Numerical Solution of the Biot/Elasticity Interface Problem Using Virtual Element Methods

Abstract

We propose, analyse and implement a virtual element discretisation for an interfacial poroelasticity/elasticity consolidation problem. The formulation of the time–dependent poroelasticity equations uses displacement, fluid pressure and total pressure, and the elasticity equations are written in displacement-pressure formulation. The construction of the virtual element scheme does not require Lagrange multipliers to impose the transmission conditions (continuity of displacement and total traction, and no-flux for the fluid) on the interface. We show the stability and convergence of the virtual element method for different polynomial degrees, and the error bounds are robust with respect to delicate model parameters (such as Lamé constants, permeability, and storativity coefficient). Finally we provide some simple numerical examples that illustrate the properties of the scheme.

Appendices

Appendix A Proof of Theorem 4.1

Thanks to the Scott–Dupont theory (see [15]) we know that for every s with $0\le s\le k$ and for every $u\in H^{1+s}(K)$, there exists $u_\pi \in {\mathbb {P}}_k(K)$, $k\ge 2$, such that

$$\begin{aligned} \Vert u - u_{\pi } \Vert _{0,K} + h_K |u -u_{\pi } |_{1,K} \lesssim h_K^{1+s} |u|_{1+s,K} \quad \text {for all }K \in {\mathcal {T}}_h. \end{aligned}$$

(A.1)

We can then write the displacement and total pressure error in terms of the poroelastic projector

$$\begin{aligned} ({\varvec{u}}- {\varvec{u}}_h)(t)&= ({\varvec{u}}- I^h_{{\varvec{u}}}{\varvec{u}})(t) + (I^h_{{\varvec{u}}} {\varvec{u}}- {\varvec{u}}_h)(t) := e_{{\varvec{u}}}^I(t) + e_{{\varvec{u}}}^A(t), \\ (\psi - \psi _h)(t)&= (\psi - I^h_{\psi } \psi )(t) + (I^h_{\psi } \psi - \psi _h)(t) := e_{\psi }^I(t) + e_{\psi }^A(t). \end{aligned}$$

Then, a combination of equations (4.1a), (3.4a) and (2.4a) gives

$$\begin{aligned} a_1^h(e_{{\varvec{u}}}^A, \varvec{v}_h)+ b_1(\varvec{v}_h, e_{\psi }^A)&= (a_1({\varvec{u}},\varvec{v}_h) -a_1^h({\varvec{u}}_h, \varvec{v}_h)) + b_1(\varvec{v}_h, \psi - \psi _h) = (F-F^h)(\varvec{v}_h), \end{aligned}$$

and taking as test function $\varvec{v}_h = \partial _t e_{{\varvec{u}}}^A$, we can write the relation

$$\begin{aligned} a_1^h(e_{{\varvec{u}}}^A, \partial _t e_{{\varvec{u}}}^A) + b_1(\partial _t e_{{\varvec{u}}}^A, e_{\psi }^A) = (F-F^h)(\partial _t e_{{\varvec{u}}}^A). \end{aligned}$$

(A.2)

Now, we write the pressure error in terms of the poroelastic projector as follows

$$\begin{aligned} (p^{\textrm{P}} - p_h^{\textrm{P}})(t) = (p^{\textrm{P}} - I^h_p p^{\textrm{P}})(t) + (I^h_p p^{\textrm{P}} - p_h^{\textrm{P}})(t) := e_{p}^{I,\textrm{P}}(t) + e_{p}^{A, \textrm{P}}(t). \end{aligned}$$

Using (4.1c), (3.4b) and (2.4b), we obtain

$$\begin{aligned}&{\tilde{a}}_2^h(\partial _t e_p^{A,\textrm{P}}, q_h^{\textrm{P}}) + a_2^h(e_p^{A, \textrm{P}}, q_h^{\textrm{P}}) - b_2(q_h^{\textrm{P}}, \partial _t e_{\psi }^{A} ) \\&\quad = ({\tilde{a}}_2^h(\partial _t I^h_p p^{\textrm{P}}, q_h^{\textrm{P}}) - {\tilde{a}}_2(\partial _t p^{\textrm{P}}, q_h^{\textrm{P}})) + b_2(q_h^{\textrm{P}}, \partial _t e_{\psi }^I) + (G- G^h)(q_h^{\textrm{P}}). \end{aligned}$$

We can take $q_h^{\textrm{P}} = e_p^{A, \textrm{P}}$, which leads to

$$\begin{aligned} \begin{aligned}&{\tilde{a}}_2^h(\partial _t e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) +a_2^h(e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) - b_2(e_p^{A, \textrm{P}}, \partial _t e_{\psi }^A) \\&\quad = ({\tilde{a}}_2^h(\partial _t I^h_p p^{\textrm{P}}, e_p^{A, \textrm{P}}) - {\tilde{a}}_2(\partial _t p^{\textrm{P}}, e_p^{A, \textrm{P}}) ) + b_2(e_p^{A, \textrm{P}}, \partial _t e_{\psi }^I) + (G- G^h)(e_p^{A, \textrm{P}}). \end{aligned} \end{aligned}$$

(A.3)

Next we use (4.1b), (3.4c) and (2.4c), and this implies

$$\begin{aligned}&b_1(e_{{\varvec{u}}}^A, \phi _h)+ b_2(e_p^{A, \textrm{P}} , \phi _h) - a_3(e_{\psi }^A, \phi _h) = -b_2(e_p^{I, \textrm{P}}, \phi _h) + a_3(e_{\psi }^I, \phi _h). \end{aligned}$$

Differentiating the above equation with respect to time and taking $\phi _h = -e_{\psi }^A$, we can assert that

$$\begin{aligned} - b_1(\partial _t e_{{\varvec{u}}}^A, e_{\psi }^A) - b_2(\partial _t e_p^{A, \textrm{P}}, e_{\psi }^A) + a_3(\partial _t e_{\psi }^A, e_{\psi }^A) = b_2(\partial _t e_p^I, e_{\psi }^A) -a_3(\partial _t e_{\psi }^I, e_{\psi }^A). \end{aligned}$$

(A.4)

Then we simply add (A.2), (A.3) and (A.4), to obtain

$$\begin{aligned} \begin{aligned}&a_1^h(e_{{\varvec{u}}}^A, \partial _t e_{{\varvec{u}}}^A) + {\tilde{a}}_2^h(\partial _t e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) +a_2^h(e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) + a_3(\partial _t e_{\psi }^A, e_{\psi }^A) \\&\qquad - b_2(e_p^{A, \textrm{P}}, \partial _t e_{\psi }^A) - b_2(\partial _t e_p^{A, \textrm{P}}, e_{\psi }^A)\\&\quad = (F- F^h)(\partial _{t} e_{{\varvec{u}}}^A ) + ({\tilde{a}}_2^h(\partial _t I^h_p p^{ \textrm{P}}, e_p^{A, \textrm{P}}) - {\tilde{a}}_2(\partial _{t} p^{\textrm{P}}, e_p^{A, \textrm{P}}))\\&\qquad + b_2(e_p^{A, \textrm{P}}, \partial _{t} e_{\psi }^I) + (G- G^h) (e_p^{A, \textrm{P}}) + b_2(\partial _{t} e_p^{I, \textrm{P}}, e_{\psi }^A) - a_3(\partial _{t} e_{\psi }^I, e_{\psi }^A). \end{aligned} \end{aligned}$$

(A.5)

Regarding the left-hand side of (A.5), repeating arguments to obtain alike to the stability proof, that is,

$$\begin{aligned}&a_1^h(e_{{\varvec{u}}}^A, \partial _t e_{{\varvec{u}}}^A) + {\tilde{a}}_2^h(\partial _t e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) +a_2^h(e_p^{A, \textrm{P}}, e_p^{A, \textrm{P}}) + a_3(\partial _t e_{\psi }^A, e_{\psi }^A)\\&\quad - b_2(e_p^{A, \textrm{P}}, \partial _t e_{\psi }^A) - b_2(\partial _t e_p^{A, \textrm{P}}, e_{\psi }^A) \\&\quad \gtrsim {\mu ^{\min }}\frac{\textrm{d}}{\textrm{d}t} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A)\Vert _0^2 + c_0\frac{\textrm{d}}{\textrm{d}t} \Vert e_p^{A, \textrm{P}}\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{2 \kappa _{\min }}{\eta } \Vert \nabla e_p^A\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{1}{{ \lambda ^{\textrm{E}}}} \frac{\textrm{d}}{\textrm{d}t} \Vert e_{\psi }^{A, \textrm{E}}\Vert _{0, \varOmega ^{\textrm{E}}}^2 \\&\quad + \frac{1}{{ \lambda ^{\textrm{P}}}} \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \biggl ( \alpha ^2\frac{\textrm{d}}{\textrm{d}t}\Vert (I-\varPi ^0_K) e_p^{A,\textrm{P}}\Vert _{0,K}^2 +\frac{\textrm{d}}{\textrm{d}t} \Vert \alpha \varPi ^0_K e_p^{A,\textrm{P}} -e_{\psi }^{A,\textrm{P}}\Vert _{0,K}^2 \biggr ) . \end{aligned}$$

Then integrating equation (A.5) in time and using the consistency of ${\tilde{a}}_2(\cdot , \cdot )$, implies the bound

$$\begin{aligned}&{\mu _{\min }} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(t))\Vert _0^2 + c_0 \Vert e_p^{A, \textrm{P}} (t)\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{1}{{ \lambda ^{\textrm{E}}}} \Vert e_{\psi }^{A, \textrm{E}}(t)\Vert _{0, \varOmega ^{\textrm{E}}}^2 + \frac{\kappa _{\min }}{\eta }\int _0^t \Vert \nabla e_p^{A, \textrm{P}}(s)\Vert _{0, \varOmega ^{\textrm{P}}}^2 \, \textrm{d}s \\&\qquad + \frac{1}{{ \lambda ^{\textrm{P}}}} \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \Bigl (\alpha ^2 \Vert (I-\varPi ^{0,k}_K) e_p^{A, \textrm{P}}(t)\Vert _{0,K}^2 + \Vert (\alpha \varPi ^{0,k}_K e_p^{A, \textrm{P}} -e_{\psi }^{A, \textrm{P}})(t)\Vert _{0,K}^2 \Bigr ) \\&\quad \lesssim {\mu _{\min }} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(0))\Vert _0^2 + c_0 \Vert e_p^{A, \textrm{P}} (0)\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{1}{{ \lambda ^{\textrm{E}}}} \Vert e_{\psi }^{A, \textrm{E}}(0)\Vert _{0, \varOmega ^{\textrm{E}}}^2\\&\qquad + \frac{1}{{ \lambda ^{\textrm{P}}}} \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \Bigl (\alpha ^2 \Vert (I-\varPi ^{0,k}_K) e_p^{A, \textrm{P}}(0)\Vert _{0,K}^2 + \Vert (\alpha \varPi ^{0,k}_K e_p^{A, \textrm{P}} -e_{\psi }^A)(0)\Vert _{0,K}^2 \Bigr ) \\&\qquad + \underbrace{ \int _0^t \bigl ((\varvec{b}- \varvec{b}_h)(s),\partial _{t} e_{{\varvec{u}}}^A(s) \bigr )_{0, \varOmega } \, \textrm{d}s}_{=:D_1} + \underbrace{\int _0^t \bigl ((\ell ^{\textrm{P}} - \ell _h^{ \textrm{P}})(s), e_p^{A, \textrm{P}}(s) \bigr )_{0, \varOmega ^{\textrm{P}}}\, \textrm{d}s}_{=:D_2} \\&\qquad + \underbrace{\int _0^t \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \Bigl ({\tilde{a}}_2^{h,K} \bigl (\partial _t(I^h_p p^{\textrm{P}} - p_{\pi }^{\textrm{P}})(s), e_p^{A, \textrm{P}}(s) \bigr ) - {\tilde{a}}^{K}_2 \bigl (\partial _t(p^{\textrm{P}} - p_{\pi }^{\textrm{P}})(s), e_p^{A, \textrm{P}}(s) \bigr ) \Bigr ) \, \textrm{d}s}_{=:D_3} \\&\qquad + \underbrace{\int _0^t \Bigl (b_2 \bigl (e_p^{A, \textrm{P}}(s), \partial _{t} e_{\psi }^I(s) \bigr ) + b_2 \bigl (\partial _{t} e_p^{I, \textrm{P}} (s), e_{\psi }^A (s)\bigr ) - a_3 \bigl (\partial _{t} e_{\psi }^I(s), e_{\psi }^A(s) \bigr ) \Bigr ) \, \textrm{d}s}_{=:D_4}. \end{aligned}$$

Then we integrate by parts in time and invoke the orthogonality property of the $L^2$-projection, which yields

$$\begin{aligned} D_1&= \sum _{K \in \mathcal {T}_h} \Bigl ( \bigl ((\textbf{I}- \varvec{\varPi }^{0,k-2}_K) \varvec{b}(t), (\textbf{I}- \varvec{\varPi }^{0,0}_K) e_{{\varvec{u}}}^A(t) \bigr )_{0, K}\\&\quad - \bigl ((\textbf{I}- \varvec{\varPi }^{0,k-2}_K) \varvec{b}(0), (\textbf{I}- \varvec{\varPi }^{0,0}_K) e_{{\varvec{u}}}^A(0) \bigr )_{0, K} \\&\quad - \int _0^t \bigl (\partial _{t} (\textbf{I}- \varvec{\varPi }^{0,k-2}_K) \varvec{b}(s), (\textbf{I}- \varvec{\varPi }^{0,0}_K) e_{{\varvec{u}}}^A(s) \bigr )_{0, K} \, \textrm{d}s \Bigr ), \end{aligned}$$

and use Cauchy–Schwarz and Young’s inequalities along with Korn’s inequality to arrive at

$$\begin{aligned} D_1&\le \frac{{\mu _{\min }} }{2} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(t))\Vert _0^2 + {C(\mu _{\min })} h^k \biggl (h^k |\varvec{b}(t)|_{k-1}^2 + |\varvec{b}(0)|_{k-1} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(0))\Vert _0 \\&\quad + \int _0^t | \partial _{t} \varvec{b}(s)|_{k-1} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(s))\Vert _{0}\, \textrm{d}s \biggr ), \end{aligned}$$

where we have used standard error estimate for the $L^2$-projection $\varvec{\varPi }_K^{0,k}$ onto piecewise constant functions. Using again Cauchy–Schwarz inequality, standard error estimates for $\varPi _K^{0,k}$ on the term $D_2$, Young’s and Poincaré inequalities readily gives

$$\begin{aligned}&D_2 {\le C} h^k \int _0^t |\ell ^{\textrm{P}} (s)|_{k-1, \varOmega ^{\textrm{P}}} \Vert \nabla e_p^{A, \textrm{P}}(s)\Vert _{0, \varOmega ^{\textrm{P}}}\, \,\textrm{d}s\le {C(\kappa _{\min },\eta )} h^{2k} \int _{0}^t |\ell ^{\textrm{P}} (s)|_{k-1, \varOmega ^{\textrm{P}}}^2 \,\textrm{d}s\\&\quad + \frac{\kappa _{\min }}{{4} \eta } \int _{0}^t \Vert \nabla e_p^{A, \textrm{P}}(s)\Vert _{0, \varOmega ^{\textrm{P}}}^2 \,\textrm{d}s. \end{aligned}$$

On the other hand, considering the polynomial approximation $p_{\pi }^{\textrm{P}}$ (cf. (A.1)) of $p^{\textrm{P}}$, utilising the triangle inequality, Young’s and Poincaré inequalities yield

$$\begin{aligned} D_3&{\le C(\kappa _{\min },\eta ) } h^{2(k+1)} \bigg (c_0 + \frac{\alpha ^2}{{\lambda ^{\textrm{P}}}}\bigg )^2 \int _0^t |\partial _{t} p^{\textrm{P}}(s)|_{k+1, \varOmega ^{\textrm{P}}}^2 \,\textrm{d}s+ \frac{\kappa _{\min }}{ {4} \eta } \int _{0}^t \Vert \nabla e_p^{A, \textrm{P}}(s)\Vert _{0, \varOmega ^{\textrm{P}}}^2 \,\textrm{d}s. \end{aligned}$$

Also,

$$\begin{aligned} D_4&\le \frac{{C}}{\lambda } h^k \int _0^t \Bigl ( \alpha ( | \partial _{t} \psi ^{\textrm{P}} (s)|_{k, \varOmega ^{\textrm{P}}} + |\partial _t {\varvec{u}}^{\textrm{P}}(s)|_{k+1,\varOmega ^{\textrm{P}}} + |\partial _t {\varvec{u}}^{\textrm{E}}(s)|_{k+1, \varOmega ^{\textrm{E}}} )\Vert e_p^{A, \textrm{P}} (s)\Vert _{0, \varOmega ^{\textrm{P}}} \\&\quad + (\alpha h |\partial _{t} p^{\textrm{P}}(s) |_{k+1, \varOmega ^{\textrm{P}}} + | \partial _{t} \psi ^{\textrm{P}} (s)|_{k, \varOmega ^{\textrm{P}}} + | \partial _{t} \psi ^{\textrm{E}} (s)|_{k, \varOmega ^{\textrm{E}}} + |\partial _t {\varvec{u}}(s)|_{k+1}) \Vert e_{\psi }^A (s)\Vert _0 \Bigr ) \, \textrm{d}s. \end{aligned}$$

Using the discrete inf-sup condition (cf. (3.3)) and a combination of equations (4.1a), (3.4a) and (2.4a), we get

$$\begin{aligned} {{\tilde{\beta }}} \Vert e_{\psi }^A(t) \Vert _0&\le \sup _{\varvec{v}_h \in \textbf{V}_h} \frac{b_1(\varvec{v}_h, e_{\psi }^A(t))}{\Vert \varvec{v}_h \Vert _1} { ~\le C \big (} h^k {\rho } | \varvec{b}(t) |_{k-1} + {\mu _{\max }} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(t))\Vert _0 {\big )}. \end{aligned}$$

(A.6)

Then, with the help of Young’s and Poincaré inequalities, the bound of $D_4$ becomes

$$\begin{aligned}&D_4 {\le } \frac{{C}}{\lambda } h^k \int _0^t \Bigl ( (\alpha {h} |\partial _{t} p^{\text {P}}(s) |_{k+1, \varOmega ^{\text {P}}} + | \partial _{t} \psi ^{\text {P}} (s)|_{k, \varOmega ^{\text {P}}} + | \partial _{t} \psi ^{\text {E}} (s)|_{k, \varOmega ^{\text {E}}} + |\partial _t {\textbf{u}}^{\text {P}}(s)|_{k+1, \varOmega ^{\text {P}}} \\ {}&\qquad \qquad \quad + |\partial _t {\textbf{u}}^{\text {E}}(s)|_{k+1, \varOmega ^{\text {E}}} ) \times ( h^k {\rho |\textbf{b}(s)|_{k-1}} + {\mu _{\max }} \Vert \mathbf {\varepsilon }(e_{{\textbf{u}}}^A(s))\Vert _0) \\ {}&\qquad \qquad \quad + \alpha \Vert e_p^{A, \text {P}} (s)\Vert _{0, \varOmega ^{\text {P}}} (| \partial _{t} \psi ^{\text {P}} (s)|_{k, \varOmega ^{\text {P}}} + |\partial _t {\textbf{u}}(s)|_{k+1} ) \Bigr ) \,\text {d}s. \end{aligned}$$

Combining the bounds of all $D_i, i=1,2,3,4$ and proceeding in a similar fashion as for the bounds in the stability proof in [18] (using Lemma 3.1 and (3.6)), we can eventually conclude that

$$\begin{aligned}&{\mu _{\min }} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(t))\Vert _0^2 + c_0 \Vert e_p^{A,\textrm{P}} (t)\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{1}{{\lambda ^{\textrm{E}}}} \Vert e_{\psi }^{A, \textrm{E}}(t)\Vert _{0, \varOmega ^{\textrm{E}}}^2 + \frac{\kappa _{\min }}{\eta } \int _0^t \Vert \nabla e_p^{A, \textrm{P}} (s)\Vert _{0, \varOmega ^{\textrm{P}}}^2 \, \textrm{d} s \\&\quad \lesssim {\mu _{\min }} \Vert \varvec{\varepsilon }(e_{{\varvec{u}}}^A(0))\Vert _0^2 + \Big (c_0 + \frac{\alpha ^2}{{\lambda ^{\textrm{P}}}} \Big ) \Vert e_p^{A, \textrm{P}} (0)\Vert _{0, \varOmega ^{\textrm{P}}}^2 + \frac{1}{{\lambda ^{\textrm{E}}}} \Vert e_{\psi }^{A,\textrm{E}}(0)\Vert _{0, \varOmega ^{\textrm{E}}}^2 \\&\qquad + {C(\mu _{\min }, \kappa _{\min }, \eta ) h^{2k} \Biggl ( \sup _{t \in [0,t_{\text {final}}] } |\varvec{b}(t)|_{k-1}^2 \!+\! \int _0^t \Bigl ( h^2 \big ( c_0 \!+\! \frac{\alpha ^2}{{\lambda ^{\textrm{P}}}}\big )^2 |\partial _{t} p^{\textrm{P}}(s) |_{k+1, \varOmega ^{\textrm{P}}}^2 \!+\! |\ell ^{\textrm{P}} (s)|_{k\!-\!1, \varOmega ^{\textrm{P}}}^2 \Bigr ) \, \textrm{d}s } \\&\qquad { + \biggl ( \int _0^t \Bigl ( |\varvec{b}(s)|_{k-1} + | \partial _{t}\varvec{b}(s)|_{k-1} + h \Big ( c_0 + \frac{\alpha ^2}{{\lambda ^{\textrm{P}}}}\Big ) |\partial _{t} p^{\textrm{P}}(s) |_{k+1, \varOmega ^{\textrm{P}}} } \\&\qquad {+ \frac{1}{{\lambda _{\min }}} \big ( | \partial _{t} \psi ^{\textrm{P}} (s)|_{k, \varOmega ^{\textrm{P}}} +| \partial _{t} \psi ^{\textrm{E}} (s)|_{k, \varOmega ^{\textrm{E}}} +|\partial _t {\varvec{u}}^{\textrm{P}}(s)|_{k+1, \varOmega ^{\textrm{P}}} +|\partial _t {\varvec{u}}^{\textrm{E}}(s)|_{k+1, \varOmega ^{\textrm{E}}} \big ) \Bigr ) \, \textrm{d}s \biggr )^2\Biggr ).} \end{aligned}$$

Then choosing ${\varvec{u}}_h(0): ={\varvec{u}}_I(0)$, $\psi _h(0): = \varPi ^{0,k-1}\psi (0)$, $p_h^{\textrm{P}}(0): = p_I^{\textrm{P}}(0)$ and applying the triangle inequality together with (A.6), completes the rest of the proof.

Appendix B: Proof of Theorem 4.2

As in the semidiscrete case we split the individual errors as

$$\begin{aligned} {\varvec{u}}(t_n) - {\varvec{u}}_h^n&= ({\varvec{u}}(t_n) - I^h_{{\varvec{u}}} {\varvec{u}}(t_n)) + (I^h_{{\varvec{u}}} {\varvec{u}}(t_n)- {\varvec{u}}_h^n)=: E_{{\varvec{u}}}^{I,n} + E_{{\varvec{u}}}^{A,n}, \\ \psi (t_n) - \psi _h^n&= (\psi (t_n) - I^h_{\psi } \psi (t_n)) + (I^h_{\psi } \psi (t_n)- \psi _h^n)=: E_{\psi }^{I,n} + E_{\psi }^{A,n}, \\ p^{\textrm{P}}(t_n) - p_h^{n,\textrm{P}}&= (p^{\textrm{P}}(t_n) - I^h_{p} p^{\textrm{P}}(t_n)) + (I^h_{p} p^{\textrm{P}}(t_n)- p_h^{n,\textrm{P}})=: E_{p}^{I,n} + E_{p}^{A,n}, \end{aligned}$$

where the error terms are $E_{p}^{I,n}:=E_{p}^{I,n} |_{\varOmega ^{\textrm{P}}}, E_{p}^{A,n}:= E_{p}^{A,n} |_{\varOmega ^{\textrm{P}}}$. Then, from estimate (4.2a) and following the steps of the proof of Theorem 4.1 we get the bounds

$$\begin{aligned} \Vert E_{{\varvec{u}}}^{I,n} \Vert _1&\lesssim h^k ( | {\varvec{u}}(0) |_{k+1} + | \psi ^{\textrm{P}}(0) |_{k,\varOmega ^{\textrm{P}}} + | \psi ^{\textrm{E}}(0) |_{k,\varOmega ^{\textrm{E}}} \nonumber \\&\quad + \Vert \partial _t{\varvec{u}}\Vert _{\textbf{L}^1(0,t_n; [H^{k+1}(\varOmega )]^2)} + \Vert \partial _t \psi \Vert _{L^1(0,t_n; k)} ), \end{aligned}$$

(B.1a)

$$\begin{aligned} \Vert E_{\psi }^{I,n} \Vert _0&\lesssim h^k ( | {\varvec{u}}(0) |_{k+1} + | \psi ^{\textrm{P}}(0) |_{k,\varOmega ^{\textrm{P}}} + | \psi ^{\textrm{E}}(0) |_{k,\varOmega ^{\textrm{E}}} \nonumber \\&\quad + \Vert \partial _t{\varvec{u}}\Vert _{\textbf{L}^1(0,t_n; [H^{k+1}(\varOmega )]^2)} + \Vert \partial _t\psi \Vert _{L^1(0,t_n; k)} ), \end{aligned}$$

(B.1b)

$$\begin{aligned} \Vert E_{p}^{I,n} \Vert _{1, \varOmega ^{\textrm{P}}}&\lesssim h^k ( | p^{\textrm{P}}(0) |_{k+1, \varOmega ^{\textrm{P}}} + \Vert \partial _t p^{\textrm{P}}\Vert _{L^1(0,t_n; H^{k+1}(\varOmega ^{\textrm{P}}))}), \end{aligned}$$

(B.1c)

where $\Vert \partial _t \psi \Vert _{L^1(0,t_n;k)}:= \Vert \partial _t \psi ^{\textrm{P}}\Vert _{L^1(0,t_n; H^k(\varOmega ^{\textrm{P}}))} + \Vert \partial _t \psi ^{\textrm{E}}\Vert _{L^1(0,t_n; H^k(\varOmega ^{\textrm{E}}))}$. From Eqs. (4.1a), (3.10a) and (2.4a), we readily get

$$\begin{aligned} a_1^h(E_{{\varvec{u}}}^{A,n},\varvec{v}_h) + b_1(\varvec{v}_h, E_{\psi }^{A,n}) = F^n(\varvec{v}_h) - F^{h,n}(\varvec{v}_h). \end{aligned}$$

(B.2)

We then use (4.1b) and (3.10c), and proceed to differentiate (2.4c) with respect to time. This implies

$$\begin{aligned} \begin{aligned}&b_1(E_{{\varvec{u}}}^{A,n} - E_{{\varvec{u}}}^{A,n-1}, \phi _h) + b_2(E_p^{A,n}-E_p^{A,n-1}, \phi _h) - a_3(E_{\psi }^{A,n} - E_{\psi }^{A,n-1}, \phi _h) \\&\quad = b_1(({\varvec{u}}(t_n) - {\varvec{u}}(t_{n-1})) - (\varDelta t) \partial _{t} {\varvec{u}}(t_n), \phi _h) + b_2((I_p^h p^{\textrm{P}}(t_n) - I_p^h p^{\textrm{P}}(t_{n-1}))\\&\qquad - (\varDelta t) \partial _{t} p^{\textrm{P}}(t_n), \phi _h) - a_3((I_{\psi }^h \psi (t_n) - I_{\psi }^h \psi (t_{n-1})) - (\varDelta t) \partial _{t} \psi (t_n), \phi _h). \end{aligned} \end{aligned}$$

(B.3)

Choosing $\varvec{v}_h = E_{{\varvec{u}}}^{A,n} - E_{{\varvec{u}}}^{A,n-1 }$ in (B.2) and $\phi _h = - E_{\psi }^{A,n}$ in (B.3) and adding the results, gives

$$\begin{aligned}&a_1^h(E_{{\varvec{u}}}^{A,n}, E_{{\varvec{u}}}^{A,n}- E_{{\varvec{u}}}^{A,n-1}) + a_3(E_{\psi }^{A,n} - E_{\psi }^{A,n-1}, E_{\psi }^{A,n}) - b_2(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{\psi }^{A,n}) \nonumber \\&\quad = ( \varvec{b}(t_n) - \varvec{b}^n_h, E_{{\varvec{u}}}^{A,n}- E_{{\varvec{u}}}^{A,n-1 } )_{0, \varOmega } - b_1(({\varvec{u}}(t_n) - {\varvec{u}}(t_{n-1})) - (\varDelta t) \partial _{t} {\varvec{u}}(t_n), E_{\psi }^{A,n}) \nonumber \\&\qquad - b_2((I_p^h p^{\textrm{P}}(t_n) - I_p^h p^{\textrm{P}}(t_{n-1})) - (\varDelta t) \partial _{t} p^{\textrm{P}}(t_n), E_{\psi }^{A,n}) \nonumber \\&\qquad + a_3((I_{\psi }^h \psi (t_n) - I_{\psi }^h \psi (t_{n-1}))- (\varDelta t) \partial _{t} \psi (t_n), E_{\psi }^{A,n}). \end{aligned}$$

(B.4)

Next, and as a consequence of using (4.1c), (3.4b) and (2.4b) with $q_h^{\textrm{P}} = E_p^{A,n}$, we are left with

$$\begin{aligned}&{\tilde{a}}_2^h(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{p}^{A,n}) + \varDelta t a_2^h(E_{p}^{A,n}, E_{p}^{A,n}) - b_2(E_{p}^{A,n}, E_{\psi }^{A,n} - E_{\psi }^{A,n-1}) \nonumber \\&\quad = \varDelta t ( \ell ^{\textrm{P}}(t_n)- \ell ^{n,\textrm{P}}_h, E_{p}^{A,n} )_{0, \varOmega ^{\textrm{P}}} + {\tilde{a}}_2^h( I^h_p p^{\textrm{P}}(t_n) - I^h_p p^{\textrm{P}}(t_{n-1}), E_{p}^{A,n}) \nonumber \\&\qquad - {\tilde{a}}_2((\varDelta t) \partial _{t} p^{\textrm{P}}(t_n), E_{p}^{A,n}) + b_2(E_{p}^{A,n}, (\varDelta t) \partial _{t} \psi - (I^h_{\psi } \psi (t_n)) - I^h_{\psi } \psi (t_{n-1})). \end{aligned}$$

(B.5)

Adding (B.4)–(B.5) and repeating the arguments used in deriving stability, we can assert that

$$\begin{aligned}&a_3(E_{\psi }^{A,n} - E_{\psi }^{A,n-1}, E_{\psi }^{A,n}) - b_2(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{\psi }^{A,n})\\&\qquad - b_2(E_{p}^{A,n}, E_{\psi }^{A,n} - E_{\psi }^{A,n-1}) + {\tilde{a}}_2^h(E_{p}^{A,n} - E_{p}^{A,n-1}, E_{p}^{A,n}) \\&\quad = (\varDelta t) \bigg ( c_0(\delta _t E_{p}^{A,n} , E_{p}^{A,n})_{0,\varOmega ^{\textrm{P}}} + \frac{1}{\lambda } \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \big (\alpha ^2 (\delta _t (I- \varPi ^{0,k}_K) E_p^{A,n}, (I- \varPi ^{0,k}_K) E_p^{A,n})_{0,K} \\&\qquad -(\delta _t (\alpha \varPi ^{0,k}_K E_p^{A,n} - E_{\psi }^{A,n}), \alpha \varPi ^{0,k}_K E_p^{A,n} - E_{\psi }^{A,n})_{0,K} \big ) + \frac{\varDelta t}{\lambda } (\delta _t E_{\psi }^{A,n} , E_{\psi }^{A,n})_{0,\varOmega ^{\textrm{E}}} \bigg ), \end{aligned}$$

The left-hand side can be bounded using the inequality

$$\begin{aligned} (f_h^n - f_h^{n-1}, f_h^n) \ge \frac{1}{2} \bigl (\Vert f_h^n \Vert _0^2 - \Vert f_h^{n-1}\Vert _0^2 \bigr ), \end{aligned}$$

and then summing over n we get

$$\begin{aligned}&{\mu _{\min }} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,n})\Vert _0^2 + c_0 \Vert E_{p}^{A,n}\Vert _{0, \varOmega ^{\textrm{P}}}^2 + (1/ {\lambda ^{\textrm{E}}}) \Vert E_{\psi }^{A,n}\Vert _{0, \varOmega ^{\textrm{E}}}^2 + (\varDelta t) \frac{\kappa _{\min }}{\eta } \sum _{j=1}^n \Vert \nabla E_{p}^{A,j} \Vert _{0, \varOmega ^{\textrm{P}}}^2 \nonumber \\&\qquad + (1/{\lambda ^{\textrm{P}}}) \sum _{K \in \mathcal {T}_h^{\textrm{P}}}\bigg ( \alpha ^2 \Vert (I- \varPi ^{0,k}_K) E_p^{A,n} \Vert _{0,K}^2 + \Vert \alpha \varPi ^{0,k}_K E_p^{A,n} - E_{\psi }^{A,n}\Vert _{0,K}^2 \bigg ) \\&\quad \le {\mu _{\min }} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,0})\Vert _0^2 + c_0 \Vert E_{p}^{A,0}\Vert _{0, \varOmega ^{\textrm{P}}}^2 + (1/ {\lambda ^{\textrm{E}}}) \Vert E_{\psi }^{A,0}\Vert _{0, \varOmega ^{\textrm{E}}}^2 \\&\qquad + (1/{\lambda ^{\textrm{P}}}) \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \bigg ( \alpha ^2 \Vert (I- \varPi ^{0,k}_K) E_p^{A,0} \Vert _{0,K}^2 + \Vert \alpha \varPi ^{0,k}_K E_p^{A,0} - E_{\psi }^{A,0} \Vert _{0,K}^2 \bigg ) \\&\qquad + \underbrace{ \sum _{j=1}^n ( \varvec{b}(t_j) - \varvec{b}^j_h, E_{{\varvec{u}}}^{A,j}- E_{{\varvec{u}}}^{A,j-1} )_{0,\varOmega }}_{=:L_1} + \underbrace{ \sum _{j=1}^n \varDelta t ( \ell ^{\textrm{P}}(t_j)- \ell ^{j,\textrm{P}}_h, E_{p}^{A,j} )_{0,\varOmega ^{\textrm{P}}}}_{=:L_2} \nonumber \\&\qquad - \underbrace{ \sum _{j=1}^n b_1(({\varvec{u}}(t_j) - {\varvec{u}}(t_{j-1})) - (\varDelta t) \partial _{t} {\varvec{u}}(t_j), E_{\psi }^{A,j})}_{=:L_3} \nonumber \\&\qquad - \underbrace{ \sum _{j=1}^n b_2((I_p^h p^{\textrm{P}}(t_j) - I_p^h p^{\textrm{P}}(t_{j-1})) - (\varDelta t) \partial _{t} p^{\textrm{P}}(t_j), E_{\psi }^{A,j})}_{=:L_4} \nonumber \\&\qquad + \underbrace{ \sum _{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} \nonumber \\&\qquad + \underbrace{ \sum _{j=1}^n ({\tilde{a}}_2^h( I^h_p p^{\textrm{P}}(t_j) - I^h_p p^{\textrm{P}}(t_{j-1}), E_{p}^{A,j}) - {\tilde{a}}_2((\varDelta t) \partial _{t} p^{\textrm{P}}(t_j), E_{p}^{A,j}) )}_{:=L_6} \nonumber \\&\qquad + \underbrace{ \sum _{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{aligned}$$

We bound the term $L_1$ with the help of the following formula

$$\begin{aligned} \sum _{j=1}^n (f_h^j - f_h^{j-1}, g_h^j )= (f_h^n, g_h^n) - (f_h^0, g_h^0)-\sum _{j=1}^n (f_h^{j-1}, g_h^j- g_h^{j-1}), \end{aligned}$$

and in combination with the orthogonality property of projection $\varvec{\varPi }^{0,k}_K$ onto piecewise polynomials, and applying Taylor expansion

$$\begin{aligned} f^j - f^{j-1} = (\varDelta t) \partial _{t} f^j - \int _{t_{j-1}}^{t_j} (s-t_{j-1})\partial _{tt} f(s) \,\textrm{d}s, \end{aligned}$$

we get

$$\begin{aligned} L_1&{ = \sum _{K \in \mathcal {T}_h} \bigg ( ((\textbf{I}- \varvec{\varPi }^{0,k}_K)\varvec{b}(t_n), (\textbf{I}- \varvec{\varPi }^{0,0}_K) E_{{\varvec{u}}}^{A,n})_{0,K} - ((\textbf{I}- \varvec{\varPi }^{0,k}_K)\varvec{b}(0), (\textbf{I}- \varvec{\varPi }^{0,0}_K)E_{{\varvec{u}}}^{A,0})_{0,K}} \\&\quad {- \sum _{j=1}^{n} \left( (\textbf{I}- \varvec{\varPi }^{0,k}_K) \bigg ((\varDelta t)\partial _t \varvec{b}^j - \int _{t_{j-1}}^{t_j} (s-t_{j-1}) \partial _{tt}\varvec{b}(s) \,\textrm{d}s\bigg ), E_{{\varvec{u}}}^{A,j-1} \right) _{0,K} \bigg ).} \end{aligned}$$

Then the Cauchy–Schwarz and Korn inequalities, the estimates of projection $\varvec{\varPi }^{0,k}_K$,and finally using the generalised Young’s inequality, gives the bound

$$\begin{aligned} L_1&{ \le }\frac{{\mu _{\min }}}{2} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,n})\Vert _0^2 +{C(\mu _{\min })} h^{2k} |\varvec{b}(t_n) |_{k-1}^2 + C(\mu _{\min }) h^k |b(t_0)|_{k-1}~ {\mu _{\min }^{{1/2}}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,0})\Vert _0 \\&\qquad +{ (\varDelta t) ~ \sum _{j=1}^n C(\mu _{\min }) \Big (\Vert \partial _{t} \varvec{b}^j \Vert _{0} + (\varDelta t)^{1/2} \Vert \partial _{tt} \varvec{b}\Vert _{\textbf{L}^2(t_{j-1},t_j;[L^2(\varOmega )]^2)} \Big )} {\mu _{\min }^{{1/2}}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j-1}) \Vert _0. \end{aligned}$$

Then the estimate satisfied by the projection $\varPi ^{0,k}_K$ along with Poincaré and Young’s inequalities yield

$$\begin{aligned} L_2 \,&{ = \sum _{j=1}^n \varDelta t ((I - \varPi ^{0,k}_K)\ell ^{\textrm{P}}(t_j), (I - \varPi ^{0,0}_K)E_{p}^{A,j} )_{0,\varOmega ^{\textrm{P}}} } \\&{\le C(\kappa _{\min },\eta )} {h^{2k} } (\varDelta t) \sum _{j=1}^n | \ell ^{\textrm{P}}(t_j)|_{k-1, \varOmega ^{\textrm{P}}}^2 + (\varDelta t) \frac{\kappa _{\min }}{6 \eta } \sum _{j=1}^n \Vert \nabla E_{p}^{A,j}\Vert _{0, \varOmega ^{\textrm{P}}}^2. \end{aligned}$$

The discrete inf-sup condition (3.3) implies that

$$\begin{aligned} \Vert E_{\psi }^{A,j} \Vert _0 \lesssim { h^k \Vert \varvec{b}^j\Vert _{k-1} + {\mu _{\max }} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j}) \Vert _0 \le h^k \Vert \varvec{b}^j\Vert _{k-1} + C(\mu ) {\mu _{\min }^{1/2}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j}) \Vert _0.} \end{aligned}$$

(B.6)

Applying an expansion in Taylor series, together with (B.6) the Cauchy–Schwarz inequality enable us to write

$$\begin{aligned} L_3 \,&{ \lesssim \sum _{j=1}^n \Vert ({\varvec{u}}(t_j) - {\varvec{u}}(t_{j-1})) - (\varDelta t) \partial _{t} {\varvec{u}}(t_j) \Vert _0 \Vert E_{\psi }^{A,j} \Vert _0 }\\&\lesssim {(\varDelta t)^{3/2}\sum _{j=1}^n \Vert \partial _{tt} {\varvec{u}}\Vert _{\textbf{L}^2(t_{j-1},t_j;[L^2(\varOmega )]^2)} \Big ( h^k \Vert \varvec{b}^j\Vert _{k-1} + C(\mu ) {\mu _{\min }^{1/2}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j}) \Vert _0 \Big ).} \end{aligned}$$

Then, applying again the Cauchy–Schwarz inequality, and after using the interpolation estimates (4.2b), the bound (B.6), we get

$$\begin{aligned} L_4 \,&{\lesssim \frac{\alpha }{\lambda ^{\textrm{P}}} \sum _{j=1}^n (\varDelta t) \Big ( \Vert I_p^h (\delta _t p^{\textrm{P}}(t_j)) - \delta _t p^{\textrm{P}}(t_j) \Vert _{0, \varOmega ^{\textrm{P}}} + \Vert \delta _t p^{\textrm{P}}(t_j)- \partial _{t} p^{\textrm{P}}(t_j) \Vert _{0, \varOmega ^{\textrm{P}}} \Big ) \Vert E_{\psi }^{A,j} \Vert _{0, \varOmega ^{\textrm{P}}}} \\&\lesssim { \frac{\alpha }{{\lambda ^{\textrm{P}}}} (\varDelta t)^{1/2} \sum _{j=1}^n \bigg ( h^{k+1} \Vert \partial _{t}p^{\textrm{P}}\Vert _{L^2(t_{j-1},t_j, H^{k+1}( \varOmega ^{\textrm{P}}))} + (\varDelta t ) \Vert \partial _{tt} p^{\textrm{P}}\Vert _{L^2(t_{j-1},t_j, L^2(\varOmega ^{\textrm{P}}))} \bigg )} \\&\quad \times {( h^k \Vert \varvec{b}^j\Vert _{k-1} + C(\mu ) {\mu _{\min }^{1/2}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j}) \Vert _{0} ).} \end{aligned}$$

On the other hand, the stability of $a_3(\cdot , \cdot )$, the proof for the bound of $L_4$ , and the interpolant estimate (4.2a) gives

$$\begin{aligned} L_5&{\lesssim \frac{1}{\lambda } \sum _{j=1}^n (\varDelta t) \Big ( \Vert I_{\psi }^h (\delta _t \psi (t_j) ) - \delta _t \psi (t_j) \Vert _{0} + \Vert \delta _t \psi (t_j) - \partial _{t} {\psi }(t_j) \Vert _{0} \Big ) \Vert E_{\psi }^{A,j} \Vert _{0} } \\&{\lesssim \frac{1}{\lambda _{\min }} (\varDelta t)^{1/2} \sum _{j=1}^n \bigg ( h^k \big (\Vert \partial _t{\varvec{u}}\Vert _{\textbf{L}^2(t_{j-1},t_j; [H^{k+1}(\varOmega )]^2)}^2 + \Vert \partial _t\psi \Vert _{L^2(t_{j-1},t_j; H^k(\varOmega ))}^2 \big )^{1/2} } \\&\quad + {(\varDelta t) \Vert \partial _{tt} \psi \Vert _{L^2(t_{j-1},t_j;L^2(\varOmega ))} \bigg ) ( h^k \Vert \varvec{b}^j\Vert _{k-1} + C(\mu ) {\mu _{\min }^{1/2}} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,j}) \Vert _0 ),} \end{aligned}$$

where $\Vert \partial _t v\Vert _{\textbf{L}^2(t_{j-1},t_j; H(\varOmega ))}^2:= \sum _{D=\varOmega ^{\textrm{P}}, \varOmega ^{\textrm{E}}} \Vert \partial _t v^D\Vert _{\textbf{L}^2(t_{j-1},t_j; H(D))}^2$ for $v={\varvec{u}}, \psi$ with space $H(D)=[H^{k+1}(D)]^2, H^k(D)$ respectively. The polynomial approximation $\varPi ^{0,k}_K p^{\textrm{P}}$ for fluid pressure, consistency of the bilinear form ${\tilde{a}}_2^h(\cdot , \cdot )$, stability of the bilinear forms ${\tilde{a}}_2(\cdot , \cdot ), {\tilde{a}}_2^h(\cdot , \cdot )$, the usage of the Cauchy–Schwarz, Poincaré and Young’s inequalities gives

$$\begin{aligned} L_6&{\le C \Big ( c_0 +\frac{\alpha ^2}{{\lambda ^{\textrm{P}}}} \Big ) \sum _{j=1}^n \sum _{K \in \mathcal {T}_h^{\textrm{P}}} (\varDelta t) \Big ( \Vert (I^h_p - \varPi ^{0,k}_K) (\delta _t p^{\textrm{P}}(t_j) ) \Vert _{0,K} + \Vert (I- \varPi ^{0,k}_K )(\delta _t p^{\textrm{P}}(t_j) ) \Vert _{0,K} } \\&\qquad \qquad \qquad \qquad \qquad \qquad {+ \Vert \delta _t p^{\textrm{P}}(t_j) - \partial _{t} p^{\textrm{P}}(t_j) \Vert _{0,K} \Big ) \Vert \nabla E_{p}^{A,j} \Vert _{0, K} }\\&{\le C(\kappa _{\min }, \eta ) \Big ( c_0 +\frac{\alpha ^2}{\lambda ^{\textrm{P}}} \Big ) \sum _{j=1}^n (\varDelta t)^{1/2}} \sum _{K \in \mathcal {T}_h^{\textrm{P}}} \! \Big ( h^{k+1} \Vert \partial _t p^{\textrm{P}} \Vert _{L^2(0,t_n;H^{k+1}(\varOmega ^{\textrm{P}}))} \\ {}&\qquad \qquad \qquad \qquad \qquad \qquad + (\varDelta t) \Vert \partial _{tt} p^{\textrm{P}} \Vert _{L^2(0,t_n;L^2(\varOmega ^{\textrm{P}}))} \Big ) \Vert \nabla E_{p}^{A,j} \Vert _{0, K} \\&{\le C(\kappa _{\min }, \eta ) (\varDelta t)} {\Big ( c_0 + \frac{\alpha ^2}{{\lambda ^{\textrm{P}}}} \Big )^2} \Big ( h^{2(k+1)} \Vert \partial _t p^{\textrm{P}} \Vert _{L^2(0,t_n;H^{k+1}(\varOmega ^{\textrm{P}}))}^2 + (\varDelta t)^2 \Vert \partial _{tt} p^{\textrm{P}} \Vert _{L^2(0,t_n;L^2(\varOmega ^{\textrm{P}}))}^2 \Big ) \\&\qquad \qquad \qquad \qquad \qquad \qquad + \varDelta t \frac{\kappa _{\min }}{6 \eta } \sum _{j=1}^n \Vert \nabla E_{p}^{A,j} \Vert _{0, \varOmega ^{\textrm{P}}}^2. \end{aligned}$$

On the other hand, the continuity of $b_2(\cdot , \cdot )$, the bound derived for the term $L_5$ and using Young’s inequality implies that

$$\begin{aligned} L_7&{\le \frac{\alpha }{\lambda }\sum _{j=1}^n \bigg ( h^k (\varDelta t)^{1/2} \big (\Vert \partial _t{\varvec{u}}\Vert _{\textbf{L}^2(t_{j-1},t_j; [H^{k+1}(\varOmega )]^2)}^2 + \Vert \partial _t\psi \Vert _{L^2(t_{j-1},t_j; H^k(\varOmega ))}^2 \big )^{1/2} } \\ {}&\qquad \qquad \qquad { + (\varDelta t)^{3/2} \Vert \partial _{tt} \psi \Vert _{L^2(t_{j-1},t_j;L^2(\varOmega ))} \bigg ) \Vert E_{p}^{A,j}\Vert _{0, \varOmega ^{\textrm{P}}} } \\&{\le C(\kappa _{\min }, \eta ) } {\Big ( \frac{\alpha }{{\lambda _{\min }}} \Big )^2}\bigg ( h^{2k} (\Vert \partial _{t} \psi \Vert _{L^2(0,t_n;{H^k(\varOmega )})}^2 + \Vert \partial _{t} {\varvec{u}}\Vert _{\textbf{L}^2(0,t_n;[H^{k+1}(\varOmega )]^2)}^2)\\ {}&\qquad \qquad \qquad + (\varDelta t )^2 \Vert \partial _{tt} \psi \Vert _{L^2(0,t_n;L^2(\varOmega ))}^2 \bigg ) +(\varDelta t) \frac{\kappa _{\min }}{6 \eta } \sum _{j=1}^n \Vert \nabla E_{p}^{A,j} \Vert _{0{, \varOmega ^{\textrm{P}}}}^2. \end{aligned}$$

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

$$\begin{aligned}&{\mu _{\min }} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,n})\Vert _0^2 + c_0 \Vert E_{p}^{A,n}\Vert _{0, \varOmega ^{\textrm{P}}}^2 + (1/{\lambda ^{\textrm{E}}}) \Vert E_{\psi }^{A,n}\Vert _{0, \varOmega ^{\textrm{E}}}^2 + (\varDelta t) \frac{\kappa _{\min }}{\eta } \sum _{j=1}^n \Vert \nabla E_{p}^{A,j} \Vert _{0, \varOmega ^{\textrm{P}}}^2 \\&\quad \lesssim {\mu _{\min }} \Vert \varvec{\varepsilon }(E_{{\varvec{u}}}^{A,0})\Vert _0^2 + (c_0 + \alpha ^2/{\lambda ^{\textrm{P}}}) \Vert E_{p}^{A,0}\Vert _{0, \varOmega ^{\textrm{P}}}^2 + (1/{\lambda ^{\textrm{E}}}) \Vert E_{\psi }^{A,0}\Vert _{0, \varOmega ^{\textrm{E}}}^2 \\&\qquad + C(\mu _{\min }) \Big (1 + \varDelta t \Big ) h^{2k} \max _{0 \le j \le n} |\varvec{b}(t_j) |_{k-1}^2 \\&\qquad + { C(\mu _{\min }, \kappa _{\min }, \eta , \alpha , \lambda _{\min }) (\varDelta t)^{1/2} }\\&\qquad { \times \biggl \{ \bigg ( \sum _{j=1}^n \Big ( (\varDelta t)^{1/2} \Vert \partial _{t} \varvec{b}^j\Vert _0 + (\varDelta t) \Vert \partial _{tt} \varvec{b}\Vert _{\textbf{L}^2(t_{j-1},t_j;[L^2(\varOmega )]^2)} } \\&\qquad { + h^k ( h\Vert \partial _{t} p^P \Vert _{L^2(t_{j-1},t_j;H^{k+1}(\varOmega ^{\textrm{P}}))} + (\Vert \partial _{t} {\varvec{u}}\Vert _{\textbf{L}^2(t_{j-1},t_j;[H^{k+1}(\varOmega )]^2)}^2 + \Vert \partial _{t} \psi \Vert _{L^2(t_{j-1},t_j;H^{k}(\varOmega ))}^2)^{1/2}) \Big )} \\&\qquad { + (\varDelta t) \Big ( \Vert \partial _{tt} {\varvec{u}}\Vert _{\textbf{L}^2(t_{j-1},t_j;[L^2(\varOmega )]^2)} + \Vert \partial _{tt} p^P \Vert _{L^2(t_{j-1},t_j;L^2(\varOmega ^{\textrm{P}}))} + \Vert \partial _{tt} \psi \Vert _{L^2(t_{j-1},t_j;L^2(\varOmega ))} \Big ) \bigg )^2}\\&\qquad { + \sum _{j=1}^n h^{2 k} \Big ( (\varDelta t)^{1/2} |\ell ^P(t_j)|_{k-1,\varOmega ^{\textrm{P}}}^2 + |\varvec{b}^j|_{k-1}^2 \Big ) } \\&\qquad { + h^{2k} \Big ( h^2 \Vert \partial _{t} p^P \Vert _{L^2(0,t_n;H^{k+1}(\varOmega ^{\textrm{P}}))}^2 + \Vert \partial _{t} {\varvec{u}}\Vert _{\textbf{L}^2(0,t_n;[H^{k+1}(\varOmega )]^2)}^2 + \Vert \partial _{t} \psi \Vert _{L^2(0,t_n;H^k(\varOmega ))}^2 \Big )} \\&\qquad { + (\varDelta t)^2 \Big ( \Vert \partial _{tt} p^P\Vert _{L^2(0,t_n;L^2(\varOmega ^{\textrm{P}}))} ^2 + \Vert \partial _{tt} {\varvec{u}}\Vert _{\textbf{L}^2(0,t_n; [L^2(\varOmega )]^2)}^2 + \Vert \partial _{tt} \psi \Vert _{L^2(0,t_n;L^2(\varOmega ))}^2 \Big ) \biggr \}. } \end{aligned}$$

And finally, the desired result (4.3) is proven after choosing ${\varvec{u}}_h^0: ={\varvec{u}}_I(0)$, $\psi _h^0: = \varPi ^{0,k-1}\psi (0)$, $p_h^{0, \textrm{P}}: = p_I^{\textrm{P}}(0)$ and applying triangle’s inequality together with (B.1a)–(B.1c) and (B.6).