We further suppose that Ω is a rectangle for \(d=2\) or a parallelepiped rectangle for \(d=3\).
Let \(\mathbb{P}_{N}(\Omega )\) the space of polynomials of degree ≤N (\(N\geq 2\)) for each variable, and let \(\mathbb{P}_{N}^{0}(\Omega )= \mathbb{P}_{N}(\Omega )\cap H^{1}_{0}( \Omega )\). We define \(\zeta _{i}\), \(0\le i \le N\), the set of nodes, roots of the polynomial \((1-x^{2})L_{N}'\), where \(L_{N}\) is the Legendre polynomial, and \(\varrho _{i}\), \(0\le i \le N\), are the weight set of the following Gauss–Lobatto quadrature formula on the interval \(]{-}1,1[\):
$$ \forall \eta _{N} \in \mathbb{P}_{2N-1} \bigl(]{-}1,1[\bigr),\quad \int _{-1}^{1} \eta _{N}(x) \,dx=\sum _{i=0}^{N} \eta _{N}(\zeta _{i})\varrho _{i}. $$
(20)
We recall the following property (see [10, 13]):
$$ \forall \eta _{N} \in \mathbb{P}_{N} \bigl(]{-}1,1[\bigr), \quad \Vert \eta _{N} \Vert _{L^{2}(]{-}1,1[)}^{2} \le \sum_{i=0}^{N} \eta _{N}^{2}(\zeta _{i}) \varrho _{i} \le 3 \Vert \eta _{N} \Vert _{L^{2}]{-}1,1[}^{2}. $$
(21)
The reference domain \(]{-}1,1[^{d}\) (\(d=2, 3\)) is transformed to the domain Ω using the affine mapping T, and the scalar product is defined on continuous functions u and v by
$$ \begin{aligned} &(u,v)_{N} \\ &\quad = \textstyle\begin{cases} \frac{\operatorname{meas}(\Omega )}{4} \sum_{i=0}^{N} \sum_{j=0}^{N} (u \circ T)(\zeta _{i},\zeta _{j}) (v \circ T)(\zeta _{i}, \zeta _{j}) \varrho _{i} \varrho _{j} & \text{if } d=2, \\ \frac{\operatorname{meas}(\Omega )}{8} \sum_{i=0}^{N} \sum_{j=0}^{N} \sum_{k=0}^{N} (u \circ T)(\zeta _{i},\zeta _{j}, \zeta _{k}) (v \circ T)(\zeta _{i},\zeta _{j},\zeta _{k}) \varrho _{i} \varrho _{j} \varrho _{k} & \text{if } d=3. \end{cases}\displaystyle \end{aligned} $$
(22)
Remark 3
For simplicity of analysis, we suppose that the spectral discretization is fixed over time.
We suppose that \(u_{0}\) and \(v_{0}\) are continuous on Ω̄. The discrete problem is deduced from (5) by applying the Galerkin method combined with numerical integration:
$$ \text{For } u_{N}^{0}= \mathfrak{I}_{N}(u_{0}) \quad \text{and}\quad u_{N}^{1}= \mathfrak{I}_{N}(u_{0})+ \delta t_{0}\mathfrak{I}_{N}(v_{0}) \quad \text{in } \Omega , $$
(23)
where \(\mathfrak{I}_{N}\) is the interpolating operator from \(L^{2}(\Omega )\) into \(\mathbb{P}_{N}(\Omega )\), find \(u_{N}^{k}\in \mathbb{P}_{N}^{0}(\Omega )\times \mathbb{P}_{N}( \Omega ) \times (\mathbb{P}_{N}^{0}(\Omega ))^{K-1} \), \(1\le k \le K\), such that for all \(v_{N} \in \mathbb{P}_{N}^{0}(\Omega )\),
$$ \biggl(\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} - \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}},v_{N} \biggr)_{N}+ \delta t_{k}\bigl( \nabla u_{N}^{k+1}, \nabla v_{N} \bigr)_{N} = 0. $$
(24)
As in (6), \(u_{N}^{k+1}\), \(1\le k \le K\), is the solution of the discrete weak problem
$$ \bigl(u_{N}^{k+1}, v_{N} \bigr)_{N} + \delta t_{k}^{2} \bigl(\nabla u_{N}^{k+1}, \nabla v_{N} \bigr)_{N}= \biggl(u_{N}^{k} + \frac{\delta t_{k}}{\delta t_{k-1}}\bigl(u_{N}^{k}-u_{N}^{k-1} \bigr),v_{N}\biggr)_{N}. $$
(25)
Proposition 3
Let the data \((u_{0},v_{0})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\). If \(u_{N}^{0}\) and \(v_{N}^{0}\) are known, then problem (25) has a unique solution \(u_{N}^{k+1}\), \(k\geq 1\), in \(H^{1}_{0}(\Omega )\). Moreover, the solution \({(u_{N}^{k})}_{0\leq k\leq K}\) of problem (23)–(24) satisfies for \(0\leq k\leq K\) the following stability condition:
$$ \begin{aligned} &\biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert ^{2} \\ &\quad \leq {\bigl(3^{d}\bigr)}^{K} \bigl( \bigl\Vert \mathfrak{I}_{N}(v_{0}) \bigr\Vert ^{2} +2 \bigl\Vert \nabla \mathfrak{I}_{N}(u_{0}) \bigr\Vert ^{2} +2\delta t_{0}^{2} \bigl\Vert \nabla \mathfrak{I}_{N}(v_{0}) \bigr\Vert ^{2} \bigr). \end{aligned} $$
(26)
Proof 4
We show that problem (25) has a unique solution using the Lax–Milgram theorem and property (21).
To prove the stability condition (26), we define \(\|\cdot \|_{d}\) the discrete norm deduced from the discrete scalar product \((\cdot ,\cdot )_{N}\). Now letting \(v_{N}=\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}}\) in (24) leads to
$$ \biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert _{d}^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert _{d}^{2}=\biggl(\frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}}, \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}}\biggr)_{N}+\bigl(\nabla u_{N}^{k+1}, \nabla u_{N}^{k}\bigr)_{N}. $$
Using the Cauchy–Schwarz inequality and (21), we have
$$ \biggl\Vert \frac{u_{N}^{k+1}-u_{N}^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u_{N}^{k+1} \bigr\Vert ^{2} \leq {3^{d}} \biggl( \biggl\Vert \frac{u_{N}^{k}-u_{N}^{k-1}}{\delta t_{k-1}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k} \bigr\Vert ^{2} \biggr). $$
Then iterating over k, we obtain
$$ \biggl\Vert \frac{u^{k+1}-u^{k}}{\delta t_{k}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{k+1} \bigr\Vert ^{2} \leq { \bigl(3^{d}\bigr)}^{K} \biggl( \biggl\Vert \frac{u^{1}-u^{0}}{\delta t_{0}} \biggr\Vert ^{2} + \bigl\Vert \nabla u^{1} \bigr\Vert ^{2} \biggr). $$
Finally, estimate (26) is deduced from (23).
Proposition 4
Let \(u_{0}\), \(v_{0}\) be continuous on Ω̅, and let \(u_{N}^{0}\), \(v_{N}^{0}\) be known. The error estimate between solutions \(u^{k+1}, k\geq 1\), and \(u_{N}^{k+1}\), \(k\geq 1\), of problems (6) and (25), respectively, is
$$ \begin{aligned} \bigl\Vert u^{k+1} - u_{N}^{k+1} \bigr\Vert & \le C \Biggl(\inf _{\chi _{N}^{k+1} \in \mathbb{P}_{N}^{0}(\Omega )} \bigl\Vert u^{k+1}-\chi _{N}^{k+1} \bigr\Vert + \Biggl[ \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert + \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert \\ &\quad {} +\sum_{j=1}^{k} \bigl(T^{1,j}+T^{2,j}+T^{3,j}\bigr) \Biggr] \Biggr), \end{aligned} $$
(27)
where
$$ \begin{aligned} &T^{1,j} = {\frac{1}{\delta t^{2}_{j}}} \sup _{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }(u^{j+1}-u^{j})v_{N} \,d{\mathbf{x}} -(\chi _{N}^{j+1}-\chi _{N}^{j},v_{N})_{N}}{ \Vert v_{N} \Vert }, \\ &T^{2,j} = \sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- (\nabla \chi _{N}^{j+1},\nabla v_{N})_{N}}{ \Vert v_{N} \Vert }, \\ &T^{3,j} = \sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }(u^{j}-u^{j-1}) v_{N} \,d{\mathbf{x}} - (\mathfrak{I}_{N}(u^{j}-u^{j-1}) , v_{N} )_{N}}{ \Vert v_{N} \Vert }, \end{aligned} $$
and C is a positive constant independent of N.
Proof 5
Consider \(\chi _{N}^{k+1}\in \mathbb{P}_{N}^{0}(\Omega )\). By the triangle inequality we have
$$ \bigl\Vert u^{k+1} - u_{N}^{k+1} \bigr\Vert \le \bigl\Vert u^{k+1} - \chi _{N}^{k+1} \bigr\Vert + \bigl\Vert \chi _{N}^{k+1} - u_{N}^{k+1} \bigr\Vert . $$
To estimate \(\|u_{N}^{k+1}- \chi _{N}^{k+1}\|\), we begin by writing problems (5) and (25) for \(v_{N}\in \mathbb{P}_{N}^{0}(\Omega )\). Then we consider \(\tau _{k} = \frac{\delta t_{k}}{\delta t_{k-1}}\) and doing the difference term by term, we obtain
$$ \bigl(u_{N}^{k+1}-\chi _{N}^{k+1},v_{N} \bigr)_{N}+ \delta t_{k}^{2}\bigl(\nabla \bigl(u_{N}^{k+1}- \chi _{N}^{k+1} \bigr),\nabla v_{N}\bigr)_{N}=\bigl(u_{N}^{k}- \chi _{N}^{k},v_{N}\bigr)_{N} + \tau _{k} \mathcalligra{K}^{k} (v_{N}), $$
where
$$ \begin{aligned} \mathcalligra{K}^{k} (v_{N})={}& \frac{1}{\delta t^{2}_{k}} \biggl( \int _{\Omega }\bigl(u^{k+1}-u^{k} \bigr)v_{N} \,d{\mathbf{x}} -\bigl(\chi _{N}^{k+1}- \chi _{N}^{k},v_{N}\bigr)_{N} \biggr) \\ &{} + \int _{\Omega }\nabla u^{k+1}\nabla v_{N} \,d{\mathbf{x}}- \bigl(\nabla \chi _{N}^{k+1},\nabla v_{N}\bigr)_{N} \\ &{} + \int _{\Omega }\bigl(u^{k}-u^{k-1}\bigr) v_{N} \,d{\mathbf{x}} - \bigl(\mathfrak{I}_{N} \bigl(u^{k}-u^{k-1}\bigr) , v_{N} \bigr)_{N}. \end{aligned} $$
Since \(\mathcalligra{K}^{k}\) is linear and continuous on \(\mathbb{P}_{N}^{0}(\Omega )\), by the Riesz theorem there exists \(\vartheta ^{k}_{N}\) in \(\mathbb{P}_{N}^{0}(\Omega )\) such that
$$ \mathcalligra{K}^{k} (v_{N})=\bigl(\vartheta ^{k}_{N},v_{N}\bigr)_{N}. $$
Applying the result proved in [14, Prop. 4.1] and [15], we get
$$ \bigl\Vert u_{N}^{k+1}- \chi _{N}^{k+1} \bigr\Vert \le C \Biggl( \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert + \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert + \sum_{j=1}^{k} \bigl\Vert \vartheta ^{j}_{N} \bigr\Vert ^{2} \Biggr)^{1/2}, $$
where C is a positive constant independent of N.
So we conclude (27), since
$$ \bigl\Vert \vartheta ^{j}_{N} \bigr\Vert \leq C \sup_{v_{N}\in \mathbb{P}_{N}^{0}( \Omega )} \frac{(\vartheta ^{j}_{N},v_{N})_{N}}{ \Vert v_{N} \Vert },$$
where C is a positive constant independent of N.
To find the order of convergence as a function of N, it is necessary to estimate each of the terms of the second member of inequality (27).
We consider \(\varpi ^{j+1}=u^{j+1} - u^{j}\) and \(\chi _{N}^{j+1}-\chi _{N}^{j}= \Pi ^{1,0}_{N-1}(\varpi ^{j+1})\). By the exactness of the Gauss–Lobatto quadrature formula of (20), \(\int _{\Omega }\Pi ^{1,0}_{N-1}(\varpi ^{j+1}) v_{N} \,d{\mathbf{x}}\) and \((\Pi ^{1,0}_{N-1}(\varpi ^{j+1}),v_{N})_{N}\) are equal, and thus
$$ T^{1,j} \leq \bigl\Vert \varpi ^{j} - \Pi ^{1,0}_{N-1}\bigl(\varpi ^{j}\bigr) \bigr\Vert , $$
(28)
where \(\Pi _{N}^{1,0}\) is the orthogonal projection operator from \(H^{1}_{0}(\Omega )\) into \(\mathbb{P}_{N}^{0}(\Omega )\) related to the inner product defined by the semi norm \(| \cdot | _{1,\Omega }\). (See ([13], Lemma VI.2.5) and [10] for all the properties of this operator.)
Since the Gauss–Lobatto quadrature formula is exact for a polynomial of degree \(\leq 2N-1\), we have
$$ \begin{aligned} &\int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- \bigl(\nabla \chi _{N}^{j+1},\nabla v_{N}\bigr)_{N} \\ &\quad = \int _{\Omega } \nabla \bigl(u^{j+1}-\Pi _{N -1}^{1,0}u^{j+1}\bigr) \nabla v_{N} \,d{\mathbf{x}} - \bigl(\nabla \bigl(\chi _{N}^{j+1}-\Pi _{N -1}^{1,0} \chi _{N}^{j+1}\bigr), \nabla v_{N} \bigr)_{N}. \end{aligned} $$
(29)
Thanks to the triangle and Cauchy–Schwarz inequalities, we have
$$ \begin{aligned} &\sup_{v_{N}\in \mathbb{P}_{N}^{0}(\Omega )} \frac{ \int _{\Omega }\nabla u^{j+1}\nabla v_{N} \,d{\mathbf{x}}- (\nabla \chi _{N}^{j+1},\nabla v_{N})_{N}}{ \Vert v_{N} \Vert } \\ &\quad \le \bigl( \bigl\vert u^{j+1}-\Pi _{N -1}^{1,0}u^{j+1} \bigr\vert _{1,\Omega } + \bigl\vert \chi _{N}^{j+1}-\Pi _{N -1}^{1,0} \chi _{N}^{j+1} \bigr\vert _{1,\Omega } \bigr). \end{aligned} $$
(30)
Thus we conclude using the properties of \(\Pi _{N-1}^{1,0}\).
Let \(\theta ^{j}=u^{j}-u^{j-1}\). We use for this estimation \(\Pi _{N -1}\) the orthogonal projection from \(L^{2}(\Omega )\) into \(\mathbb{P}_{N-1}(\Omega )\). By the same argument above for the Gauss–Lobatto formula, we have
$$ \begin{aligned} &\int _{\Omega }\theta ^{j}({\mathbf{x}})v_{N}({ \mathbf{x}}) \,d{\mathbf{x}} - \bigl( \mathfrak{I}_{N} \theta ^{j},v_{N}\bigr)_{N} \\ &\quad = \int _{\Omega }\bigl(\theta ^{j}- \Pi _{N-1} \theta ^{j}\bigr) ({\mathbf{x}}) v_{N} ({\mathbf{x}}) \,d{\mathbf{x}} -\bigl( \mathfrak{I}_{N} \theta ^{j} - \Pi _{N -1} \theta ^{j}, v_{N}\bigr)_{N}. \end{aligned} $$
Using inequality (21) in each direction leads to
$$ \int _{\Omega }\theta ^{j}({\mathbf{x}})v_{N}({ \mathbf{x}}) \,d{\mathbf{x}} - \bigl( \mathfrak{I}_{N}\theta ^{j},v_{N}\bigr)_{N}\le \bigl[ \bigl\Vert \theta ^{j}-\Pi _{N -1} \theta ^{j} \bigr\Vert ^{2} + 9 \bigl\Vert \theta ^{j}- \mathfrak{I}_{N} \theta ^{j} \bigr\Vert ^{2} \bigr] \Vert v_{N} \Vert . $$
Thanks to the approximation properties of operator \(\Pi _{N -1}\) (see [10, Thm. 7.1]) and \(\mathfrak{I}_{N}\) (see [10, Thm. 14.2]), for \(\theta ^{j}\in H^{s} (\Omega )\); \(s>1\), we obtain
$$ \sup_{v_{N}\in \mathbb{P}_{N}(\Omega )} \frac{ \int _{\Omega } \theta ^{j}({\mathbf{x}})v_{N}({\mathbf{x}}) \,d{\mathbf{x}} - (\theta ^{j},v_{N})_{N}}{ \Vert v_{N} \Vert } \le C N^{-2s} \bigl\Vert \theta ^{j} \bigr\Vert ^{2}_{s,\Omega } . $$
(31)
Finally, to estimate
$$ \inf_{\chi _{N}^{k+1}\in \mathbb{P}_{N}^{0}(\Omega )} \bigl\Vert u^{k+1}- \chi _{N}^{k+1} \bigr\Vert , \qquad \bigl\Vert u_{0}-u_{N}^{0} \bigr\Vert \quad \text{and} \quad \bigl\Vert v_{0}-v_{N}^{0} \bigr\Vert , $$
(32)
we choose, respectively, \(\chi _{N}^{k+1}=\Pi _{N}^{1,0}u^{k+1}\), \(u_{N}^{0}=\Pi _{N}^{1,0}u_{0}\), and \(v_{N}^{0}=\Pi _{N} v_{0}\). Then we conclude using properties of operators \(\Pi _{N}^{1,0}\) and \(\Pi _{N}\).
So, from estimates (28), (30), (31), and (32) we obtain the following main theorem.
Theorem 2
For \((u_{0},v_{0})\) continuous on Ω̄, solution \((u^{k})_{0\leq k\leq K}\) of problem (5) belongs to \(H^{s}(\Omega )\); \(s> 1\). The error between solutions \(u^{k+1}\) and \((u_{N}^{k+1})\) of problems (6) and (25), respectively, satisfies
$$ \begin{aligned} \bigl\Vert u^{k+1}-u_{N}^{k+1} \bigr\Vert \le{}& C \Biggl[N^{-s} \Biggl( \bigl\Vert u^{k+1} \bigr\Vert _{s,\Omega }+ \sum _{j=1}^{k} \bigl(\delta t^{-2}_{j} \bigl\Vert u^{j+1}-u^{j} \bigr\Vert _{s,\Omega } + \bigl\Vert u^{j}-u^{j-1} \bigr\Vert _{s,\Omega } \bigr) \Biggr) \\ &{} + N^{1-s} \sum_{j=1}^{k} \bigl\Vert u^{j+1} \bigr\Vert _{s,\Omega } \Biggr], \end{aligned} $$
(33)
where C is a positive constant independent of N.