In order to establish an error estimate, we introduce the following properties. There is an important inequality in the finite element spaces \(\mathcal {P}^{h}( {\varOmega})\times\mathcal{Q}^{h}( {\varOmega}) \), which allows the \(H^{1}\)-norm to be bounded above by the \(L^{2}\)-norm. Such an inequality is called an inverse inequality. Let us introduce the broken Sobolev space of \(\mathcal{T}_{h}\) of the domain Ω,
$$H^{s}( \mathcal{T}_{h}):= \bigl\{ p\in L^{2}( { \varOmega}): p|_{K} \in H^{s}(K)\ \forall K\in \mathcal{T}_{h} \bigr\} $$
with the broken Sobolev norm and seminorm, respectively,
$$\Vert p \Vert _{H^{s}( \mathcal{T}_{h})}:= \biggl(\sum_{K\in \mathcal{T}_{h}} \Vert p \Vert _{H^{s}(K) } \biggr)^{\frac{1}{2}}, \qquad \vert p \vert _{H^{s}( \mathcal{T}_{h})}:= \biggl(\sum_{K\in\mathcal {T}_{h}} \vert p \vert_{H^{s}(K) } \biggr)^{\frac{1}{2}}. $$
Then, the following result can be proved.
Lemma 3.1
(Trace theorem)
Let
\(p \in\mathcal{P}^{h}( {\varOmega})\)
with shape-regularity mesh. Then there exists a constant
\(C_{\mathrm{inv}}>0\)
such that
$$\begin{aligned} \Vert p \Vert _{L^{2}(\partial K)} \leq C_{\mathrm{inv}} \bigl( \Vert p \Vert _{L^{2}(K)} \bigl( h_{K}^{-1} \Vert p \Vert _{L^{2}(K)} + \Vert \nabla p \Vert _{L^{2}(K)} \bigr) \bigr)^{\frac{1}{2}}. \end{aligned}$$
Proof
See Lemma A.3 in [26] for the proof and further details. □
We now consider the following semi-discrete DG approximation for the spatial discretization of (2.1): Find \((p^{h},{\vec{q}}^{ h} ): \bar{I} \times\bar{I} \rightarrow\mathcal{P}^{h} ( {\varOmega })\times \mathcal{Q}^{h}( {\varOmega}) \) such that
$$ \begin{aligned} & \biggl(\frac{1}{c^{2}} p^{h}_{t},r^{h} \biggr)+a_{h} \bigl(p^{h},r^{h} \bigr)-{b}_{h}' \bigl({\vec{q}}^{ h},r^{h} \bigr) = 0 \quad\forall r^{h} \in\mathcal{P}^{h}( {\varOmega}), t\in I, \\ & \bigl(\vec{q}^{ h}_{t},\vec{v}^{ h} \bigr)+b_{h} \bigl(p^{h},\vec{v}^{ h} \bigr)+c_{h} \bigl(\vec{q}^{ h}, \vec{v}^{ h} \bigr) = 0 \quad \forall{\vec{v}}^{ h} \in\mathcal{Q}^{h}( { \varOmega}), t \in I, \end{aligned} $$
(3.1)
with
$$p^{h}(\cdot, 0)={\varPi}_{h} p_{0},\qquad \vec{q}^{ h}(\cdot,0)=\boldsymbol {\varPi}_{h}\vec{q}_{0},\qquad p^{h}(x,\cdot)=0\quad \forall x \in\partial\varOmega. $$
Here \(\varPi_{h}\) and \(\boldsymbol {\varPi}_{h}\) denote the \(L^{2}\)-projections of p and q⃗ in \(L^{2}(\varOmega)\) and \(\mathbb{L}^{2}(\varOmega)\) onto \(\mathcal {P}^{h} (\varOmega)\) and \(\mathcal{Q}^{h}(\varOmega)\), respectively, that is, for any \(p \in L^{2}(\varOmega), {\vec{q}} \in\mathbb{L}^{2}(\varOmega)\)
$$\begin{aligned} \bigl(\varPi_{h} p, r^{h} \bigr) = \bigl(p,r^{h} \bigr) \quad\text{and}\quad \bigl(\boldsymbol {\varPi}_{h} {\vec{q}} , {\vec{v}}^{ h} \bigr)= \bigl( {\vec{q}}, {\vec{v}}^{ h} \bigr) \quad\forall r^{h}\in\mathcal{P}^{h}( {\varOmega}), { \vec{v}}^{ h} \in\mathcal{Q}^{h}( {\varOmega}), \end{aligned}$$
(3.2)
and the discrete forms \(a_{h},b_{h}\), and \(c_{h}\) are given by (2.8). For the simplicity of notations, we let \(\frac{1}{c^{2}}\mathcal{R}_{p}+ \mathcal{A}_{h}: \mathcal{P}^{h}( {\varOmega}) \rightarrow[\mathcal {P}^{h}(\varOmega)]{'}\), \(\mathcal{B}_{h}: \mathcal{P}^{h}( {\varOmega}) \rightarrow[\mathcal{Q}^{h}( {\varOmega})]{'}\), and \(\mathcal{R}_{\vec{q}}+ \mathcal{C}_{h}:\mathcal{Q}^{h}( {\varOmega}) \rightarrow[\mathcal{Q}^{h}( {\varOmega})]{'} \) given by
$$\begin{aligned} &\frac{1}{c^{2}}\mathcal{R}_{p}p^{h} \bigl(r^{h} \bigr)= \biggl(\frac{1}{c^{2}}p^{h} ,r^{h} \biggr),\qquad \mathcal{R}_{\vec{q}}{\vec{q}}^{ h} \bigl({\vec{v}}^{h} \bigr) = \bigl({\vec{q}}^{ h}, { \vec{v}}^{ h} \bigr), \\ & \mathcal{A}_{h} p^{h} \bigl(r^{h} \bigr) = a_{h} \bigl(p^{h},r^{h} \bigr),\qquad \mathcal{B}_{h} p^{h} \bigl({\vec{v}}^{ h} \bigr) = b_{h} \bigl(p^{h}, {\vec{v}}^{ h} \bigr),\qquad \mathcal{C}_{h} { \vec{q}}^{ h} \bigl({\vec{v}}^{ h} \bigr) = c_{h} \bigl({\vec{q}}^{ h}, {\vec{v}}^{ h} \bigr). \end{aligned}$$
Then it can be seen that the dual operator of \(\mathcal{B}_{h}\), \(\mathcal {B}'_{h}:\mathcal{Q}^{h} (\varOmega) \rightarrow[\mathcal {P}^{h}(\varOmega )]{'} \) satisfies
$$\begin{aligned} \mathcal{B}'_{h} {\vec{q}}^{ h} \bigl( r^{h} \bigr) = \mathcal{B}_{h} r^{h} \bigl({\vec {q}}^{ h} \bigr) = b_{h}' \bigl( { \vec{q}}^{ h}, r^{h} \bigr), \end{aligned}$$
which follows from the trace identity (2.5).
Lemma 3.2
There is a unique semi-discrete solution
\((p^{h},{\vec {q}}^{ h} )\)
of (3.1) satisfying
$$\bigl(p^{h},{\vec{q}}^{ h} \bigr) \in C^{1} \bigl(I; \mathcal{P}^{h}( {\varOmega}) \times\mathcal{Q}^{h}( { \varOmega}) \bigr). $$
Proof
Theorem 2.1 is used for the proof. We use the operator notations of (3.1) to obtain
where
Then we show that \(L_{h}\) is monotone by the definition and the trace identity (2.5),
To obtain \(\operatorname{Rg}(\mathcal{M}_{h}+{L}_{h})= [\mathcal{P}^{h} ( {\varOmega})\times \mathcal {Q}^{h}( {\varOmega}) ] '\), it is sufficient to show that \(\operatorname{Ker} (\mathcal {M}_{h}+{L}_{h})=\{(0, \vec{{0}})\}\). Since
$$\begin{aligned} \mathcal{M}_{h} \bigl( p ^{h}, {\vec{q}} ^{ h} \bigr)^{ T} \bigl( \bigl( p ^{h}, {\vec{q}} ^{ h} \bigr)^{T} \bigr) = \int_{ {\varOmega}} \biggl( \frac{1}{c^{2}} \bigl(p^{h} \bigr) ^{2}+ \bigl({\vec{q}}^{ h} \bigr){^{2}} \biggr)\,dx \geq C \int_{ {\varOmega}} \bigl( \bigl(p^{h} \bigr) ^{2}+ \bigl({\vec{q}}^{ h} \bigr){^{2}} \bigr)\,dx \end{aligned}$$
for some \(C=\min\{ \frac{1}{{c^{*}}^{2}},1 \}\), we can have the surjection, which provides the conclusion. □
To estimate of the difference of the semi-discrete LDG solution \((p^{h} ,{\vec{q}}^{ h})\) in (3.1) with analytical solutions \((p ,{\vec{q}})\) in (2.1), we want to extend to a larger space which contains both solutions. In the next section we show the error estimates.
Extension of LDG form. We define the spaces
$$\mathcal{P}(h):= H^{1}_{0}( {\varOmega}) + \mathcal{P}^{h}( {\varOmega}) \quad\text{and}\quad \mathcal{Q}(h):= {H}_{\mathrm{div}}( {\varOmega}) + \mathcal{Q}^{h}( {\varOmega}) $$
with the DG energy norm on \(\mathcal{P}(h) \times\mathcal{Q}(h)\),
$$\bigl\Vert (p,\vec{q}) \bigr\Vert _{h}^{2}:= \Vert p \Vert ^{2} _{\mathcal {P}(h)}+ \Vert \vec{q} \Vert ^{2}_{\mathcal{Q}(h)}, $$
where
$$\begin{aligned} \Vert p \Vert _{\mathcal{P}(h) }^{2}&:=\sum _{K\in\mathcal{T}_{h}} \Vert p \Vert _{H^{1}(K)}^{2}+\sum _{e\in\mathcal{E}} \bigl\Vert \mathtt{C}_{11} [\hspace{-2pt}[p]\hspace{-2pt}] \bigr\Vert _{L^{2}(e)}^{2}, \Vert {\vec{q}} \Vert _{\mathcal{Q}(h) }^{2}\\ &:=\sum_{K\in \mathcal{T}_{h}} \Vert {\vec{q}} \Vert _{{H}_{\mathrm{div}}(K) } ^{2}+\sum _{e\in\mathcal{E}} \bigl\Vert \mathtt{C}_{22} [\hspace{-2pt}[{ \vec{q}}]\hspace{-2pt}] \bigr\Vert _{L^{2}(e)}^{2}, \end{aligned}$$
and the norm \(\|\vec{ {q}}\|_{{H}_{\mathrm{div}}(K)} ^{2}:=\|\vec{q}\|_{\mathbb{L}^{2} (K) } ^{2}+\|\nabla\cdot{\vec{q}}\|_{{L}^{2}(K) } ^{2}\). For the convenience of notation, let us denote
$$\Vert \cdot \Vert _{0,\mathcal{E}}:=\sum_{e\in\mathcal{E}} \Vert \cdot \Vert _{L^{2}(e)},\qquad \Vert \cdot \Vert _{s,K}:= \Vert \cdot \Vert _{H^{s}(K)},\qquad \Vert \cdot \Vert _{s,\varOmega}:= \Vert \cdot \Vert _{H^{s}(\varOmega)}, $$
and the same as \(\mathbb{H}^{s}(K)\) and \(\mathbb{H}^{s}(\varOmega) \), respectively. Furthermore, for \(1\leq p\leq\infty\) we use the Bochner space \(L^{p}(I;\mathcal{P}(h)\times\mathcal{Q}(h)) \)
$$\begin{aligned} \bigl\Vert ( p, \vec{{q}} ) \bigr\Vert _{ L^{p}(I;\mathcal{P}(h) \times \mathcal{Q}(h)) }:= \textstyle\begin{cases} (\int_{I} \Vert p \Vert ^{p} _{ \mathcal{P}(h)}+ \Vert {\vec {q}} \Vert ^{p} _{ \mathcal {Q}(h)} \,dt )^{1/p} & \text{if } 1\leq p< \infty,\\ \operatorname{ess} \sup_{t\in I} ( \Vert p \Vert _{ \mathcal{P}(h)}+ \Vert {\vec{q}} \Vert _{ \mathcal{Q}(h)}) & \text{if } p=\infty, \end{cases}\displaystyle \end{aligned}$$
and we denote
$$\begin{aligned} & \Vert \cdot \Vert _{\mathscr{L}^{p}_{s, {\varOmega}}}:= \Vert \cdot \Vert _{L^{p}(I;{H}^{s}( {\varOmega}))},\qquad \Vert \cdot \Vert _{\mathscr {L}^{\infty}_{s, {\varOmega}}}:= \Vert \cdot \Vert _{L^{\infty }(I;{H}^{s}( {\varOmega}))}, \\ & \Vert \cdot \Vert _{\mathscr{L}^{p}_{s, \mathcal{E}}}:= \Vert \cdot \Vert _{L^{p}(I;{H}^{s}( \mathcal{E}))}, \qquad \Vert \cdot \Vert _{\mathscr {L}^{\infty}_{s, \mathcal {E}}}:= \Vert \cdot \Vert _{L^{\infty}(I;{H}^{s}(\mathcal{E}))}. \end{aligned}$$
The main result of this section is to establish the \(L^{2}( {\varOmega })\)-norm error estimate. It also gives a bound in the \(L^{2}( {\varOmega })\)-norm of the first time derivative.
Theorem 3.3
Let the analytical solution
\((p,{\vec{q}} ) \)
of (2.1) satisfies
$$ ( p, {\vec{q}} ) \in L^{\infty} \bigl( I; H^{1+s} _{0}( {\varOmega})\times\mathbb{H}^{1+s} ( { \varOmega}) \bigr),\qquad ( p_{t}, {\vec{q}}_{t} ) \in L^{1} \bigl( I; H^{s} ( {\varOmega})\times\mathbb {H}^{s} ( {\varOmega}) \bigr) $$
(3.3)
for a regularity exponent
\(s>\frac{1}{2}\), and let
\((p^{h},{\vec{q}}^{ h} ) \)
be the semi-discrete DG approximation obtained by (3.1). Then we have the estimate for the error
\(e^{p}=p-p^{h} \)
and
\(e^{\vec {{q}} }=\vec{ {q}}-\vec{ {q}}^{h}\)
$$\begin{aligned} &\sup_{t\in I} \bigl( \bigl\Vert e^{p} \bigr\Vert _{0,\varOmega} + \bigl\Vert e^{{\vec{q}}} \bigr\Vert _{0,\varOmega} \bigr) +\sup_{t\in I} \bigl( \bigl\Vert \bigl[ \hspace{-2pt}\bigl[ e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal{E}} + \bigl\Vert \bigl[ \hspace{-2pt}\bigl[e^{{\vec{q}}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal{E}} \bigr) \\ &\quad\leq C_{0} \bigl( \bigl\Vert e^{p}(0) \bigr\Vert _{0,\varOmega} + \bigl\Vert e^{{\vec {q}}}(0) \bigr\Vert _{0,\varOmega} \bigr) \\ &\qquad{}+C_{p} h^{\min\{s,k+\frac{1}{2}\}} \bigl( \Vert p \Vert _{\mathscr {L}^{\infty}_{1+s,\varOmega}}+ \Vert p_{t} \Vert _{\mathscr {L}^{1}_{s,\varOmega}} \bigr)+ C_{\vec{q}}h^{\min\{ s,k'+\frac {1}{2} \} } \bigl( \Vert {\vec{q}} \Vert _{\mathscr{L}^{\infty }_{1+s,\varOmega}} + \Vert { \vec{q}}_{t} \Vert _{\mathscr {L}^{1}_{s,\varOmega}} \bigr) \end{aligned}$$
with positive constants
\(C_{0}, C_{p}, C_{\vec{q}}\)
depending on the bounds
\(c_{*}\), \(c^{*}\), and
\(\sigma^{*}\), which are independent of the mesh size
h, where
k
and
\(k'\)
are the order of approximation polynomials
\((p^{h}, \vec{q}^{ h})\), respectively.
Remark 3.4
The condition (3.3) implies that \(( p, {\vec{q}} )\in C ( \bar{I}; H^{s} ( {\varOmega})\times\mathbb{H}^{s} ( {\varOmega}) ) \), thus it is required to have the initial condition \(( p_{0}, {\vec{q}}_{0} ) \in H^{s} ( {\varOmega})\times\mathbb{H}^{s} ( {\varOmega})\) and also
$$\begin{aligned} & \bigl\Vert e^{p}(0) \bigr\Vert _{0,\varOmega} =\bigl\| ( {p}- \varPi_{h} p) (0)\bigr\| _{0,\varOmega} \leq C h^{\min\{s,k +1\}} \Vert p \Vert _{s,\varOmega}, \\ & \bigl\Vert e^{{\vec{q}}}(0) \bigr\Vert _{0,\varOmega} =\bigl\| ({{ \vec{q}}}- \boldsymbol {\varPi}_{h} {{\vec{q}}}) (0)\bigr\| _{0,\varOmega} \leq C h^{\min\{s,k'+1 \}} \Vert {{\vec{q}}} \Vert _{s,\varOmega}. \end{aligned}$$
Therefore, Theorem 3.3 implies
$$\begin{aligned} \sup_{t\in I} \bigl( \bigl\Vert e^{p} \bigr\Vert _{0,\varOmega} + \bigl\Vert e^{{\vec{q}}} \bigr\Vert _{0,\varOmega } \bigr) +\sup_{t\in I} \bigl( \bigl\Vert \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal{E}} + \bigl\Vert \bigl[ \hspace{-2pt}\bigl[e^{{\vec{q}}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal {E}} \bigr) \leq Ch^{\min\{s,k+\frac{1}{2},k'+\frac{1}{2} \}}. \end{aligned}$$
It can be noted that, for smooth solutions, Theorem 3.3 yields the convergence rates in the \(L^{2}\)-norm:
$$\begin{aligned} \sup_{t\in I} \bigl( \bigl\Vert e^{p} \bigr\Vert _{0, {\varOmega}} + \bigl\Vert e^{{\vec{q}}} \bigr\Vert _{0, {\varOmega}} \bigr) \leq Ch^{\min\{k,k'\}+\frac{1}{2} }. \end{aligned}$$
Following [25], we introduce lifting operators in order to extend the numerical flux to the entire space \(\mathcal{P}(h)\times \mathcal{Q}(h)\). We define the lifting operator \(\mathcal{L}_{h}^{+} p \in\mathcal{Q}^{h} ( {\varOmega}) \) for \(p \in\mathcal{P}(h) \) by
$$\begin{aligned} \int_{ {\varOmega}} \mathcal{L}_{h}^{+} p\cdot{ \vec{q}}^{ h} \,dx = \sum_{e\in\mathcal{E} } \int_{e} [\hspace{-2pt}[p ]\hspace{-2pt}] \bigl( \vec{\mathbf {C}}_{12} \bigl[ \hspace{-2pt}\bigl[{ \vec{q}}^{ h} \bigr]\hspace{-2pt}\bigr]+ \bigl\{ \hspace{-2pt}\bigl\{ {\vec{q}}^{ h} \bigr\} \hspace{-2pt}\bigr\} \bigr) \,ds, \quad\forall{ \vec{q}}^{ h} \in\mathcal{Q}^{h}(\varOmega), \end{aligned}$$
(3.4)
and also \(\mathcal{L}^{-}_{h} {\vec{q}} \in\mathcal{P}^{h}( {\varOmega })\) for \({\vec{q}} \in\mathcal{Q}(h) \) by
$$\begin{aligned} \int_{ {\varOmega}} \mathcal{L}_{h}^{-}{\vec{q}} p^{h} \,dx = \sum_{e\in \mathcal {E} } \int_{e} [\hspace{-2pt}[{\vec{q}}]\hspace{-2pt}] \cdot\bigl( \vec{\mathbf {C}}_{12} \bigl[ \hspace{-2pt}\bigl[p^{h} \bigr]\hspace{-2pt}\bigr]- \bigl\{ \hspace{-2pt}\bigl\{ p^{h} \bigr\} \hspace{-2pt}\bigr\} \bigr) \,ds,\quad \forall p^{h} \in \mathcal{P}^{h}( {\varOmega}). \end{aligned}$$
(3.5)
It can be seen that by the definition of the \(L^{2}\)-projection (3.2) we have
$$\begin{aligned} &\int_{\varOmega} \mathcal{L}^{+}_{h} p \cdot\vec{{q} } \,dx = \int_{\varOmega} \mathcal{L}^{+}_{h} p \cdot\boldsymbol { \varPi}_{h} {\vec{q}}\,dx,\\ &\int_{ {\varOmega }} \mathcal{L}^{-}_{h} {\vec{q}} p \,dx = \int_{\varOmega} \mathcal{L}^{-}_{h} {\vec{q}} {{ \varPi}}_{h} p \,dx\quad \forall p \in\mathcal{P}(h), {\vec{q}} \in \mathcal{Q}(h). \end{aligned}$$
We now extend (3.1) using the two lifting functions
$$ \begin{aligned} & \biggl(\frac{1}{c^{2}} p_{t},r \biggr) + \tilde{a}_{h}(p,r) - \tilde{b}'_{h}({\vec {q}},r) = 0\quad \forall r\in\mathcal{P}(h), t\in I, \\ &({\vec{q}}_{t},{\vec{v}}) + \tilde{b}_{h}(p,{\vec{v}}) + \tilde{c}_{h}({\vec{q}},{\vec{v}}) = 0\quad \forall{\vec{q}} \in\mathcal {Q}(h), t\in I, \end{aligned} $$
where the bilinear forms are given by
$$ \begin{aligned} &\tilde{a}_{h}(p,r) = \sum _{K \in\mathcal{T}_{h}} \int_{K} \frac {\sigma _{p}}{c^{2}}pr \,dx + \sum _{e \in\mathcal{E} } \int_{e} \mathtt{C}_{11} [\hspace{-2pt}[p ]\hspace{-2pt}]\cdot [\hspace{-2pt}[r ]\hspace{-2pt}]\,ds,\\ & \tilde{b}_{h}(p,{\vec{v}}) = -\sum _{K \in\mathcal{T}_{h}} \int_{K} p\nabla\cdot\vec{{v} } \,dx - \int_{\varOmega} p \mathcal{L}^{-}_{h} {\vec{v}} \,dx, \\ &\tilde{b}'_{h}(\vec{{q}},r) = \sum _{K \in\mathcal{T}_{h}} \int_{K}\vec{{q} } \cdot\nabla r \,dx - \int_{\varOmega} \vec{{q}} \cdot\mathcal{L}^{+}_{h} r \,dx,\\ &\tilde{c}_{h}(\vec{{q}},\vec{{v}}) = \sum_{K \in\mathcal{T}_{h}} \int_{K}\sigma_{\vec{q}} \vec{{q}}\cdot\vec{{v}} \,dx + \sum_{e \in \mathcal {E}} \int_{e} \mathtt{C}_{22} [\hspace{-2pt}[{q} ]\hspace{-2pt}] [\hspace{-2pt}[{v} ]\hspace{-2pt}]\,ds. \end{aligned} $$
(3.6)
Error equations. To derive the error equations we define for \(r\in\mathcal{P}(h) \), \(\vec{{v}}\in\mathcal{Q}(h) \) and \(p \in{H}^{1}_{0}{( {\varOmega })}\), \(\vec{{q}}\in\mathbb{H}^{1}( {\varOmega}) \)
$$ \begin{aligned} &\mathcal{R}^{p} (p, \vec{{v}} ):= \sum_{e\in\mathcal{E} } \int_{e} [\hspace{-2pt}[\vec{{v}} ]\hspace{-2pt}] \bigl( - \vec{\mathtt {C}}_{12}\cdot[\hspace{-2pt}[ \varPi_{h} p -p ]\hspace{-2pt}]+ \{\hspace{-2pt}\{ \varPi_{h} p -p \}\hspace{-2pt}\} \bigr) \,ds, \\ &\mathcal{R}^{\vec{q}} (\vec{{q}},r):= \sum_{e\in\mathcal{E} } \int_{e} [\hspace{-2pt}[r]\hspace{-2pt}]\cdot\bigl( \vec{\mathtt {C}}_{12} [\hspace{-2pt}[ \boldsymbol { \varPi}_{h} \vec{{q}} - \vec{{q}} ]\hspace{-2pt}]+ \{\hspace{-2pt}\{\boldsymbol {\varPi}_{h} \vec{{q}} -\vec{{q}} \}\hspace{-2pt}\} \bigr) \,ds. \end{aligned} $$
(3.7)
The assumption that \(p \in H^{1}_{0}( {\varOmega}), \vec{{q}} \in\mathbb {H}^{1}( {\varOmega})\) ensures that \(\mathcal{R}^{p}(p,\vec{{v}}), \mathcal {R}^{\vec{q}} ( \vec{{q}},r)\) are well defined since the trace map of \(p, \vec{{q}}\) are uniquely defined on all \(e\in\mathcal{E}\). From the definition (2.4) of the jump it directly follows that \(\mathcal{R}^{p}(p,\vec{{v}})=0, \mathcal{R}^{\vec{q}} ( \vec{{q}} ,r)=0 \) when \(r\in H^{1}_{0}( {\varOmega}), \vec{{v}}\in\mathbb{H}^{1} ( {\varOmega})\). Using the definition of the error equations, we have the following property.
Lemma 3.5
Let the analytical solution
\((p,\vec{{q}})\)
of (2.1) satisfy
$$( p, \vec{{q}} ) \in L^{\infty} \bigl( I; H^{1} _{0}( {\varOmega})\times\mathbb{H}^{1} ( {\varOmega}) \bigr),\qquad ( p_{t}, \vec{{q}}_{t} ) \in L^{1} \bigl( I; L^{2} ( { \varOmega})\times\mathbb{L}^{2} ( {\varOmega}) \bigr). $$
Let
\((p^{h},\vec{{q}}^{h})\)
be the semi-discrete DG approximation obtained by (3.1). Then the error
\(e^{p}=p-p^{h}, e^{\vec{{q}}}=\vec{{q}}-\vec{{q}}^{h}\)
satisfy
$$ \begin{aligned} & \biggl(\frac{1}{c^{2}}e^{p}_{t},r^{h} \biggr)+\tilde{a}_{h} \bigl(e^{p},r^{h} \bigr)- \tilde{b}_{h}' \bigl(e^{\vec{ {q}}},r^{h} \bigr)=\mathcal{R}^{\vec{q}} \bigl({\vec{ {q}}},r ^{h} \bigr),\quad \forall r^{h}\in\mathcal{P}^{h}( {\varOmega}) \textit{ a.e. in } I, \\ & \bigl(e^{\vec{{q}}}_{t},\vec{v}^{ h} \bigr) + \tilde{b}_{h} \bigl(e^{p},{\vec{v}^{ h}} \bigr)+ \tilde{c}_{h} \bigl(e^{\vec{{q}}},\vec{v}^{ h} \bigr) = \mathcal{R} ^{p} \bigl(p, \vec{v}^{ h} \bigr) ,\quad\forall \vec{v}^{ h} \in\mathcal{Q}^{h}(\varOmega) \textit{ a.e. in } I. \end{aligned} $$
(3.8)
Proof
Let \(p^{h}\in\mathcal{P}^{h}(\varOmega)\) and \(\vec{v}^{ h} \in \mathcal{Q}^{h}(\varOmega) \). Then we obtain using the discrete formulation in (3.1)
$$\begin{aligned} & \biggl(\frac{1}{c^{2}}e^{p}_{t},r^{h} \biggr)+ \tilde{a}_{h} \bigl(e^{p},r^{h} \bigr)- \tilde{b}_{h}' \bigl(e^{\vec {{q}}},r^{h} \bigr) \\ &\quad = \biggl(\frac{1}{c^{2}}p_{t},r^{h} \biggr)+ \tilde{a}_{h} \bigl(p,r^{h} \bigr)-\tilde{b}_{h}' \bigl({\vec{{q}}},r^{h} \bigr) \quad \text{a.e. in } I, \\ & \bigl(e^{\vec{{q}}}_{t},\vec{v}^{ h} \bigr)+ \tilde{b}_{h} \bigl(e^{p},\vec{v}^{ h} \bigr)+\tilde {c}_{h} \bigl(e^{\vec{{q}}},\vec{v}^{ h} \bigr) \\ &\quad= \bigl({ \vec{{q}}}_{t},\vec{v}^{ h} \bigr)+\tilde{b}_{h} \bigl(p,\vec{v}^{ h} \bigr)+\tilde{c}_{h} \bigl({\vec{{q}}}, \vec{v}^{ h} \bigr) \quad\text{a.e. in } I. \end{aligned}$$
By the definition of \(\tilde{b}_{h} \), the property (3.2) of \(L^{2}\)-projection \(\varPi_{h},\boldsymbol {\varPi}_{h}\), and the definitions (3.4), (3.5) of the lifted elements \(\mathcal {L}^{+}_{h}\), \(\mathcal{L}^{-}_{h}\), we obtain
$$\begin{aligned} &\tilde{b}_{h} \bigl(p,\vec{v}^{ h} \bigr)=- \sum _{K \in\mathcal{T}_{h}} \int_{K} p \nabla\cdot\vec{v}^{ h} \,dx -\sum _{e\in\mathcal{E}} \int_{e} \bigl[ \hspace{-2pt}\bigl[\vec{v}^{ h} \bigr]\hspace{-2pt}\bigr] \bigl( \vec{\mathtt{C}}_{12}\cdot[\hspace{-2pt}[\varPi_{h} p]\hspace{-2pt}]- \{\hspace{-2pt}\{\varPi_{h} p\}\hspace{-2pt}\} \bigr) \,ds, \\ &\tilde{b}_{h}' \bigl({\vec{{q}}},r^{h} \bigr)= \sum_{K \in\mathcal{T}_{h}} \int_{K}\vec{{q}}\cdot\nabla r^{h} \,dx -\sum _{e\in\mathcal{E} } \int_{e} \bigl[ \hspace{-2pt}\bigl[r^{h} \bigr]\hspace{-2pt}\bigr]\cdot \bigl( \vec{\mathtt{C}}_{12} [\hspace{-2pt}[\boldsymbol {\varPi}_{h} \vec {{q}}]\hspace{-2pt}] + \{\hspace{-2pt}\{\boldsymbol {\varPi}_{h} \vec{{q}} \}\hspace{-2pt}\} \bigr) \,ds. \end{aligned}$$
Since \(( p_{t}, \vec{{q}} _{t}) \in L^{1} ( I; L^{2} (\varOmega)\times \mathbb{L}^{2} (\varOmega) ) \), we have \(\nabla\cdot\vec{{q}} \in L^{2}(\varOmega) \) and \(\nabla p\in\mathbb{L}^{2}(\varOmega)\) almost everywhere in I, which implies that p and q⃗ have continuous normal components across all interior faces. By integration by parts in element-wise and combination with the trace operators, we obtain
$$\begin{aligned} &\tilde{b}_{h} \bigl(p,\vec{v}^{ h} \bigr)=\sum _{K \in\mathcal{T}_{h}} \int_{K}\nabla p \cdot\vec{v}^{ h} \,dx -\sum _{\mathcal{E}} \int_{e} \bigl[ \hspace{-2pt}\bigl[\vec{v}^{ h} \bigr]\hspace{-2pt}\bigr] \bigl( \{\hspace{-2pt}\{p\}\hspace{-2pt}\}+\vec{\mathtt{C}}_{12} \cdot[\hspace{-2pt}[\varPi _{h} p] \hspace{-2pt}]- \{\hspace{-2pt}\{\varPi_{h} p\}\hspace{-2pt}\} \bigr) \,ds, \\ &\tilde{b}_{h}' \bigl({\vec{{q}}},r^{h} \bigr)=- \sum_{K \in\mathcal{T}_{h}} \int_{K} \nabla\cdot\vec{{q}} r^{h} \,dx +\sum _{\mathcal{E}} \int_{e} \bigl[ \hspace{-2pt}\bigl[r^{h} \bigr]\hspace{-2pt}\bigr]\cdot \bigl( \{\hspace{-2pt}\{\vec{{q}}\}\hspace{-2pt}\} - \vec{\mathtt{C}}_{12} [\hspace{-2pt}[\boldsymbol { \varPi}_{h} \vec{{q}} ] \hspace{-2pt}] - \{\hspace{-2pt}\{\boldsymbol { \varPi}_{h} \vec{{q}} \}\hspace{-2pt}\} \bigr)\,ds. \end{aligned}$$
From the definition of \(\mathcal{R}^{\vec{q}}({\vec{{q}}},r^{h})\) and \(\mathcal{R}^{p}(p,\vec{v}^{ h}) \) in (3.7) we have
$$\begin{aligned} & \biggl(\frac{1}{c^{2}}p_{t},r^{h} \biggr)+ \tilde{a}_{h} \bigl(p,r^{h} \bigr)-\tilde{b}_{h}' \bigl({\vec{{q}}},r^{h} \bigr) = \biggl(\frac{1}{c^{2}}p_{t} + \frac{\sigma_{p}}{c^{2}}p+ \nabla\cdot{\vec{{q}}},r^{h} \biggr) + \mathcal{R}^{\vec{q}} \bigl({\vec{{q}}},r^{h} \bigr), \\ & \bigl({\vec{{q}}}_{t},\vec{v}^{ h} \bigr)+ \tilde{b}_{h} \bigl(p,\vec{v}^{ h} \bigr)+\tilde {c}_{h} \bigl({\vec{{q}}},\vec{v}^{ h} \bigr) = ( { \vec{{q}}}_{t} +\sigma_{\vec{q}} {\vec{{q}}} +\nabla p,{\vec {{v}}}) + \mathcal{R}^{p} \bigl(p,\vec{v}^{ h} \bigr), \end{aligned}$$
and we obtain
$$\begin{aligned} & \biggl(\frac{1}{c^{2}}e^{p}_{t},r^{h} \biggr)+ \tilde{a}_{h} \bigl(e^{p},r^{h} \bigr)- \tilde{b}_{h}' \bigl(e^{\vec {{q}}},r^{h} \bigr) = \mathcal{R}^{\vec{q}} \bigl({\vec{{q}}},r^{h} \bigr), \\ & \bigl(e^{\vec{{q}}}_{t},\vec{v}^{ h} \bigr)+ \tilde{b}_{h} \bigl(e^{p},\vec{v}^{ h} \bigr)+ \tilde{c}_{h} \bigl(e^{\vec{{q}}},\vec{v}^{ h} \bigr)= \mathcal{R}^{p} \bigl(p,\vec{v}^{ h} \bigr), \end{aligned}$$
where we have used the differential equations in (2.1). □
There is also an important relation between \(\tilde{b}_{h}\) and \(\tilde {b}'_{h}\) from the dual property of ∇ and \(-\nabla\cdot\) in the following lemma.
Lemma 3.6
Let the analytical solution
\((p,{\vec {q}})\)
of (2.1) satisfy
$$( p, {\vec{q}} ) \in L^{\infty} \bigl( I; H^{1} _{0}( \varOmega)\times\mathbb{H}^{1} (\varOmega) \bigr),\qquad ( p_{t}, { \vec{q}}_{t} ) \in L^{1} \bigl( I; L^{2} (\varOmega) \times\mathbb{L}^{2} (\varOmega) \bigr). $$
Let
\((p^{h},{\vec{q}}^{ h})\)
be the semi-discrete DG approximation obtained by (3.1). Then the following property holds for all
\((r^{h}, {\vec{v}}^{h}) \in \mathcal{P}^{h}(\varOmega)\times\mathcal{Q}^{h}(\varOmega)\):
$$\begin{aligned} &{-}\tilde{b}_{h}' \bigl(e^{ {\vec{q}}}, \varPi_{h} p-p^{h} \bigr)+\tilde{b}_{h} \bigl(e^{p}, \boldsymbol {\varPi}_{h} {\vec{q}}- {\vec{q}}^{ h} \bigr)= 0, \\ & {-}\tilde{b}_{h}' \bigl(\vec{v}^{ h} ,r^{h} \bigr) + \tilde{b}_{h} \bigl(r^{h}, \vec{v}^{ h} \bigr) = 0. \end{aligned}$$
Proof
By the definition of \(\tilde{b}_{h}'\), the property (3.4) of lifted element, and the property of \(L^{2}\)-projection, we obtain
$$\begin{aligned} \tilde{b}_{h}' \bigl({ \vec{q}}-\boldsymbol { \varPi}_{h} \vec{q},r^{h} \bigr)&=0. \end{aligned}$$
(3.9)
Here, we have used the definition of \(L^{2}\)-projection, \(\boldsymbol {\varPi}_{h} ({ \vec{q}}-\boldsymbol {\varPi}_{h} \vec{q} )=\boldsymbol {\varPi}_{h} { \vec{q}}-\boldsymbol {\varPi}_{h} \vec {q}=\vec{0}\). In the similar way, \(\tilde{b}_{h}( p- {\varPi}_{h} p, \vec{q} ^{h}) =0. \) For \((r^{h},{\vec{v}} ^{h} ) \in\mathcal{P}^{h}(\varOmega) \times \mathcal {Q}^{h}(\varOmega)\), we use definition of \(\tilde{b}_{h}'\), element-wise integration by parts, and the trace identity (2.5) to obtain
$$\begin{aligned} \tilde{b}_{h}' \bigl( \vec{v}^{ h} ,r^{h} \bigr)=-\sum_{K \in\mathcal{T}_{h}} \int_{K} \nabla\cdot\vec{v}^{ h} r^{h} \,dx -\sum_{e\in\mathcal{E} } \int_{e} \bigl( \bigl[ \hspace{-2pt}\bigl[r^{h} \bigr]\hspace{-2pt}\bigr] \cdot\vec{\mathtt{C}}_{12} - \bigl\{ \hspace{-2pt}\bigl\{ r^{h} \bigr\} \hspace{-2pt}\bigr\} \bigr) \bigl[ \hspace{-2pt}\bigl[\vec{v}^{ h} \bigr]\hspace{-2pt}\bigr]\,ds, \end{aligned}$$
and from the definition of \(\tilde{b}_{h}\),
$$\begin{aligned} \tilde{b}_{h} \bigl( r^{h}, \vec{v}^{ h} \bigr) =- \sum_{K \in\mathcal{T}_{h}} \int_{K} r^{h} \nabla\cdot\vec{v}^{ h} \,dx -\sum_{e\in\mathcal{E} } \int_{e} \bigl( \bigl[ \hspace{-2pt}\bigl[r^{h} \bigr]\hspace{-2pt}\bigr] \cdot\vec{\mathtt{C}}_{12}- \bigl\{ \hspace{-2pt}\bigl\{ r^{h} \bigr\} \hspace{-2pt} \bigr\} \bigr) \bigl[ \hspace{-2pt}\bigl[ \vec{v}^{ h} \bigr]\hspace{-2pt}\bigr]\,ds. \end{aligned}$$
Subtracting \(\tilde{b}'_{h}\) from \(\tilde{b}_{h}\) we have
$$ -\tilde{b}_{h}' \bigl( \vec{v}^{ h},r^{h} \bigr) +\tilde{b}_{h} \bigl( r^{h}, \vec{v}^{ h} \bigr)=0\quad \forall\bigl(r^{h}, \vec{v}^{ h} \bigr) \in\mathcal{P}^{h}(\varOmega)\times \mathcal{Q}^{h}(\varOmega). $$
(3.10)
Using the definition of error \(e^{p}\) and \(e^{\vec{q}}\) with the properties (3.9) and (3.10) we obtain
$$\begin{aligned} \tilde{b}_{h}' \bigl(e^{ \vec{q}}, \varPi_{h} p-p^{h} \bigr)-\tilde{b}_{h} \bigl(e^{p}, \boldsymbol { \varPi}_{h} \vec{q}- \vec{q}^{h} \bigr) &=0, \end{aligned}$$
which completes the proof. □
Approximation properties. Let \(\varPi_{h}\) and \(\boldsymbol {\varPi}_{h}\) denote the \(L^{2}\)-projections onto \(\mathcal {V}^{h}\) and \(\mathcal{Q}^{h}\), respectively. We note the following \(L^{2}\)-projection approximation properties; see [7]. Using the approximation properties in [7], we introduce the following results.
Lemma 3.7
Let
\(p\in H^{1+s}(\varOmega), s>\frac{1}{2}\). Then the following holds:
$$\begin{aligned} &\bigl\Vert \{\hspace{-2pt}\{\varPi_{h}p-p\}\hspace{-2pt}\} \bigr\Vert _{0,\mathcal{E}} \leq Ch^{\min\{ s,k\} +\frac{1}{2}} \Vert p \Vert _{1+s,\varOmega}, \\ &\bigl\Vert [\hspace{-2pt}[\varPi_{h}p-p]\hspace{-2pt}] \bigr\Vert _{0,\mathcal {E}} \leq Ch^{\min\{ s,k\} +\frac{1}{2}} \Vert p \Vert _{1+s,\varOmega} \end{aligned}$$
with a constant
C
that is independent of the local mesh size
\(h_{K}\)
and depends only on the shape-regularity of the mesh, the approximation order
k, the dimension
d, and the regularity exponent
s.
Proof
It is directly obtained from the properties in [7] and definition of jump and average on faces of elements K. □
Lemma 3.8
Let
\((p,\vec{q})\in H^{1+s}(\varOmega) \times\mathbb{H}^{1+s}(\varOmega) \)
with
\(s>\frac{1}{2}\). Then, the following holds:
-
(i)
For
\(( r,\vec{{v}}) \in\mathcal{P}(h) \times \mathcal{Q}(h)\), the forms (3.7) can be bounded by
$$\begin{aligned} & \bigl\vert \mathcal{R}^{p} (p, \vec{{v}}) \bigr\vert \leq C_{R}^{p} h^{\min\{ s,k \} + \frac{1}{2}} \bigl\Vert {\mathtt{C} ^{\frac{1}{2}}_{22} } [\hspace{-2pt}[\vec{{v}} ]\hspace{-2pt}] \bigr\Vert _{0,\mathcal{E}} \Vert p \Vert _{{1+s},\varOmega}, \\ & \bigl\vert \mathcal{R}^{\vec{q}} (\vec{q},r) \bigr\vert \leq C_{R}^{\vec{q}} h^{\min\{ s,k' \} + \frac{1}{2}} \bigl\Vert \mathtt{C}_{11} ^{\frac {1}{2}} [\hspace{-2pt}[r ]\hspace{-2pt}] \bigr\Vert _{0,\mathcal{E}} \Vert \vec{ {q}} \Vert _{ {1+s},\varOmega}, \end{aligned}$$
with constants
\(C_{R}^{p}\)
and
\(C_{R}^{\vec{q}}\)
independent of
h, which depend only on the stabilization parameters
\(\boldsymbol {\alpha }_{11}, \boldsymbol {\alpha}_{12}, \boldsymbol {\alpha}_{22}\)
given in (2.3), and the constant in the approximation properties in [7].
-
(ii)
The bilinear forms are estimated by the following:
$$\begin{aligned} &\tilde{a}_{h} \bigl(e^{p},\varPi_{h} p-p \bigr) \leq C_{a} h^{\min\{s, k\}+\frac{1}{2} } \bigl( h^{\frac{1}{2} } \bigl\Vert e^{p} \bigr\Vert _{0,\varOmega}+ \bigl\Vert \mathtt {C}_{11}^{\frac {1}{2}} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal {E}} \bigr) \Vert p \Vert _{ {1+s},\varOmega}, \\ &\tilde{c}_{h} \bigl(e^{\vec{q}},\boldsymbol {\varPi}_{h} \vec{q}- \vec{q} \bigr) \leq C_{c} h^{\min \{s,k'\}+\frac{1}{2}} \bigl( h^{\frac{1}{2} } \bigl\Vert e^{\vec {q}} \bigr\Vert _{0,\varOmega} + \bigl\Vert \mathtt{C}_{22}^{\frac{1}{2}} \bigl[ \hspace{-2pt}\bigl[e^{\vec {q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{0,\mathcal{E}} \bigr) \Vert \vec{q} \Vert _{ {1+s},\varOmega}, \end{aligned}$$
with constants
\(C_{a}\)
and
\(C_{c}\)
independent of
h, which depend only on
\(\boldsymbol {\alpha}_{11}, \boldsymbol {\alpha}_{22}\), and the constant in the approximation properties in [7].
Proof
(i) To show the first estimate we begin with the definition of \(\mathcal{R}^{p} \) in (3.7) and apply the Cauchy–Schwartz inequality and approximation properties in [7] to obtain
$$\begin{aligned} \bigl\vert \mathcal{R}^{p} (p, \vec{{v}} ) \bigr\vert ^{2} \leq{}& \sum_{e \in\mathcal{E} } \int_{e} \bigl\vert {\mathtt{C}^{\frac {1}{2}}_{22} } [\hspace{-2pt}[\vec{{v}} ]\hspace{-2pt}] \bigr\vert ^{2} \,ds \sum _{e \in\mathcal{E} } \int_{e} {\mathtt{C} ^{-1}_{22} } \bigl\vert \bigl( \vec{\mathtt{C}}_{12}\cdot[\hspace{-2pt}[ \varPi_{h} p -p ]\hspace{-2pt}]+ \{\hspace{-2pt}\{ \varPi_{h} p -p \}\hspace{-2pt}\} \bigr) \bigr\vert ^{2} \,ds \\ \leq{}&\alpha_{22} ^{-1} \bigl\Vert {\mathtt{C}^{\frac{1}{2}}_{22} } [\hspace{-2pt}[ \vec{{v}} ]\hspace{-2pt}] \bigr\Vert _{0,\mathcal{E}}^{2} \sum _{K \in \mathcal{T}_{h} } \bigl(1+ \vert\vec{\mathtt{C}}_{12} \vert\bigr) \Vert p - { \varPi}_{h} p \Vert ^{2} _{0,\partial K} \\ \leq{}& { \bigl({C_{R}^{p}} \bigr)}^{2} h^{2\min\{ s,k \}+ 1 } \bigl\Vert {\mathtt{C}^{\frac {1}{2}}_{22} } [\hspace{-2pt}[\vec{{v}} ]\hspace{-2pt}] \bigr\Vert ^{2}_{0,\mathcal{E}} \Vert p \Vert ^{2} _{1+s,\varOmega} \end{aligned}$$
for a positive constant \(C_{R}^{p}\) that depends on \(\boldsymbol {\alpha}_{12}, \boldsymbol {\alpha}_{22}\). This completes the first estimate. Similarly, we can have the second bound in (i).
(ii) From the definition of \(\tilde{a}_{h}\) and \(\tilde{c}_{h}\) in (3.6) we apply Hölder’s inequality, the definition of \(\alpha _{11}\), the Cauchy–Schwartz inequality, and the approximation properties in [7] to obtain the estimates of (ii) with the same order of h. □
Proof of Theorem
3.3
.
Proof
From Theorem 2.1, we have
$$e^{p} \in C^{0} \bigl(\bar{I};\mathcal{P}(h) \bigr)\cap C^{1} \bigl( \bar{I}; L^{2}( {\varOmega}) \bigr) \quad\text{and}\quad e^{\vec{q} } \in C^{0} \bigl(\bar{I};\mathcal{Q}(h) \bigr)\cap C^{1} \bigl( \bar{I}; \mathbb{L}^{2}( {\varOmega}) \bigr). $$
Since \(e^{p}= p- \varPi_{h} p+ \varPi_{h} p-p^{h}\), \(e^{\vec{q}} =\vec{q} -\boldsymbol {\varPi }_{h} \vec{q} +\boldsymbol {\varPi}_{h} \vec{q}-\vec{q} ^{h}\), using the error equations (3.8), we have
$$\begin{aligned} &\frac{1}{2} \biggl(\frac{d}{dt} \biggl\Vert \frac{1}{c}e^{p} \biggr\Vert _{0,\varOmega}^{2}+ \frac {d}{dt} \bigl\Vert e^{\vec{q}} \bigr\Vert _{0,\varOmega}^{2} \biggr)\\ &\quad = \biggl( \frac{1}{c^{2}} e_{t}^{p},p-\varPi_{h} p \biggr)- \tilde{a}_{h} \bigl(e^{p},\varPi_{h}p-p^{h} \bigr) + \bigl(e_{t}^{\vec{q}},\boldsymbol {\varPi}_{h} \vec{q} - \vec{q}^{h} \bigr) \\ &\qquad{}-\tilde{c}_{h} \bigl(e^{\vec{q}},\boldsymbol {\varPi}_{h}\vec{q}- \vec{q}^{h} \bigr) +\mathcal{R}^{\vec{q}} \bigl({\vec{q}}, \varPi_{h}p-p^{h} \bigr) +\mathcal{R}^{p} \bigl(p,\boldsymbol { \varPi}_{h}\vec{q}-\vec{q}^{h} \bigr), \end{aligned}$$
by the property in (2.2). Now we fix \(\tau\in I\) and integrate over the time interval \((0,\tau)\), which yields
$$\begin{aligned} &\frac{1}{2} \biggl( \biggl\Vert \frac{1}{c}e^{p}(\tau) \biggr\Vert _{0,\varOmega}^{2} + \bigl\Vert e^{\vec{q}}(\tau) \bigr\Vert _{0,\varOmega}^{2} \biggr)+ \int_{0}^{\tau} \bigl( \tilde{a}_{h} \bigl(e^{p},e^{p} \bigr)+\tilde{ c}_{h} \bigl(e^{\vec{q} },e^{\vec{q}} \bigr) \bigr) \,dt \\ &\quad =\frac{1}{2} \biggl( \biggl\Vert \frac{1}{c}e^{p}(0) \biggr\Vert _{0,\varOmega }^{2} + \bigl\Vert e^{\vec {q}}(0) \bigr\Vert _{0,\varOmega}^{2} \biggr) + \int_{0}^{\tau} \biggl( \biggl(\frac{1}{c^{2}}e_{t}^{p},p- \varPi_{h} p \biggr)+ \bigl(e_{t}^{\vec{q}}, \vec{q}-\boldsymbol { \varPi} _{h}\vec{q} \bigr) \biggr)\,dt \\ &\qquad{} + \int_{0}^{\tau} \bigl(\tilde{a}_{h} \bigl(e^{p},p- \varPi_{h} p \bigr) +\tilde{c}_{h} \bigl(e^{\vec{q}}, \vec{q}- \boldsymbol {\varPi}_{h} \vec{q} \bigr) + \mathcal{R}^{p} \bigl(p,\boldsymbol {\varPi}_{h} \vec{q}- \vec{q}^{ h} \bigr) \\ &\qquad{}+\mathcal{R}^{\vec {q}} \bigl({\vec{q}}, \varPi_{h} p-p^{h} \bigr) \bigr) \,dt. \end{aligned}$$
(3.11)
Integration by parts in the first integral on the right hand side and the standard Hölder inequality yield
$$\begin{aligned} &\int_{0}^{\tau} \biggl( \biggl( \frac{1}{c^{2}}e_{t}^{p},p- \varPi_{h} p \biggr)+ \bigl(e_{t}^{\vec {q}}, \vec{q}-\boldsymbol { \varPi} _{h}\vec{q} \bigr) \biggr)\,dt \\ &\quad= \biggl[ \biggl(\frac{1}{c^{2}}e^{p},p-\varPi p \biggr)+ \bigl( e^{\vec{q}},\vec{q}-\boldsymbol {\varPi}_{h} \vec{q} \bigr) \biggr]_{t=0}^{t=\tau} \\ &\qquad{}- \int_{0}^{\tau} \biggl( \biggl(\frac{1}{c^{2}}e^{p},(p- \varPi_{h} p)_{t} \biggr)+ \bigl(e^{\vec {q}}, (\vec{q}- \boldsymbol { \varPi}_{h} \vec{q})_{t} \bigr) \biggr)\,dt \leq T_{1}, \end{aligned}$$
where
$$\begin{aligned} T_{1}:={}& \biggl\Vert \frac{1}{c} e^{p} \biggr\Vert _{\mathscr{L}^{\infty}_{0, {\varOmega}}} \biggl( 2 \biggl\Vert \frac{1}{c}(p- \varPi_{h} p) \biggr\Vert _{\mathscr{L}^{\infty}_{0, {\varOmega}}} + \biggl\Vert \frac{1}{c}(p-\varPi_{h} p)_{t} \biggr\Vert _{\mathscr{L}^{1}_{0, {\varOmega}}} \biggr)\\ &{}+ \bigl\Vert e^{\vec {q}} \bigr\Vert _{\mathscr{L}^{\infty}_{0, {\varOmega}}} \bigl( 2 \Vert \vec{q}-\boldsymbol {\varPi}_{h} \vec {q} \Vert _{\mathscr{L}^{\infty }_{0, {\varOmega}}} + \bigl\Vert (\vec{q}-\boldsymbol {\varPi}_{h} \vec{q})_{t} \bigr\Vert _{\mathscr {L}^{1}_{0, {\varOmega }}} \bigr). \end{aligned}$$
From the definition of \(\tilde{a}_{h}\) and \(\tilde{c}_{h}\) and the standard Hölder inequality in the second integral on the right hand side in (3.11), we have
$$\begin{aligned} \int_{0}^{\tau} \bigl(\tilde{a}_{h} \bigl(e^{p},p- \varPi_{h} p \bigr) +\tilde{c}_{h} \bigl(e^{\vec{q}}, \vec{q}- \boldsymbol {\varPi}_{h} \vec{q} \bigr) \bigr)\,dt \leq T_{2}, \end{aligned}$$
where
$$\begin{aligned} T_{2}:={}& {\sigma^{*}}T \biggl( \biggl\Vert \frac{1}{c}e^{p} \biggr\Vert _{{\mathscr {L}^{\infty }(0,\varOmega)}} \bigl\Vert c(p- \varPi_{h} p) \bigr\Vert _{\mathscr{L}^{\infty }_{0, {\varOmega}}}\\ &{}+ \bigl\Vert e^{\vec{q}} \bigr\Vert _{\mathscr{L}^{\infty}(0,( {\varOmega}))} \Vert \vec{q}- \boldsymbol {\varPi}_{h} \vec{q} \Vert _{\mathscr{L}^{\infty}_{0, {\varOmega}}} \biggr) \\ &{}+ \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr{L}^{1}(0,\mathcal {E})} \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} [\hspace{-2pt}[p- \varPi_{h} p]\hspace{-2pt}] \bigr\Vert _{\mathscr {L}^{\infty}(0,\mathcal {E})} \\ &{}+ \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec{q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr {L}^{1}(0,\mathcal{E})} \bigl\Vert \mathtt{C}^{\frac {1}{2}}_{22} [\hspace{-2pt}[ \vec{q}- \boldsymbol {\varPi} _{h}\vec{q}]\hspace{-2pt}] \bigr\Vert _{\mathscr{L}^{\infty}(0,\mathcal{E})}. \end{aligned}$$
Next, we combine \(T_{1}\) and \(T_{2}\) and rewrite the left hand side (3.11) with the new bounds
$$\begin{aligned} &\frac{1}{2} \biggl( \biggl\Vert \frac{1}{c}e^{p}( \tau) \biggr\Vert _{0,\varOmega }^{2} + \bigl\Vert e^{\vec {q}}(\tau) \bigr\Vert _{0,\varOmega}^{2} \biggr)\\ &\qquad{} + \int_{0}^{\tau} \biggl( \biggl\Vert \frac{\sigma_{p}}{c^{2}} e^{p} \biggr\Vert ^{2}_{0, {\varOmega}} + \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{0,\mathcal{E}} + \bigl\Vert {\sigma_{\vec{q}}} e^{\vec{q}} \bigr\Vert ^{2}_{0, {\varOmega}} + \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec{q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{0,\mathcal{E}} \biggr) \,dt \\ &\quad \leq\frac{1}{2} \biggl( \biggl\Vert \frac{1}{c}e^{p}(0) \biggr\Vert _{0, {\varOmega }}^{2} + \bigl\Vert e^{\vec {q}}(0) \bigr\Vert _{0, {\varOmega}}^{2} \biggr) \\ &\qquad{}+ \int_{0}^{\tau} \bigl( \bigl\vert \mathcal{R}^{p} \bigl(p,\boldsymbol {\varPi}_{h} \vec{q}-\vec {q}^{ h} \bigr) \bigr\vert + \bigl\vert \mathcal{R}^{\vec{q}} \bigl({\vec{q}},\varPi_{h} p-p^{h} \bigr) \bigr\vert \bigr) \,dt+T_{1} +T_{2}. \end{aligned}$$
Since this inequality holds for any \(\tau\in I\), it also holds for the supremum over I, that is,
$$\begin{aligned} &\frac{1}{2}\sup_{t\in I} \biggl( \biggl\Vert \frac{1}{c} e^{p}(t) \biggr\Vert ^{2}_{0, {\varOmega}} + \bigl\Vert e^{\vec{q}}(t) \bigr\Vert ^{2}_{ 0, {\varOmega }} \biggr)\\ &\qquad{} + \biggl\Vert \frac{\sigma_{p}}{c^{2}} e^{p} \biggr\Vert ^{2}_{\mathscr{L}^{1}_{0, {\varOmega}}} + \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{\mathscr {L}^{1}_{0,{\mathcal{E}}}} + \bigl\Vert {\sigma_{\vec{q}}} e^{\vec{q}} \bigr\Vert ^{2}_{\mathscr {L}^{1}_{0, {\varOmega}}} + \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec {q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{\mathscr{L}^{1}_{0, {\mathcal{E}}}} \\ &\quad \leq\frac{1}{2} \biggl( \biggl\Vert \frac{1}{c}e^{p}(0) \biggr\Vert _{0, {\varOmega }}^{2} + \bigl\Vert e^{\vec {q}}(0) \bigr\Vert _{0, {\varOmega}}^{2} \biggr) \\ &\qquad{}+ \int_{I} \bigl( \bigl\vert \mathcal{R}^{p} \bigl(p, \boldsymbol {\varPi}_{h} \vec{q}-\vec{q}^{ h} \bigr) \bigr\vert + \bigl\vert \mathcal{R}^{\vec{q}} \bigl({\vec{q}},\varPi_{h} p-p^{h} \bigr) \bigr\vert \bigr)\,dt+T_{1}+T_{2}. \end{aligned}$$
Using the geometric–arithmetic mean inequality \(|ab|\leq\frac {1}{2\varepsilon} a^{2} + \frac{\varepsilon}{2}b^{2}\) valid for \(\varepsilon>0\), \((a+b)^{2}\leq2(a^{2}+b^{2}) \), and the approximation results in Lemma 3.7, we obtain
$$\begin{aligned} T_{1}+T_{2}\leq{}& \frac{1}{ \varepsilon} \biggl( \biggl\Vert \frac{1}{c} e^{p} \biggr\Vert _{\mathscr{L}^{\infty }_{0, {\varOmega}}} ^{2} + \bigl\Vert e^{\vec{q}} \bigr\Vert _{\mathscr {L}^{\infty}_{0, {\varOmega }}} ^{2} \biggr) + \frac{1 }{2\varepsilon'} \bigl( \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr{L}^{1}_{0, {\mathcal{E}}}} ^{2} + \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec{q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr{L}^{1}_{0, {\mathcal{E}}}}^{2} \bigr) \\ &{}+C\varepsilon h^{2\min\{s,k\} +2} \biggl( \frac{ 1}{2} \sigma^{*} {^{2}}T^{2} {c^{*}}^{2}+ \frac{4}{{c_{*}^{2}} } \biggr) \Vert p \Vert _{\mathscr {L}^{\infty}_{1+s, {\varOmega}}}^{2} \\ &{}+ C\varepsilon h^{2\min\{s,k'\} +2} \biggl( \frac{ 1}{2} \sigma^{*} {^{2}}T^{2} + 4 \biggr) \Vert \vec{q} \Vert _{\mathscr{L}^{\infty}_{1+s, {\varOmega}}}^{2} \\ &{}+C h^{2\min\{s,k\}+1 } \biggl(\frac{\varepsilon'}{2} \Vert p \Vert _{\mathscr{L}^{\infty}_{1+s, {\varOmega}}} ^{2} +{\varepsilon}\frac { 1}{{c_{*}^{2}} } \Vert p_{t} \Vert _{\mathscr{L}^{1}_{s, {\varOmega}}}^{2} \biggr)\\ &{}+Ch^{2\min\{s,k'\}+1 } \biggl( \frac{\varepsilon'}{2} \Vert \vec {q} \Vert _{\mathscr {L}^{\infty}_{1+s, {\varOmega}}} ^{2}+ \varepsilon \Vert \vec{q}_{t} \Vert _{\mathscr {L}^{1}_{s, {\varOmega }}} ^{2} \biggr). \end{aligned}$$
Using the approximation properties in [7] and Lemma 3.8 we can also bound the error equations
$$\begin{aligned} \int_{I} \bigl\vert \mathcal{R}^{p} \bigl(p,\boldsymbol { \varPi} \vec{q}-\vec{q}^{ h} \bigr) \bigr\vert \,dt \leq{}& \frac{1}{2\varepsilon'} \bigl( \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec{q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr {L}^{1}_{0, \mathcal{E}}}^{2} + \alpha_{22} h^{2\min\{ s,k' \}+1 } \Vert \vec{q} \Vert _{\mathscr{L}^{\infty }_{1+s, \varOmega}}^{2} \bigr) \\ &{} + \frac{\varepsilon'}{2} \bigl({C_{R}^{p}} \bigr)^{2} \bigl(1+T^{2} \bigr) h^{2\min\{ s,k \} +1} \Vert p \Vert _{\mathscr{L}^{\infty}_{1+s, {\varOmega}}}^{2} \end{aligned}$$
and
$$\begin{aligned} \int_{I} \bigl| \mathcal{R}^{q} \bigl({\vec{q}}, \varPi_{h} p-p^{h} \bigr)\bigr|\,dt \leq{}& \frac{1}{2\varepsilon'} \bigl( \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert _{\mathscr {L}^{1}_{0, \mathcal{E}}}^{2} + \alpha_{11} h^{2\min\{ s,k \}+1 } \Vert p \Vert _{\mathscr {L}^{\infty}_{1+s,\varOmega}}^{2} \bigr) \\ &{} + \frac{\varepsilon'}{2} \bigl({C_{R}^{\vec{q}}} \bigr)^{2} \bigl(1+T^{2} \bigr) h^{2\min\{ s,k' \} +1} \Vert \vec{q} \Vert _{\mathscr{L}^{\infty}_{1+s, {\varOmega}}}^{2}. \end{aligned}$$
Combining the above estimates and \(T_{1}, T_{2} \) with \(\varepsilon=4\) and \(\varepsilon'=2\), we have
$$\begin{aligned} &\frac{1}{4}\sup_{t\in I} \biggl( \biggl\Vert \frac{1}{c}e^{p} \biggr\Vert ^{2}_{0, {\varOmega}} + \bigl\Vert e^{\vec{q}} \bigr\Vert ^{2}_{0,\varOmega } \biggr) +\frac{1}{2} \bigl( \bigl\Vert \mathtt{C}^{\frac{1}{2}}_{11} \bigl[ \hspace{-2pt}\bigl[e^{p} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{\mathscr {L}^{1}_{0, \mathcal{E}}}+ \bigl\Vert \mathtt{C}^{\frac {1}{2}}_{22} \bigl[ \hspace{-2pt}\bigl[e^{\vec{q}} \bigr]\hspace{-2pt}\bigr] \bigr\Vert ^{2}_{\mathscr{L}^{1}_{0, \mathcal{E}}} \bigr)\\ &\quad \leq\frac{1}{2} \biggl( \biggl\Vert \frac {1}{c}e^{p}(0) \biggr\Vert _{0,\varOmega}^{2}+ \bigl\Vert e^{\vec{q}}(0) \bigr\Vert _{0, {\varOmega }}^{2} \biggr) \\ &\qquad{}+C h^{2\min\{s,k+\frac{1}{2}\}} \bigl( \Vert p \Vert _{\mathscr {L}^{\infty }_{{1+s,\varOmega}}}^{2}+ \Vert p_{t} \Vert _{{\mathscr {L}^{1}_{s,\varOmega}}}^{2} \bigr)+C h^{2\min\{ s,k'+\frac{1}{2} \} } \bigl( \Vert \vec{q} \Vert _{{\mathscr {L}^{\infty}_{1+s,\varOmega}}}^{2} + \Vert \vec{q}_{t} \Vert _{{\mathscr {L}^{1}_{s,\varOmega}}}^{2} \bigr) \end{aligned}$$
with a constant that is independent of the mesh size h. Using the bound \(\frac{1}{{c^{*}}^{2}}\|e^{p}\|^{2}_{0,\varOmega} \leq\| \frac {1}{c}e^{p}\|^{2}_{0,\varOmega} \), we conclude the proof of Theorem 3.3. □
Remark 3.9
In our LDG method the parameters are independent of mesh size h, which gives higher accuracy of the \(L^{2}\)-norms of errors in p and q⃗ with \(k+\frac{1}{2}\), \(k'+\frac{1}{2}\), respectively, for smooth solutions.
Remark 3.10
In this paper, we consider a first-order hyperbolic system of acoustic wave equation in a bounded domain with lower-order damping terms and present a priori error analysis introducing a LDG method for the system. The system (2.1) with the time-dependent damping terms \(\sigma_{p}(x,t)\) and \(\sigma_{\vec{q}}(x,t)\) can also be considered and the well-posedness with the initial condition \((p_{0},{\vec{q}}_{0}) \) in \(D(L)\) is obtained from Theorem 4.10, page 245 in [5].