varDelta} t) ~ h \sum\limits_{j=1}^{n} \frac{\rho}{\mu} \left( | \partial_{t} \boldsymbol{b}^{j} |_{1} + \left( {\varDelta} t {\int}_{t_{j-1}}^{t_{j}} | \partial_{tt} \boldsymbol{b} (s)|_{1}^{2} \mathrm{d}s \right)^{1/2} \right) \mu \| \boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j-1}) \|_{0} \Big). \end{array} $$

Then, the estimate satisfied by the projection ${\varPi ^{0}_{K}}$ along with Poincaré and Young’s inequalities yields:

$$ \begin{array}{@{}rcl@{}} L_{2} & \le& C_{2} \sum\limits_{j=1}^{n} ({\varDelta} t) h | \ell(t_{j})|_{1} \|\nabla E_{p}^{A,j}\|_{0} \le C_{2} \sum\limits_{j=1}^{n} ({\varDelta} t) \frac{\eta }{\kappa_{\min}} h^{2} | \ell(t_{j})|_{1}^{2} \\&&+ ({\varDelta} t) \frac{\kappa_{\min}}{6 \eta} \sum\limits_{j=1}^{n} \| \nabla E_{p}^{A,j}\|_{0}^{2}. \end{array} $$

The discrete inf-sup condition (3.6) implies that

$$ \begin{array}{@{}rcl@{}} \|E_{\psi}^{A,j} \|_{0} \le C (h |\boldsymbol{b}(t_{j})|_{1} + \|\boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j}) \|_{0} ). \end{array} $$

(A.6)

Applying an expansion in Taylor series, together with Eq. A.6, the Cauchy-Schwarz, and Young inequalities, enables us to write

$$ \begin{array}{@{}rcl@{}} L_{3} & \le C \sum\limits_{j=1}^{n} \left( ({\varDelta} t)^{3} {\int}_{t_{j-1}}^{t_{j}} \| \partial_{tt} \boldsymbol{u} (s)\|_{0}^{2} \right)^{1/2} (h |\boldsymbol{b}(t_{j})|_{1} + \|\boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j}) \|_{0} ). \end{array} $$

Then, after using the estimates of the projection ${I_{p}^{h}}$ (4.2b), (A.6), and applying again Cauchy-Schwarz inequality, we get

$$ \begin{array}{@{}rcl@{}} L_{4} \!&\le&\! C \frac{\alpha}{\lambda} \sum\limits_{j=1}^{n} \left( \| {I_{p}^{h}} (p(t_{j}) - p(t_{j-1})) - (p(t_{j}) - p(t_{j-1})) \|_{0} + \| (p(t_{j}) - p(t_{j-1})) \right.\\&&\left.- ({\varDelta} t) \partial_{t} p(t_{j}) \|_{0}\vphantom{I_{p}^{h}} \right) \|E_{\psi}^{A,j} \|_{0} \\ \!&\le&\! C \frac{\alpha}{\lambda} \sum\limits_{j=1}^{n} \left( h^{2} \left( ({\varDelta} t) {\int}_{t_{j-1}}^{t_{j}}| \partial_{t}p(s)|_{2}^{2} \mathrm{d}s \right)^{1/2} + \left( ({\varDelta} t )^{3} {\int}_{t_{j-1}}^{t_{j}} \| \partial_{tt} p(s ) \|_{0}^{2} \mathrm{d}s\right)^{1/2} \right) \\&&\times\|E_{\psi}^{A,j} \|_{0}\\ \!&\le&\! C \frac{\alpha}{\lambda} \sum\limits_{j=1}^{n} \left( h^{2} \left( ({\varDelta} t) {\int}_{t_{j-1}}^{t_{j}}| \partial_{t}p(s)|_{2}^{2} \mathrm{d}s \right)^{1/2} + \left( ({\varDelta} t )^{3} {\int}_{t_{j-1}}^{t_{j}} \| \partial_{tt} p(s ) \|_{0}^{2} \mathrm{d}s\right)^{1/2} \right)\\&&\times\left( \rho h |\boldsymbol{b}(t_{j})|_{1} + \|\boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j}) \|_{0} \right). \end{array} $$

The stability of a₃(⋅,⋅) and the proof for the bound of L₄ gives

$$ \begin{array}{@{}rcl@{}} L_{5}&\le& C(1/ \lambda ) \sum\limits_{j=1}^{n} \| (I_{\psi}^{h} \psi(t_{j}) - I_{\psi}^{h} \psi(t_{j-1})) \\&&- ({\varDelta} t) \partial_{t} \psi (t_{j}) \|_{0} (\rho h |\boldsymbol{b}(t_{j})|_{1} + \|\boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j}) \|_{0} ) \\ &\le& C(1/ \lambda ) \sum\limits_{j=1}^{n} \left( h^{2} \left( ({\varDelta} t) {\int}_{t_{j-1}}^{t_{j}}(| \partial_{t}\boldsymbol{u}(s)|_{2}^{2} + | \partial_{t}\psi(s)|_{1}^{2}) \mathrm{d}s \right)^{1/2} \right.\\&&\left.+ \left( ({\varDelta} t)^{3} {\int}_{t_{j-1}}^{t_{j}} \| \partial_{tt} \psi (s)\|_{0}^{2} \mathrm{d}s \right)^{1/2} \right) \\ &&\times (\rho h |\boldsymbol{b}(t_{j})|_{1} + \|\boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,j}) \|_{0} ). \end{array} $$

The polynomial approximation p_π for fluid pressure, consistency of the bilinear form $\tilde {a}_{2}^{h}(\cdot , \cdot )$, stability of the bilinear forms $\tilde {a}_{2}(\cdot , \cdot ), \tilde {a}_{2}^{h}(\cdot , \cdot )$, and the Cauchy-Schwarz, Poincaré and Young’s inequalities gives

$$ \begin{array}{@{}rcl@{}} L_{6} &=& \sum\limits_{j=1}^{n} \Big(\tilde{a}_{2}^{h}(({I^{h}_{p}} p(t_{j}) - {I^{h}_{p}} p(t_{j-1})) - (p_{\pi}(t_{j}) - p_{\pi}(t_{j-1})), E_{p}^{A,j}) \\ &&+ \tilde{a}_{2}((p_{\pi}(t_{j}) - p_{\pi}(t_{j-1})) - (p(t_{j}) - p(t_{j-1})), E_{p}^{A,j}) + \tilde{a}_{2}((p(t_{j}) \\&&- p(t_{j-1})) - ({\varDelta} t) \partial_{t} p(t_{j}), E_{p}^{A,j}) \Big) \\ &\le& C \Big(c_{0} + \frac{\alpha^{2}}{\lambda} \Big) \sum\limits_{j=1}^{n} \bigg(h^{2} \Big(({\varDelta} t) {\int}_{t_{j-1}}^{t_{j}} |\partial_{t} p(s) |_{2}^{2} \mathrm{d}s \Big)^{1/2} \\&&+ \Big(({\varDelta} t)^{3}{\int}_{t_{j-1}}^{t_{j}} \| \partial_{tt} p(s) \|_{0}^{2} \mathrm{d}s \Big)^{1/2} \bigg) \| \nabla E_{p}^{A,j} \|_{0} \\ &\le& C {\Big(c_{0} + \frac{\alpha^{2}}{\lambda} \Big)^{2}} \Big(h^{4} \|\partial_{t} p \|_{L^{2}(0,t_{n};H^{2}({\varOmega}))}^{2} + ({\varDelta} t)^{2} \| \partial_{tt} p \|_{L^{2}(0,t_{n};L^{2}({\varOmega}))}^{2} \Big) \\&&+ {\varDelta} t \frac{\kappa_{\min}}{6 \eta} \sum\limits_{j=1}^{n} \| \nabla E_{p}^{A,j} \|_{0}^{2}. \end{array} $$

The continuity of b₂(⋅,⋅), the bound derived for the term L₅ and using the Young’s inequality, gives

$$ \begin{array}{@{}rcl@{}} L_{7} & \le& \frac{\alpha}{\lambda} \sum\limits_{j=1}^{n} \| ({\varDelta} t) \partial_{t} \psi(t_{j}) - (I^{h}_{\psi} \psi(t_{j}) - I^{h}_{\psi} \psi(t_{j-1}))\|_{0} \| E_{p}^{A,j}\|_{0} \\ & \le& C {\Big(\frac{\alpha}{\lambda} \Big)^{2}}\bigg(h^{2} (\| \partial_{t} \psi \|_{L^{2}(0,t_{n};H^{1}({\varOmega}))}^{2} + \| \partial_{t} \boldsymbol{u} \|_{\boldsymbol{L}^{2}(0,t_{n};[H^{2}({\varOmega})]^{2})}^{2})\\&&\qquad\qquad+ ({\varDelta} t )^{2} \| \partial_{tt} \psi \|_{L^{2}(0,t_{n};L^{2}({\varOmega}))}^{2} \bigg) \\ && +({\varDelta} t) \frac{\kappa_{\min}}{6 \eta} \sum\limits_{j=1}^{n} \| \nabla E_{p}^{A,j} \|_{0}^{2}. \end{array} $$

In turn, putting together the bounds obtained for all L_i’s, $i=1, \dots , 7$, using the Young’s inequality and Lemma 3.2 concludes that

$$ \begin{array}{@{}rcl@{}} &&\mu \| \boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,n})\|_{0}^{2} + c_{0} \| E_{p}^{A,n}\|_{0}^{2} + ({\varDelta} t) \frac{\kappa_{\min}}{\eta} \sum\limits_{j=1}^{n} \| \nabla E_{p}^{A,j} \|_{0}^{2} \\ && \le C \bigg(\mu \| \boldsymbol{\varepsilon}(E_{\boldsymbol{u}}^{A,0})\|_{0}^{2} + (c_{0} + \alpha^{2}/\lambda) \| E_{p}^{A,0}\|_{0}^{2} + (1/\lambda) \| E_{\psi}^{A,0}\|_{0}^{2} \\&&\quad+ \Big(1 + {\varDelta} t \Big) h^{2} \max\limits_{0 \le j \le n} |\boldsymbol{b}(t_{j}) |_{1}^{2} \\ && \quad + h^{2} {\varDelta} t \sum\limits_{j=1}^{n} (|(t_{j})|_{1}^{2} + ({\varDelta} t) | \partial_{t} \boldsymbol{b} |_{1}^{2} + | \ell(t_{j})|_{1}^{2}) + ({\varDelta} t)^{2} h^{2} \| \partial_{tt} \boldsymbol{b} \|_{L^{2}(0,t_{n};[H^{1}({\varOmega})]^{2})} \\ &&\quad + ({\varDelta} t )^{2} \big((c_{0} + \alpha^{2}/\lambda )^{2} \| \partial_{tt} p \|_{L^{2}(0,t_{n};L^{2}({\varOmega}))}^{2} + \| \partial_{tt} \boldsymbol{u} \|_{\boldsymbol{L}^{2}(0,t_{n};[L^{2}({\varOmega})]^{2})}^{2} \\&&\quad+ {\frac{\alpha^{2}}{\lambda^{2}}} \| \partial_{tt} \psi \|_{L^{2}(0,t_{n};L^{2}({\varOmega}))}^{2} \big)\\ && \quad + h^{2} \big({\frac{\alpha^{2}}{\lambda^{2}}} \| \partial_{t} \psi \|_{L^{2}(0,t_{n};H^{1}({\varOmega}))}^{2} + {\frac{\alpha^{2}}{\lambda^{2}}} \| \partial_{t} \boldsymbol{u}\|_{\boldsymbol{L}^{2}(0,t_{n};[H^{2}({\varOmega})]^{2})}^{2} \\&&\quad+ {(c_{0} + \alpha^{2}/\lambda )^{2}} h^{2} \| \partial_{t}p \|_{L^{2}(0,t_{n};H^{2}({\varOmega}))}^{2} \big) \bigg). \end{array} $$

And finally, the desired result (4.9) holds after choosing ${\boldsymbol {u}_{h}^{0}}: =\boldsymbol {u}_{I}(0)$, ${\psi _{h}^{0}}: = \varPi ^{0,0}\psi (0)$, ${p_{h}^{0}}: = p_{I}(0)$ and applying triangle’s inequality together with Eqs. 1– 3, and A.6.