$\displaystyle\quad+(a_{2}^{h}(e_{I}^{p},u_{h})-a_{2}(e^{p},J_{2}^{*}u_{h}))+(a_{3}^{h}(p_{h},e_{I}^{p})-a_{3}(J_{4}p_{h},e^{p}))=:T_{1}+T_{2}+T_{3}+T_{4}.$

Theorem 6.7(b)-6.7(d) imply
$(\Pi_{k-2}\nabla^{2}u_{h},\Pi_{k-2}\nabla^{2}e_{I}^{u})_{\Omega}=(\nabla^{2}_{\mathrm{pw}}\Pi^{\nabla^{2\\!}}_{k}u_{h},\nabla^{2}J_{2}^{*}e_{I}^{u})_{\Omega}$
and
$(\Pi_{k}u_{h},\Pi_{k}e_{I}^{u})_{\Omega}=(\Pi_{k}u_{h},J_{2}^{*}e_{I}^{u})_{\Omega}$.
This results in

$\displaystyle T_{1}$
$\displaystyle=-(\Pi_{k}u_{h},v)_{\Omega}+(\Pi_{k}u_{h}-J_{2}^{*}u_{h},e^{u})_{\Omega}-(\nabla^{2}_{\mathrm{pw}}\Pi^{\nabla^{2\\!}}_{k}u_{h},\nabla^{2}_{\mathrm{pw}}v)_{\Omega}$
$\displaystyle\quad+(\nabla^{2}_{\mathrm{pw}}(\Pi^{\nabla^{2\\!}}_{k}u_{h}-J_{2}^{*}u_{h}),\nabla^{2}e^{u})_{\Omega}+S_{1,0}((1-\Pi_{k})u_{h},(1-\Pi_{k})e_{I}^{u})$
$\displaystyle\quad+S_{\nabla^{2}}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})e_{I}^{u}).$

An integration by parts and the fact that $v\in V$, lead to

$\displaystyle-(\nabla^{2}_{\mathrm{pw}}\Pi^{\nabla^{2\\!}}_{k}u_{h},\nabla^{2}_{\mathrm{pw}}v)_{\Omega}$
$\displaystyle=-\sum_{K\in\mathcal{T}_{h}}(\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h},v)_{K}-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{s}}([\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})],\partial_{\boldsymbol{n}}v)_{e}$
$\displaystyle\quad-\sum_{e\in\mathcal{E}^{i}}([T(\Pi^{\nabla^{2\\!}}_{k}u_{h})],v)_{e}.$

This simplifies $T_{1}$ to

$\displaystyle T_{1}$
$\displaystyle=\sum_{K\in\mathcal{T}_{h}}\Big{(}(-\Pi_{k}u_{h}-\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h},v)_{K}+(\Pi_{k}u_{h}-J_{2}^{*}u_{h},e^{u})_{K}$
$\displaystyle\qquad+(\nabla^{2}(\Pi^{\nabla^{2\\!}}_{k}u_{h}-J_{2}^{*}u_{h}),\nabla^{2}e^{u})_{K}+S_{1,0}^{K}((1-\Pi_{k})u_{h},(1-\Pi_{k})e_{I}^{u})$
$\displaystyle\qquad+S_{\nabla^{2}}^{K}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})e_{I}^{u})\Big{)}-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{s}}([\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})],\partial_{\boldsymbol{n}}v)_{e}$
$\displaystyle\qquad-\sum_{e\in\mathcal{E}^{i}}([T(\Pi^{\nabla^{2\\!}}_{k}u_{h})],v)_{e}.$

Theorem 6.7(c) and the $L^{2}$-orthogonality of $\Pi_{k-1}$ imply
$(\Pi_{\ell-1}\nabla p_{h},\nabla J_{2}^{*}e_{I}^{u}-\Pi_{k-1}\nabla
e_{I}^{u})_{\Omega}=0$. This and an integration by parts show

(6.10) $\displaystyle\alpha^{-1}T_{2}$ $\displaystyle=(\nabla
J_{4}p_{h}-\Pi_{\ell-1}\nabla p_{h},\nabla
e^{u})_{\Omega}-\sum_{K\in\mathcal{T}_{h}}(\nabla\cdot\Pi_{\ell-1}\nabla
p_{h},v)_{K}+\sum_{e\in\mathcal{E}^{i}}([\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}_{e}],v)_{e}.$

Theorem 5.4(b), the $L^{2}$-orthogonality of $\Pi_{\ell-1}$, and again an
integration by parts prove that

$\displaystyle\alpha^{-1}T_{3}$
$\displaystyle=\sum_{K\in\mathcal{T}_{h}}\Big{(}(\Pi_{\ell-1}\nabla
e_{I}^{p}-\nabla J_{4}e_{I}^{p},\Pi_{k-1}\nabla u_{h}-\Pi_{\ell-1}\nabla
u_{h})_{K}+(q,\nabla\cdot\Pi_{k-1}\nabla u_{h})_{K}$ (6.11)
$\displaystyle\quad+(\nabla e^{p},\Pi_{k-1}\nabla u_{h}-\nabla
J_{2}^{*}u_{h})_{K}\Big{)}-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{c}}(q,[\Pi_{k-1}\nabla
u_{h}\cdot\boldsymbol{n}_{e}])_{e}.$

The identities $(\Pi_{\ell-1}\nabla p_{h},\Pi_{\ell-1}\nabla
e_{I}^{p})_{\Omega}=(\Pi_{\ell-1}\nabla p_{h},\nabla J_{4}e_{I}^{p})_{\Omega}$
and
$(\Pi_{\ell}p_{h},\Pi_{\ell}e_{I}^{p})_{\Omega}=(\Pi_{\ell}p_{h},J_{4}e_{I}^{p})_{\Omega}$
follow from Theorem 5.4(b)-5.4(d). This in the first step and an integration
by parts in the next step lead to

$\displaystyle T_{4}$
$\displaystyle=\beta((\Pi_{\ell}p_{h}-J_{4}p_{h},e^{p})_{\Omega}-(\Pi_{\ell}p_{h},q)_{\Omega})+\gamma((\Pi_{\ell-1}\nabla
p_{h}-\nabla J_{4}p_{h},\nabla e^{p})_{\Omega}$
$\displaystyle\quad-(\Pi_{\ell-1}\nabla p_{h},\nabla
q)_{\Omega})+S_{2,0}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})e_{I}^{p})$
$\displaystyle\quad+S_{\nabla}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})e_{I}^{p})$
$\displaystyle=\sum_{K\in\mathcal{T}_{h}}\Big{(}\beta(\Pi_{\ell}p_{h}-J_{4}p_{h},e^{p})_{K}+\gamma(\Pi_{\ell-1}\nabla
p_{h}-\nabla J_{4}p_{h},\nabla e^{p})_{K}$
$\displaystyle\quad+(\gamma\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}-\beta\Pi_{\ell}p_{h},q)_{K}+S_{2,0}^{K}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})e_{I}^{p})$
$\displaystyle\quad+S_{\nabla}^{K}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})e_{I}^{p})\Big{)}-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{c}}\gamma([\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}_{e}],q)_{e}.$

The rearrangement of the terms results in

(6.12) $\displaystyle\|\vec{\boldsymbol{e}}\|^{2}_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim T_{5}+\dots+T_{10},$

where

$\displaystyle T_{5}$
$\displaystyle:=(\tilde{f}-\Pi_{k}\tilde{f},J_{2}^{*}e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u})_{\Omega}+(\tilde{f}-\Pi_{k}v_{h}-\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h}-\alpha\nabla\cdot\Pi_{\ell-1}\nabla
p_{h},v)_{\Omega},$ $\displaystyle T_{6}$
$\displaystyle:=(\tilde{g}-\Pi_{\ell}\tilde{g},J_{4}e_{I}^{p}-\Pi^{\nabla}_{\ell}e_{I}^{p})_{\Omega}+(\tilde{g}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}-\beta\Pi_{\ell}p_{h}+\alpha\nabla\cdot\Pi_{k-1}\nabla
u_{h},q)_{\Omega},$ $\displaystyle T_{7}$
$\displaystyle:=(\Pi_{k}u_{h}-J_{2}^{*}u_{h},e^{u})_{\Omega}+(\nabla^{2}_{\mathrm{pw}}(\Pi^{\nabla^{2\\!}}_{k}u_{h}-J_{2}^{*}u_{h}),\nabla^{2}e^{u})_{\Omega}$
$\displaystyle\quad+\alpha(\nabla J_{4}p_{h}-\Pi_{\ell-1}\nabla p_{h},\nabla
e^{u})_{\Omega}+\alpha(\Pi_{\ell-1}\nabla e_{I}^{p}-\nabla
J_{4}e_{I}^{p},\Pi_{k-1}\nabla u_{h}-\Pi_{\ell-1}\nabla u_{h})_{K}$
$\displaystyle\quad+\alpha(\nabla e^{p},\Pi_{k-1}\nabla u_{h}-\nabla
J_{2}^{*}u_{h})_{\Omega}+\beta(\Pi_{\ell}p_{h}-J_{4}p_{h},e^{p})_{\Omega}$
$\displaystyle\quad+\gamma(\Pi_{\ell-1}\nabla p_{h}-\nabla J_{4}p_{h},\nabla
e^{p})_{\Omega},$ $\displaystyle T_{8}$
$\displaystyle:=S_{1,0}((1-\Pi_{k})u_{h},(1-\Pi_{k})e_{I}^{u})+S_{\nabla^{2}}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})e_{I}^{u})$
$\displaystyle\quad+S_{2,0}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})e_{I}^{p})+S_{\nabla}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})e_{I}^{p}),$
$\displaystyle T_{9}$
$\displaystyle:=-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{c}}(q,[\alpha\Pi_{k-1}\nabla
u_{h}\cdot\boldsymbol{n}_{e}+\gamma\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}_{e}])_{e},$ $\displaystyle T_{10}$
$\displaystyle:=-\sum_{e\in\mathcal{E}^{i}\cup\mathcal{E}^{s}}([\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})],\partial_{\boldsymbol{n}}v)_{e}-\sum_{e\in\mathcal{E}^{i}}([T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}_{e}],v)_{e}.$

The Poincaré–Friedrichs inequality implies that
$h_{P}^{-2}\|J_{2}^{*}e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u}\|_{L^{2}(K)}\lesssim|J_{2}^{*}e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u}|_{2,K}$
and $h_{P}^{-2}\|v\|_{L^{2}(K)}\lesssim|v|_{2,K}$. Then the triangle
inequality $|J_{2}^{*}e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u}|_{2,K}$
$\leq|J_{2}^{*}e_{I}^{u}-e_{I}^{u}|_{2,K}+|e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u}|_{2,K}$
and
$|v|_{2,K}\leq|e^{u}-e_{I}^{u}|_{2,K}+|e_{I}^{u}-J_{2}^{*}e_{I}^{u}|_{2,K}$
followed by propositions 5.1 and 3.1, and Theorem 6.7(e) show

(6.13) $\displaystyle
T_{5}\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}\eta_{1,K}\Big{)}|e^{u}|_{2,\Omega}.$

Similarly we can prove that
$T_{6}\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}\eta_{2,K}\Big{)}|e^{p}|_{1,\Omega}\leq\Big{(}\sum_{K\in\mathcal{T}_{h}}\eta_{2,K}\Big{)}\gamma^{1/2}|e^{p}|_{1,\Omega}$.
Then we proceed to rewrite the terms in $T_{7}$ using the
$L^{2}$-orthogonality $\Pi_{k},\Pi_{\ell}$ and Theorem 6.7(d)-5.4(c) as

$\displaystyle(\Pi_{k}u_{h}-J_{2}^{*}u_{h},e^{u})_{\Omega}$
$\displaystyle=(\Pi_{k}u_{h}-J_{2}^{*}u_{h},e^{u}-\Pi_{k}e^{u})_{\Omega}=(\Pi^{\nabla^{2\\!}}_{k}u_{h}-J_{2}^{*}u_{h},e^{u}-\Pi_{k}e^{u})_{\Omega},$
$\displaystyle(\Pi_{\ell}p_{h}-J_{4}p_{h},e^{p})_{\Omega}$
$\displaystyle=(\Pi_{\ell}p_{h}-J_{4}p_{h},e^{p}-\Pi_{\ell}e^{p})_{\Omega}=(\Pi^{\nabla}_{\ell}p_{h}-J_{4}p_{h},e^{p}-\Pi_{\ell}e^{p})_{\Omega}.$

Then we combine Cauchy–Schwarz and triangle inequalities, which results in

$\displaystyle T_{7}$
$\displaystyle\lesssim(\|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}\|_{\Omega}+|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,h}+\|u_{h}-J_{2}^{*}u_{h}\|_{\Omega}+|u_{h}-J_{2}^{*}u_{h}|_{2,h}$
$\displaystyle\quad+|u_{h}-\Pi^{\nabla}_{\ell}u_{h}|_{1,h}+\|p_{h}-\Pi^{\nabla}_{\ell}p_{h}\|_{\Omega}+|p_{h}-\Pi^{\nabla}_{\ell}p_{h}|_{1,h}+\|p_{h}-J_{4}p_{h}\|_{\Omega}$
$\displaystyle\quad+|p_{h}-J_{4}p_{h}|_{1,h})\|\vec{\boldsymbol{e}}\|_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\sum_{K\in\mathcal{T}_{h}}\Big{(}(1+\alpha^{1/2}h_{K}+h_{K}^{2})|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,K}+\alpha^{1/2}|u_{h}-\Pi^{\nabla}_{\ell}u_{h}|_{1,h}$
$\displaystyle\qquad+\beta^{1/2}\|p_{h}-\Pi^{\nabla}_{\ell}p_{h}\|_{K}+\gamma^{1/2}|p_{h}-\Pi^{\nabla}_{\ell}p_{h}|_{1,K}+\eta_{8,K}\Big{)}\|\vec{\boldsymbol{e}}\|_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\sum_{K\in\mathcal{T}_{h}}(\eta_{6,K}+\eta_{7,K}+\eta_{8,K})\|\vec{\boldsymbol{e}}\|_{\mathbf{H}_{\epsilon}}.$

The second step follows from the Poincaré–Friedrichs inequality and Theorem
5.1-5.4, and the last step from (3.8a)-(3.8c) and the equivalence
$|u_{h}-\Pi^{\nabla}_{\ell}u_{h}|_{1,K}\approx\|\mathrm{Dof}^{k,\mathrm{nc}}(u_{h}-\Pi^{\nabla}_{\ell}u_{h})\|_{\ell^{2}}$
(see [30] for a proof). Cauchy–Schwarz inequalities for inner products and
(3.8b) lead to

$\displaystyle T_{8}$
$\displaystyle\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}\|u_{h}-\Pi_{k}u_{h}\|_{L^{2}(K)}+S_{\nabla^{2}}^{1/2}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})u_{h})$
$\displaystyle\quad+S_{2,0}^{1/2}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})p_{h})+S_{\nabla}^{1/2}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})p_{h})\Big{)}\|\vec{\boldsymbol{e}}\|_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}\eta_{6,K}\Big{)}\|\vec{\boldsymbol{e}}\|_{\mathbf{H}_{\epsilon}},$

with
$\|u_{h}-\Pi_{k}u_{h}\|_{K}\leq\|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}\|_{K}\lesssim
h_{K}^{2}|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,K}$ followed by (3.8a) in the
last estimate. The trace inequality shows $\|q\|_{e}\lesssim
h_{e}^{-1/2}\|q\|_{\omega_{e}}+h_{e}^{1/2}|q|_{1,\omega_{e}}\lesssim
h_{e}^{1/2}|q|_{1,\omega_{e}}$ with the Poincaré–Friedrichs inequality in the
last bound. This and the Cauchy–Schwarz inequality prove that

$\displaystyle
T_{9}\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}\eta_{5,K}\Big{)}\gamma^{1/2}|e^{p}|_{1,\Omega}.$

Finally, analogous arguments as those used in $T_{9}$ show that
$T_{10}\lesssim\Big{(}\sum_{K\in\mathcal{T}_{h}}(\eta_{3,K}+\eta_{4,K})\Big{)}|e^{u}|_{2,\Omega}$.
The previous bounds in (6.12) conclude the proof. ∎

###  6.5. Efficiency

######  Theorem 6.9 (Efficiency up to stabilisation and data oscillation).

Under the assumption $\ell\leq k\leq\ell+2$, the local error estimators are
bounded above as follows:

(6.14a) $\displaystyle\eta_{1,K}$
$\displaystyle\lesssim\|u-u_{h}\|_{K}+|u-u_{h}|_{2,K}+\gamma^{1/2}|p-p_{h}|_{1,K}+\eta_{6,K}+\mathrm{osc}_{2}(\tilde{f},K),$
(6.14b) $\displaystyle\eta_{2,K}$
$\displaystyle\lesssim|u-u_{h}|_{2,K}+\beta\|p-p_{h}\|_{K}+\gamma|p-p_{h}|_{1,K}+(\beta^{1/2}+\gamma^{1/2})\eta_{6,K}+\mathrm{osc}_{1}(\tilde{g},K),$
(6.14c) $\displaystyle\eta_{3,K}$
$\displaystyle\lesssim\sum_{e\in\mathcal{E}_{K}}\sum_{K^{\prime}\in\omega_{e}}\Big{(}\|u-u_{h}\|_{K^{\prime}}+|u-u_{h}|_{2,K^{\prime}}+|p-p_{h}|_{1,K^{\prime}}+\eta_{6,K^{\prime}}+\mathrm{osc}_{2}(\tilde{f},K^{\prime})\Big{)},$
(6.14d) $\displaystyle\eta_{4,K}$
$\displaystyle\lesssim\sum_{e\in\mathcal{E}_{K}}\sum_{K^{\prime}\in\omega_{e}}\Big{(}\|u-u_{h}\|_{K^{\prime}}+|u-u_{h}|_{2,K^{\prime}}+|p-p_{h}|_{1,K^{\prime}}+\eta_{6,K^{\prime}}+\mathrm{osc}_{2}(\tilde{f},K^{\prime})\Big{)},$
$\displaystyle\eta_{5,K}$
$\displaystyle\lesssim\sum_{e\in\mathcal{E}_{k}}\sum_{K^{\prime}\in\omega_{e}}\Big{(}|u-u_{h}|_{1,K^{\prime}}+\beta\|p-p_{h}\|_{K^{\prime}}+\gamma|p-p_{h}|_{1,K^{\prime}}+(\beta^{1/2}+\gamma^{1/2})\eta_{6,K^{\prime}}$
(6.14e)
$\displaystyle\qquad\qquad+\mathrm{osc}_{1}(\tilde{g},K^{\prime})\Big{)},$
(6.14f) $\displaystyle\eta_{7,K}$
$\displaystyle\lesssim|u-u_{h}|_{1,K}+|u-\Pi^{\nabla}_{\ell}u|_{1,K},$ (6.14g)
$\displaystyle\eta_{8,K}$
$\displaystyle\lesssim|u-u_{h}|_{2,K}+|p-p_{h}|_{1,K}+\eta_{6,K}.$

###### Proof.

Recall the element bubble-function $b_{2,K}:=b_{K}\in H^{2}_{0}(K)$ supported
in $K\in\mathcal{T}_{h}$ and $\ell\leq k$. Let

$v_{k}:=\Pi_{k}\tilde{f}-\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h}-\Pi_{k}u_{h}-\alpha\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}\in\mathbb{P}_{k}(K)\;\text{and}\;v:=v_{k}b_{2,K}\in
H^{2}_{0}(\Omega)\subset V.$

This in the first equation of the continuous problem (2.3a), and
$a^{K}(\Pi^{\nabla^{2\\!}}_{k}u_{h},v)=(\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h},v)_{K}$
and $(\Pi_{\ell-1}\nabla p_{h},\nabla v)_{K}=-(\nabla\cdot\Pi_{\ell-1}\nabla
p_{h},v)_{K}$ from an integration by parts lead to

$\displaystyle(u-\Pi_{k}u_{h},v)_{K}+a^{K}(u-\Pi^{\nabla^{2\\!}}_{k}u_{h},v)-\alpha(\nabla
p-\Pi_{\ell-1}\nabla p_{h},\nabla v)_{K}$
$\displaystyle\quad=(\tilde{f}-\Pi_{k}\tilde{f},v)_{K}+(v_{k},v)_{K}.$

Hence $v=v_{k}b_{2,K}$, the inequalities (5.8), and $\alpha\leq 1\leq\gamma$
show that

$\displaystyle h_{K}^{2}\|v_{k}\|_{K}$ $\displaystyle\lesssim
h_{K}^{2}\|\tilde{f}-\Pi_{k}\tilde{f}\|_{K}+h_{K}^{2}\|u-\Pi_{k}u_{h}\|_{K}+|u-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,K}$
$\displaystyle\quad+\gamma^{1/2}h_{K}\|\nabla p-\Pi_{\ell-1}\nabla
p_{h}\|_{K}$
$\displaystyle\lesssim\|u-u_{h}\|_{K}+|u-u_{h}|_{2,K}+\gamma^{1/2}|p-p_{h}|_{1,K}+\eta_{6,K}+\mathrm{osc}_{2}(\tilde{f},K)$

with triangle inequalities and (3.8b)-(3.8d) in the last estimate. This and
the triangle inequality
$\eta_{1,K}\leq\mathrm{osc}_{2}(\tilde{f},K)+h_{K}^{2}\|v_{k}\|_{K}$ conclude
the proof of (6.14a). The bubble-function $b_{1,K}\in H^{1}_{0}(K)$ supported
in $K$ is constructed as in [15] and it satisfies, for any
$\chi\in\mathbb{P}_{\ell}(K)$, that

(6.15) $\displaystyle\|\chi\|_{K}\lesssim$
$\displaystyle\sum_{m=0}^{1}h_{K}^{m}|b_{1,K}\chi|_{m,K}\lesssim\|\chi\|_{K}.$

Let $q_{\ell}:=\Pi_{\ell}\tilde{g}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}-\beta\Pi_{\ell}p_{h}+\alpha\nabla\cdot\Pi_{k-1}\nabla u_{h}$ and
$q:=q_{\ell}b_{1,K}.$ This in the second equation of the continuous problem
(2.3b), and $(\nabla q,\Pi_{k-1}\nabla
u_{h})_{K}=-(q,\nabla\cdot\Pi_{k-1}\nabla u_{h})_{K}$ and $(\Pi_{\ell-1}\nabla
p_{h},\nabla q)_{K}=-(\nabla\cdot\Pi_{\ell-1}\nabla p_{h},q)_{K}$ show

$\displaystyle\beta(p-\Pi_{\ell}p_{h},q)_{K}+\alpha(q,\nabla u-\Pi_{k-1}\nabla
u_{h})_{K}+\gamma(\nabla p-\Pi_{\ell-1}\nabla p_{h},\nabla q)_{K}$
$\displaystyle=(\tilde{g}-\Pi_{\ell}\tilde{g},q)_{K}+(q_{\ell},q)_{K}.$

Hence (6.15) in the above equation allows us to assert that

$\displaystyle
h_{K}\|q_{\ell}\|_{K}\lesssim|u-u_{h}|_{2,K}+\beta\|p-p_{h}\|_{K}+\gamma|p-p_{h}|_{1,K}+|u_{h}-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,K}$
$\displaystyle\qquad+\gamma|p_{h}-\Pi^{\nabla}_{\ell}p_{h}|_{1,K}+\mathrm{osc}_{1}(\tilde{g},K)$
$\displaystyle\quad\lesssim|u-u_{h}|_{2,K}+\beta\|p-p_{h}\|_{K}+\gamma|p-p_{h}|_{1,K}+(\beta^{1/2}+\gamma^{1/2})\eta_{6,K}+\mathrm{osc}_{1}(\tilde{g},K).$

This concludes the proof of (6.14b). It follows from [21] that
$v:=\phi_{e}[\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})]$
satisfies the first inequality

$\displaystyle\|[\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})]\|_{e}^{2}$
$\displaystyle\lesssim([\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})],\partial_{\boldsymbol{n}}v)_{e}=a^{\omega_{e}}(\Pi^{\nabla^{2\\!}}_{k}u_{h},v)-(\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h},v)_{\omega_{e}}$
$\displaystyle=a^{\omega_{e}}(\Pi^{\nabla^{2\\!}}_{k}u_{h}-u,v)+(\tilde{f}-\Pi_{k}\tilde{f},v)_{\omega_{e}}+(v_{k},v)_{\omega_{e}}+(\Pi_{k}u_{h}-u,v)_{\omega_{e}}$
(6.16) $\displaystyle\quad+\alpha(\nabla p-\Pi_{\ell-1}\nabla p_{h},\nabla
v)_{\omega_{e}}$

with the second equality from an integration by parts and the last equality
from (2.3a). The Cauchy–Schwarz inequality in (6.16) and the inverse estimate
result in

$\displaystyle\|[\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})]\|_{e}^{2}$
$\displaystyle\lesssim\sum_{K^{\prime}\in\omega_{e}}\Big{(}h_{K^{\prime}}^{-2}(|u-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,K^{\prime}}+\eta_{1,K})+\|u-\Pi_{k}u_{h}\|_{K^{\prime}}$
$\displaystyle\quad+\alpha h_{K^{\prime}}^{-1/2}\|\nabla p-\Pi_{\ell-1}\nabla
p_{h}\|_{K^{\prime}}\Big{)}\|v\|_{K^{\prime}}.$

Refer to [21] for the estimate $\|v\|_{\omega_{e}}\lesssim
h_{e}^{3/2}\|[\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})]\|_{e}$.
This and (6.14a) conclude the proof of (6.14c). Since
$[T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}]$ is a polynomial along an edge $e$, the analogous
arguments as in the bound of $\eta_{3,K}$ for
$w:=\psi_{e}[T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}]$ lead to

$\displaystyle([T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}],w)_{e}=(u-\Pi_{k}u_{h},w)_{\omega_{e}}-\alpha(\nabla
p-\Pi_{\ell-1}\nabla p_{h},\nabla w)_{\omega_{e}}$
$\displaystyle-(\tilde{f}-\Pi_{k}u_{h}-\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h}-\alpha\nabla\cdot\Pi_{\ell-1}\nabla
p_{h},w)_{\omega_{e}}-a^{\omega_{e}}(\Pi^{\nabla^{2\\!}}_{k}u_{h}-u,w)$
$\displaystyle+([\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})],\partial_{\boldsymbol{n}}w)_{e}.$

The Cauchy–Schwarz inequality, the inverse estimates
$\sum_{m=0}^{2}h_{K}^{m-2}|w|_{m,K}\lesssim\|w\|_{K}\lesssim
h_{e}^{-3/2}\|[T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}]\|_{e}$, and (6.14a)-(6.14c) conclude the proof of
(6.14d). Let $b_{e}\in H^{1}_{0}(\omega_{e})$ be the edge-bubble function
constructed as in [13, Lemma 9] with the estimates

(6.17)
$\displaystyle\|\chi\|^{2}_{e}\lesssim(b_{e},\chi^{2})_{e}\lesssim\|\chi\|^{2}_{e}\quad\text{and}\quad\sum_{m=0}^{1}h_{K}^{m-1/2}\|b_{e}\chi\|_{m,K}\lesssim\|\chi\|_{e}$

for $\chi\in\mathbb{P}_{\ell}(e)$ with the constant elongation of $\chi$ in
the normal direction of $e\in\mathcal{E}_{K}$. The test function
$q=b_{e}[\alpha\nabla\cdot\Pi_{k-1}\nabla
u_{h}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla p_{h}]$ in (2.3b) and an integration
by parts show

$\displaystyle\beta(p-\Pi_{\ell}p_{h},q)_{\omega_{e}}+\alpha(\nabla q,\nabla
u-\Pi_{k-1}\nabla u_{h})_{\omega_{e}}+\gamma(\nabla p-\Pi_{\ell-1}\nabla
p_{h},\nabla q)_{\omega_{e}}$
$\displaystyle=(\tilde{g}-\Pi_{\ell}\tilde{g},q)_{\omega_{e}}+(\Pi_{\ell}\tilde{g}-\Pi_{\ell}p_{h}+\alpha\nabla\cdot\Pi_{k-1}\nabla
u_{h}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla p_{h},q)_{\omega_{e}}$
$\displaystyle\quad-([\alpha\nabla\cdot\Pi_{k-1}\nabla
u_{h}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla p_{h}],q)_{e}.$

The Cauchy–Schwarz inequality, $\chi=[\alpha\nabla\cdot\Pi_{k-1}\nabla
u_{h}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla p_{h}]$ in (6.17), and the estimate
(6.14b) for $\eta_{2,K}$ conclude the proof of (6.14e). Then we can invoke the
equivalence
$|u_{h}-\Pi^{\nabla}_{\ell}u_{h}|_{1,K}\approx\|\text{Dof}^{k}_{K}(u_{h}-\Pi^{\nabla}_{\ell}u_{h})\|_{\ell^{2}}$
again as in the reliability, and then the definition of
$\Pi^{\nabla^{2\\!}}_{\ell}$ and the triangle inequality allow us to prove
that

$\displaystyle\eta_{7,K}\lesssim|u_{h}-\Pi^{\nabla}_{\ell}u|_{1,K}\leq|u-u_{h}|_{1,K}+|u-\Pi^{\nabla}_{\ell}u|_{1,K}.$

The estimate (6.14g) immediately follows from the arguments involved in the
proof of Theorem 5.1(d). ∎

######  Remark 6.10 (Higher degrees $k\geq 3$ and $\ell\leq k\leq\ell+2$).

If we introduce $J\vec{\boldsymbol{u}_{h}}=(J_{2}u_{h},J_{4}p_{h})$ in the
proof of Theorem 6.8 for $k\geq 3$, then the proof follows analogously with
only difference in the estimate (6.10) of the term $T_{2}$. There the
arguments utilise the $H^{1}$-orthogonality
$\nabla_{\mathrm{pw}}(v_{h}-J_{2}^{*}v_{h})\perp(\mathbb{P}_{\ell-1}(\mathcal{T}_{h}))^{2}$
and hence the same estimator works if one can construct $J_{2}^{*}$ for $k\geq
3$ with this orthogonality (which is possible but not trivial). Still for
higher $k$, we can invoke the $H^{1}$-orthogonality Theorem 5.2(b) and this
leads to

$\displaystyle\alpha(\Pi_{\ell-1}\nabla p_{h},\nabla
J_{2}e_{I}^{u}-\Pi_{k-1}\nabla e_{I}^{u})_{K}=\alpha(\Pi_{\ell-1}\nabla
p_{h}-\Pi_{k-3}\nabla p_{h},\nabla J_{2}e_{I}^{u}-\Pi_{k-1}\nabla
e_{I}^{u})_{K}$
$\displaystyle\qquad\lesssim\alpha(|p_{h}-\Pi^{\nabla}_{\ell}p_{h}|_{1,K}+|p_{h}-\Pi^{\nabla}_{k-2}p_{h}|_{1,K})h_{K}|e_{I}^{u}|_{1,K}$

with the Cauchy–Schwarz inequality, the triangle inequality, Theorem 5.2(e),
and Proposition 3.1 in the last estimate. Consequently, we assume that
$k-2\leq\ell$ and obtain an additional contribution, say $\eta_{9,K}$, in the
error estimator. The equivalence of norms show

$\eta_{9,K}^{2}:=\alpha
h_{K}^{2}\|\text{Dof}^{\,k}_{K}(p_{h}-\Pi^{\nabla}_{k-2}p_{h})\|_{\ell^{2}}^{2},$

and also the efficiency

$\eta_{9,K}\lesssim h_{K}|p_{h}-\Pi^{\nabla}_{k-2}p_{h}|_{1,K}\leq
h_{K}|p_{h}-\Pi^{\nabla}_{k-2}p|_{1,K}\leq|p-p_{h}|_{1,K}+h_{K}|p-\Pi^{\nabla}_{k-2}p|_{1,K}.$

######  Remark 6.11 (Conforming VEM).

As companion operators are not required in the conforming case, the proof of
reliability and efficiency follows analogously assuming $J=I$, where $I$
denotes the identity operator. Note that the local contributions $\eta_{8,K}$
and $\eta_{9,K}$ arise due to the noncoformity of the method and hence the
error estimator in the conforming case is

$\|u-u_{h}^{c}\|_{\mathbf{H}_{\epsilon}}^{2}\lesssim\sum_{i=1}^{7}\eta_{i}^{2}.$

######  Remark 6.12 (Choices of projection operators).

Note that the projection $\Pi_{k-2}\nabla v_{h}$ for $v_{h}\in
V_{h}^{k,\mathrm{c}}$ (or $V_{h}^{k,\mathrm{nc}}$) is also computable in terms
of the DoFs, and both a priori and a posteriori error analysis hold with this
choice. We prefer to use $\Pi_{k-2}\nabla v_{h}$ instead of $\Pi_{k-1}\nabla
v_{h}$ in the numerical experiments below. Also from the theoretical analysis,
observe that if we set $\ell=k$ (one degree higher for pressure) and modify
the term $(\nabla p,\nabla u)_{K}\approx(\Pi_{k-1}\nabla p,\Pi_{k-1}\nabla
u)_{K}$ for all $K\in\mathcal{T}_{h}$ in the discrete approximation, then the
error estimator component $\eta_{7}$ will disappear. But higher approximation
of pressure may not be a good choice from a numerical perspective.

##  7\. Numerical results

We now present a number of computational tests that confirm the theoretical a
priori and a posteriori error estimates from Sections 4-5 and Section 6, and
we also include typical benchmark solutions for Kirchhoff–Love plates that we
modify to include the coupling with filtration in porous media. All meshes
were generated with the library PolyMesher [41].

###  7.1. Example 1: Accuracy verification with smooth solutions

In order to investigate numerically the error decay predicted by Theorems 4.1
and 5.5, we follow the approach of manufactured solutions. We set the
parameters $\alpha=\beta=\gamma=1$ in all the examples below.

We construct a transverse load and a source function $f,g$, respectively, as
well as homogeneous and non-homogeneous boundary data for $u$ and $p$, such
that the problem has the following smooth deflection and fluid pressure moment
exact solutions

$u(x,y)=\sin^{2}(\pi x)\sin^{2}(\pi y),\quad p(x,y)=\cos(\pi xy),$

on the square domain $\Omega=(0,1)^{2}$ with mixed boundary conditions
$\Gamma_{c}:=\\{x=0\\}\cup\\{y=0\\}$ and
$\Gamma_{s}:=\partial\Omega\setminus\Gamma_{c}$. Then we employ a sequence of
successively refined meshes $\mathcal{T}_{h}^{i}$ and compute the projected
virtual element solution
$(\Pi^{\nabla^{2\\!}}_{k}u_{h},\Pi^{\nabla}_{\ell}p_{h})$ on each mesh
refinement $h_{i}$, and monitor the norms
$|u-\Pi^{\nabla^{2\\!}}_{k}u_{h}|_{2,h}$ for displacement approximation,
$|p-\Pi^{\nabla}_{\ell}p_{h}|_{1,h}$ for pressure approximation and the
combined energy norm $\|\cdot\|_{\mathbf{H}_{\epsilon}^{h}}$. The experimental
order of convergence $\texttt{r}_{i}$ is computed from the formula

$\texttt{r}_{i}=\frac{\log(\frac{\texttt{e}_{i+1}}{\texttt{e}_{i}})}{\log(\frac{h_{i+1}}{h_{i}})},$

where $\texttt{e}_{i}$ denotes a norm of the error on the mesh
$\mathcal{T}_{h}^{i}$.

We impose the appropriate boundary conditions for both clamped and simply
supported boundaries. In case of the conforming VEM, note that the degrees of
freedom include the gradient values at vertices and
$u(z)=0=\partial_{\boldsymbol{n}\boldsymbol{n}}u(z)=0$ implies $\nabla
u(z)=\boldsymbol{0}$ for a corner $z$ along the boundary $\Gamma^{s}$. Hence,
we have to impose the zero gradient values at the corners on simply supported
part in addition to the clamped part.

Figure 3. Approximation $u_{h}$ of displacement $u$ for $k=2$ (left) and
$p_{h}$ of pressure $p$ for $\ell=1$ (right) on a smooth Voronoi mesh of 400
elements.

We take $\ell=k-1$ in numerical experiments to obtain the expected optimal
convergence rates for both conforming and nonconforming VEM. See Table 3
(resp. 4) for $k=2$ and $\ell=1$ (resp. for $k=3$ and $\ell=2$). Tables 3-4
display the errors and the convergence rates on a sequence of Voronoi meshes.

$h$ | $|u-\Pi_{2}^{\nabla^{2}}u_{h}|_{2,h}$ | $\texttt{r}_{i}$ | $|p-\Pi_{1}^{\nabla}p_{h}|_{1,h}$ | $\texttt{r}_{i}$ | $|\vec{\boldsymbol{u}}-\vec{\Pi}\vec{\boldsymbol{u}}_{h}|_{\mathbf{H}_{\epsilon}^{h}}$ |  $\texttt{r}_{i}$$\displaystyle\int$
---|---|---|---|---|---|---
Conforming VE discretisation
0.6801 | 13.530 | 0.575 | 0.8743 | 0.106 | 14.848 | 0.551
0.3124 | 8.6492 | 1.213 | 0.8051 | 1.247 | 9.6714 | 1.212
0.1558 | 3.7205 | 0.988 | 0.3382 | 1.450 | 4.1621 | 1.036
0.0795 | 1.9133 | 0.908 | 0.1275 | 1.328 | 2.0729 | 0.941
0.0390 | 1.0023 | 1.148 | 0.0495 | 1.126 | 1.0608 | 1.151
0.0193 | 0.4483 | * | 0.0225 | * | 0.4734 | *
Nonconforming VE discretisation
0.7147 | 24.738 | 0.966 | 5.1202 | 0.855 | 33.475 | 1.001
0.3625 | 12.837 | 1.197 | 2.8642 | 1.360 | 16.960 | 1.223
0.1862 | 5.7833 | 0.943 | 1.1574 | 1.798 | 7.5073 | 1.092
0.0938 | 3.0282 | 1.181 | 0.3372 | 1.825 | 3.5485 | 1.261
0.0476 | 1.3600 | 1.207 | 0.0978 | 1.632 | 1.5092 | 1.252
0.0244 | 0.6077 | * | 0.0329 | * | 0.6544 | *
Table 3. Error $\vec{\boldsymbol{u}}-\vec{\Pi}\vec{\boldsymbol{u}}_{h}=(u-\Pi_{2}^{\nabla^{2}}u_{h},p-\Pi^{\nabla}_{1}p_{h})$ in the energy norm $\|\bullet\|_{\mathbf{H}_{\epsilon}^{h}}$ with $k=2$ and $\ell=1$ on a sequence of smooth Voronoi meshes of 5,25,100,400,1600, and 6400 elements. $h$ | $|u-\Pi_{3}^{\nabla^{2}}u_{h}|_{2,h}$ | $\texttt{r}_{i}$ | $|p-\Pi_{2}^{\nabla}p_{h}|_{1,h}$ | $\texttt{r}_{i}$ | $|\vec{\boldsymbol{u}}-\vec{\Pi}\vec{\boldsymbol{u}}_{h}|_{\mathbf{H}_{\epsilon}^{h}}$ |  $\texttt{r}_{i}$$\displaystyle\int$
---|---|---|---|---|---|---
Conforming VE discretisation
0.6801 | 8.2153 | 1.986 | 0.4883 | 1.712 | 8.9142 | 1.984
0.3096 | 1.7206 | 2.058 | 0.1269 | 2.861 | 1.8713 | 2.114
0.1508 | 0.3916 | 2.219 | 0.0162 | 2.313 | 0.4091 | 2.226
0.0794 | 0.0942 | 1.979 | 0.0036 | 2.069 | 0.0980 | 1.984
0.0393 | 0.0234 | 2.109 | 0.0008 | 2.166 | 0.0243 | 2.111
0.0203 | 0.0058 | * | 0.0002 | * | 0.0060 | *
Nonconforming VE discretisation
0.7147 | 10.748 | 0.7089 | 3.6144 | -0.3735 | 15.115 | 0.3286
0.3406 | 6.3548 | 1.5061 | 4.7674 | 1.9128 | 11.847 | 1.7228
0.1875 | 2.5874 | 2.0609 | 1.5229 | 2.5617 | 4.2385 | 2.2546
0.0969 | 0.6639 | 1.9931 | 0.2808 | 2.8056 | 0.9570 | 2.2010
0.0484 | 0.1663 | 2.2325 | 0.0340 | 2.6055 | 0.2075 | 2.3010
0.0239 | 0.0345 | 2.0891 | 0.0064 | 2.2739 | 0.0410 | 2.1178
0.0123 | 0.0086 | * | 0.0014 | * | 0.0101 | *
Table 4. Error
$\vec{\boldsymbol{u}}-\vec{\Pi}\vec{\boldsymbol{u}}_{h}=(u-\Pi_{3}^{\nabla^{2}}u_{h},p-\Pi^{\nabla}_{2}p_{h})$
in the energy norm $\|\bullet\|_{\mathbf{H}_{\epsilon}^{h}}$ with $k=3$ and
$\ell=2$ on a sequence of smooth Voronoi meshes of 5,25,100,400,1600, and 6400
elements.

Figures 4-5 display the error and the error estimator convergence rates for
both uniform and adaptive refinements. In this example, we choose a smooth
Voronoi mesh of 25 elements as an initial partition and follow the standard
adaptive algorithm

$\text{SOLVE}\longrightarrow\text{ESTIMATE}\longrightarrow\text{MARK}\longrightarrow\text{REFINE}$

In all the adaptive experiments below, we first solve the discrete problem
(3.11) (resp. (3.12)) for conforming (resp. nonconforming), compute the upper
bound $\eta$ in Theorem 6.8, consider the Dörfler marking strategy with
$\theta=0.5$, and divide a marked polygon into quadrilaterals by connecting
vertices to the centroid of the respective polygon. The same refinement
strategy is utilised to divide all the elements in case of uniform refinement.
The additional error estimator component $\eta_{9}$ from Remark 6.10 is
incorporated in the experiment of the nonconforming VEM with degree $k=3$ and
$\ell=2$.

Figure 4. Left (resp. right) panel displays NDof vs error in energy norm
(resp. error estimator) in both uniform and adaptive refinements for
conforming VEM.

Figure 5. Left (resp. right) panel displays NDof vs error in energy norm
(resp. error estimator) in both uniform and adaptive refinements for
nonconforming VEM.

###  7.2. Example 2: Convergence rates with non-smooth solutions

We consider the L-shaped domain
$\Omega=(-1,1)^{2}\setminus([0,1)\times[-1,0))$ and the exact solution

$u(r,\theta)=r^{5/3}\sin\Big{(}\frac{5\theta}{3}\Big{)},\quad
p(r,\theta)=r^{2/3}\sin\Big{(}\frac{2\theta}{3}\Big{)}$

with clamped boundary conditions for $u$ and Dirichlet boundary condition for
$p$ on $\partial\Omega$ (observe that we can take Dirichlet boundary condition
instead of Neumann for $p$ on $\Gamma^{c}$ without affecting the well-
posedness and error analysis of the model problem). Since both the
displacement $u\in H^{(8/3)-\epsilon}(\Omega)$ and the pressure $p\in
H^{(5/3)-\epsilon}(\Omega)$ for all $\epsilon>0$ have corner singularities,
the lowest-order scheme $k=2$ and $\ell=1$ suffices to achieve the optimal
convergence rates with respect to the regularity of $u$ and $p$.

When the adaptive algorithm is run, we see more refinement around the singular
corner as displayed in Figures 6-7. Figure 8 shows that the method with unifom
refinement leads to suboptimal rates whereas adaptive refinement recovers the
optimal convergence rates, and the error estimator mirrors the behaviour of
the actual error. We observe from the plots of error estimator components that
$\eta_{7}$ (resp. $\eta_{8}$) dominates the remaining contributions for the
case of conforming (resp. nonconforming) VEM.

Figure 6. Approximation $u_{h}$ of displacement $u$ on adaptive meshes
$\mathcal{T}_{1},\mathcal{T}_{5},\mathcal{T}_{10}$.

Figure 7. Approximation $p_{h}$ of pressure $p$ on adaptive meshes
$\mathcal{T}_{1},\mathcal{T}_{5},\mathcal{T}_{10}$.

Figure 8. Left panel displays NDof vs error in energy norm and estimator in
both uniform and adaptive refinements, and right panel displays estimator
components in adaptive refinement for conforming VEM.

Figure 9. Left panel displays NDof vs error in energy norm and estimator in
both uniform and adaptive refinements, and right panel displays estimator
components in adaptive refinement for nonconforming VEM.

### Conclusions

This paper analyses conforming and nonconforming VEMs for the poroelastic
plate model (2.2) referred from [31]. The well-posedness of both continuous
and discrete problems follow straightforwardly. Theorem 4.1 provides the a
priori error estimate in the best-approximation form for the conforming case.
In the nonconforming case, we first developed a key tool, the so-called
companion operator, which maps from the nonconforming VE space to the
continuous space ($V$ for displacement space and $Q$ for pressure space). We
stress that the construction of the companion operator in this paper is novel
and substantially different from those already present in the literature [30,
15, 16], in the sense that it is designed for general degree VE spaces
satisfying orthogonality and best-approximation properties. Furthermore, this
operator is an independent vital technical argument that can be useful in
other nonconforming VE methods for different second and fourth-order elliptic
problems. The VE functions locally need not be polynomials and this paper
contributes to the non-standard proofs of basic estimates such as Poincaré-
type inequalities, inverse estimates, and norm-equivalences. This paper also
presents a residual-based reliable and efficient a posteriori error estimator,
which is robust with respect to the model parameters. To support the
robustness of the error analysis with respect to the model parameters, we also
perform uniform and adaptive numerical tests for moderate and extreme values
of model parameters $\alpha=10^{-6},\beta=10^{6}=\gamma$ and observe from
Figure 10 that although the magnitude of errors and error estimators become
large for large values of $\beta$ and $\gamma$, the convergence rates remain
optimal as predicted by the theory.

We emphasise here that both conforming and nonconforming VEM have great
advantages over standard FEM from the perspective that the $C^{1}$-conforming
FEM requires higher dofs compared to conforming VEM and nonconforming FEM for
higher degrees can not be easily put in one framework as we can do in VEM.
However since the model problem is a fourth-order PDE, we require $C^{1}$
continuity in the conforming case, and consequently the nonconforming VE can
be implemented with less computational effort compared to its conforming
counterpart.

Recall that the model problem (2.2) analysed in this paper is a scaled version
of the original model problem (2.1). Certainly, one can start with (2.1)
instead of the normalised version, but the coercivity of $\mathcal{A}$
(uniformly with respect to the parameters) is not straightforward and one may
have to invoke a different abstract result (for example a global inf-sup
condition or similar arguments) to prove the well-posedness of the problem.
Note that our analysis (well-posedness as well as the robustness with respect
to model parameters) majorly depends on having the same coefficient $\alpha$
for the coupled terms and the relation $\alpha\leq 1\leq\gamma$. It will be
interesting to track all model parameters, but the formulation is
substantially modified leading to a completely different analysis and we keep
it open for a future work. Also the limiting case $d\to 0$ indeed requires a
change in the model to consider nonlinear effects, and altogether implying a
different physical phenomenon to be regarded and studied separately. Finally,
we mention that as an important extension of this work we are developing new
mixed formulations for the coupling of Biot–Kirchhoff plates interacting with
a bulk free flow.

Figure 10. Left (resp. right) panel displays NDof vs error in energy norm and
estimator in both uniform and adaptive refinements and both conforming and for
nonconforming VEM in Example 1 (resp. Example 2). The superscript $c$ (resp.
$nc$) denotes the respect term in the conforming (resp. nonconforming) case.
The triangle in left (resp. right) panel represents slope $0.5$ (resp. $0.33$
and $0.5$).

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