Open Access

Full discretization of wave equation

  • Abderrazak Chaoui1Email author,
  • Fateh Ellaggoune2 and
  • Assia Guezane-Lakoud3
Boundary Value Problems20152015:133

https://doi.org/10.1186/s13661-015-0396-3

Received: 14 April 2015

Accepted: 19 July 2015

Published: 7 August 2015

Abstract

Rothe’s method for time discretization and Crouseix-Raviart nonconforming finite element method to the spatial variable. After introducing error estimators, we prove the equivalence between the error and its indicators.

Keywords

Rothe’s method Crouseix-Raviart nonconforming finite element method wave equation a posteriori estimate

MSC

35K55 35A35

1 Introduction

Among commonly used methods for the numerical approach of problems which arise in engineering, for example, Laplace equation and Maxwell system, the finite element method is one of the most relied on methods because it is much more interested in the analysis of the error committed between the exact solution and the approximate solution. In many of these applications, adaptive techniques using a posteriori error estimators have become an indispensable tool. These estimators allow to measure the quality of the computed solution and provide information to control the mesh adaptation algorithm. There are a lot of works on the a posteriori estimators for the elliptic partial differential equations and dynamic partial differential equations. Of these works, it is possible to refer to [1] where the authors considered an elliptic second order boundary value problem approximated by a discontinuous Galerkin method. Time dependent Stokes equations in [2] and second order wave equations in [3] are discretized by Euler’s implicit scheme in time and standard finite elements in space. Using Rothe’s method in [4] and [5], the authors studied the equation of telegraph and integrodifferential equation with integral conditions (resp.).

The purpose of this work is to combine Rothe’s method with nonconforming finite element method of Crouseix-Raviart and to introduce a posteriori error estimators suitable for the wave equation assumptions on the mesh. These indicators can give a good overview of the local distribution of the error and a useful tool for mesh adaptation.

Let Ω be a bounded open domain of \(\mathbb{R}^{d}\), \(d=2 \mbox{ or }3\) with Lipschitz boundary Γ that we suppose to be polygonal (\(d=2\)) or polyhedral (\(d=3\)). We further assume that Ω is simply connected and that Γ is connected. Let T be a fixed positive number,
$$ \begin{aligned} &\partial_{tt}u-\triangle u=f\quad \text{in }\Omega\times\, ] 0,T [, \\ &u ( x,0 ) =u_{0} ( x )\quad \text{in }\Omega\text{ at }t=0, \\ &u=0\quad \text{at }\Gamma\times I,I= ( 0,T ), \\ &\partial_{t}u ( x,0 ) =u_{1} ( x ) \quad \text{in }\Omega \text{ at }t=0, \end{aligned} $$
(1)
where \(f\in L^{2} ( ( 0,T ) ,L^{2} ( \Omega ) ) \), \(U_{0}\in H_{0}^{1} ( \Omega ) \) and \(U_{1}\in L^{2} ( \Omega ) \). Under these conditions, problem (1) is equivalent to
$$ ( \partial_{tt}u,v ) + ( \nabla u,\nabla v ) = ( f,v ) , \quad \forall v\in H_{0}^{1} ( \Omega ) \text{ a.e. }t\in ( 0,T ) $$
(2)
has a unique weak solution \(C ( ( 0,T ) ,H_{0}^{1} ( \Omega ) ) \cap C^{1} ( ( 0,T ) ,L^{2} ( \Omega ) ) \). If we put \(U=\bigl( {\scriptsize\begin{matrix}{} u \cr \partial_{t}u \end{matrix}} \bigr) \) and \(F=\bigl( {\scriptsize\begin{matrix}{} 0 \cr f\end{matrix}} \bigr) \), then problem (1) can be rewritten as follows:
$$ \left \{ \textstyle\begin{array}{l} \partial_{t}U- \bigl({\scriptsize\begin{matrix}{} 0 &1 \cr \triangle&0 \end{matrix}} \bigr) U=F\quad \text{in }\Omega\times \, ] 0,T [, \\ U=0\quad \text{on }\Gamma\times I, \\ U ( \cdot,0 ) =U_{0} \end{array}\displaystyle \right . $$
(3)
with \(U_{0}=\bigl( {\scriptsize\begin{matrix}{} u_{0} \cr u_{1} \end{matrix}} \bigr) \).

2 Time discretization using Rothe’s method

We divide the interval \(( 0,T ) \) into subintervals of length \(\tau=\frac{T}{n}\) and denote \(u^{j}=u ( j\tau,x ) \), \(x\in\Omega\), \(j=1,\ldots,n\). Successively, for \(j=1,\ldots,n\), we solve the linear stationary boundary value problem
$$ \left \{ \textstyle\begin{array}{l} \frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}}-\triangle u^{j}=f^{j}\quad \text{in }\Omega, \\ u^{j}=0\quad \text{on }\Gamma, \\ u^{0} ( x,0 ) =u_{0}\quad \text{in }\Omega, \end{array}\displaystyle \right . $$
(4)
where \(f^{j}=f ( x,t_{j} ) =f ( x,j\tau )\). Setting \(u^{-1} ( x ) =u_{0} ( x ) -\tau u_{1} ( x )\), define \(\delta u^{j}=\frac{u^{j}-u^{j-1}}{\tau}\), \(\delta^{2}u^{j}=\frac{\delta u^{j}-\delta u^{j-1}}{\tau}\), \(j=1,\ldots,n\).
This problem has a unique weak solution \(u^{j}\in H_{0}^{1} ( \Omega ) \) by the Lax-Milgram lemma whose variational formulation is
$$ \bigl( \delta^{2}u^{j},v \bigr) + \bigl( \nabla u^{j},\nabla v \bigr) = \bigl( f^{j},v \bigr),\quad \forall v \in H_{0}^{1} ( \Omega ). $$
(5)
We define Rothe’s functions by a piecewise linear interpolation with respect to the time t,
$$\begin{aligned}& u^{n} ( t,x ) =u^{j-1}+ ( t-t_{j-1} ) \delta u^{j}\quad \text{for }t_{j-1}\leq t\prec t_{j},j=1,\ldots,n, \end{aligned}$$
(6)
$$\begin{aligned}& \delta u^{n} ( t,x ) =\delta u^{j-1}+ ( t-t_{j-1} ) \delta^{2}u^{j}\quad \text{for }t_{j-1}\leq t\prec t_{j},j=1,\ldots,n, \end{aligned}$$
(7)
the auxiliary functions are
$$\begin{aligned}& \overline{u ( t,x ) ^{n}}=\left \{ \textstyle\begin{array}{l@{\quad}l} u^{j}&\text{if }t\in (t_{j-1},t_{j} ), j=1,\ldots,n, \\ u_{0}&\text{if }t\in [ -h,0 ], \end{array}\displaystyle \right . \end{aligned}$$
(8)
$$\begin{aligned}& \overline{\delta u ( t,x ) ^{n}}=\left \{ \textstyle\begin{array}{l@{\quad}l} \delta u^{j}&\text{if }t\in ( t_{j-1},t_{j} ), j=1,\ldots,n, \\ u_{1}&\text{if }t\in [ -h,0 ]. \end{array}\displaystyle \right . \end{aligned}$$
(9)

3 Full discretization

We consider the following nonconforming finite element method to approximate our problem. For all \(j=1,\ldots,n\), we consider a triangulation \(\Upsilon_{jh}\) made of triangles T if \(d=2\) and of tetrahedra if \(d=3\) whose edges/faces are denoted by e. We assume that this triangulation is regular in Ciarlet’s sense ([6], p.124), i.e. , \(\exists\sigma \succ0\) such that \(\frac{h_{T}}{\rho_{T}}\leq\sigma\), \(\forall T\in \Upsilon_{jh}\), where \(h_{T}\) is the diameter of T and \(\rho _{T}\) is the diameter of its largest inscribed bull. We define \(h_{j}=\max_{T\in\Upsilon_{jh}}h_{T}\). Let \(\zeta _{jh}^{\mathrm{int}}\) be the set of interior edges/faces of \(\Upsilon _{jh}\) and \(\zeta_{T}\) be the set of edges/faces of the element T. For an edge/face \(e\in\zeta_{T}\cap\zeta_{K}\), we denote by \(h_{e}=\frac{1}{2} ( \frac{d ( T ) }{\vert e\vert }+\frac{d ( K ) }{\vert e \vert } ) \) its mean height. Problem (5) is approximated by the Crouseix-Raviart nonconforming finite element space
$$\begin{aligned} X_{jh}^{0} =& \biggl\{ v\in L^{2} ( \Omega ) ;v \backslash T\in P_{1}, \forall T\in\Upsilon_{jh},\int _{e}v\backslash T=\int_{e}v\backslash K, \forall e\in \zeta_{T}\cap\zeta_{K}\cap \zeta_{jh}^{\mathrm{int}}, \\ &T, K\in\Upsilon_{jh}, \int_{e}v\backslash T=0, \forall e\in\zeta_{T}\cap\Gamma, T\in\Upsilon_{jh} \biggr\} . \end{aligned}$$
We consider the fully discrete scheme for problem (1): for each \(j=1,\ldots,n\), find \(u_{h}^{j}\in X_{jh}^{0}\) such that
$$ \tau^{2}\sum_{T\in\Upsilon_{jh}}\int _{T}\nabla u_{h}^{j}\nabla v_{h}=\tau^{2}\int_{\Omega}f^{j}v_{h}- \int_{\Omega} \bigl( u_{h}^{j}-2u_{h}^{j-1}+u_{h}^{j-2} \bigr) v_{h}. $$
(10)
We will use the following Crouseix-Raviart property:
$$ \int_{e} [ u_{h} ] _{e}=0,\quad \forall e\in\zeta_{jh}, \forall u_{h}\in X_{jh}^{0}, $$
(11)
where the jump of some function v across an edge/face at point x is defined by
$$ \bigl[ v ( x ) \bigr] _{e}=\left \{ \textstyle\begin{array}{l@{\quad}l} \lim_{\alpha\rightarrow0^{+}}v ( x+\alpha \eta_{e} ) -v ( x-\alpha\eta_{e} )& \text{if }e\in \zeta _{jh}^{\mathrm{int}}, \\ v ( x )& \text{if }e\in\zeta_{jh}\backslash \zeta _{jh}^{\mathrm{int}}, \end{array}\displaystyle \right . $$
\(\eta_{e}\) denotes the outward normal vector for a boundary edge/face e and \(t_{e}= ( -\eta_{e_{2}},\eta _{e_{1}} ) \) is the tangent vector if \(\eta_{e}= ( \eta _{e_{1}},\eta_{e_{2}} )\).

Since \([ v_{h} ] \) is linear on e, the condition \(\int_{e} [ v_{h} ] _{e}=0\) is equivalent to the continuity of \(v_{h} \) at e barycenter.

For \(v_{h}\in X_{jh}^{0}\), we define its broken gradient \(\nabla_{h}v\) in Ω by
$$ ( \nabla_{h}v ) \backslash T=\nabla ( v\backslash T ),\quad \forall T \in\Upsilon_{jh}. $$

We will need local subdomain, also called patches. For any \(T\in \Upsilon_{jh}\), let \(w_{T}\) be the union of all elements having a common edge/face with T. Similarly let \(w_{e}\) be the union of all elements having e as an edge/face. Finally, let \(w_{x}\) be the union of elements having x as a node, and \(\widetilde{w}_{T}\) (resp. \(\widetilde{w}_{e}\)) be the union of all triangles sharing a node with T (resp. e).

Later on, we also need the standard \(P_{1}\) conforming finite element spaces:
$$\begin{aligned}& V_{jh}= \bigl\{ v\in H^{1} ( \Omega ) ;v\backslash T\in P_{1}, \forall T\in\Upsilon _{jh} \bigr\} , \\& V_{jh}^{0}=V_{jh}\cap H_{0}^{1} ( \Omega ) . \end{aligned}$$
We further need
$$\begin{aligned}& Y_{jh}= \biggl\{ v\in L^{2} ( \Omega ) ; v\backslash T\in H^{1} ( \Omega ) ,\forall T\in \Upsilon_{jh}, \int _{e}v\backslash T=\int_{e}v\backslash K, \forall e\in\zeta_{T}\cap\zeta_{K}\cap\zeta _{jh}^{\mathrm{int}}, \\& \hphantom{Y_{jh}={}}T, K\in\Upsilon_{jh} \biggr\} , \\& Y_{jh}^{0}= \biggl\{ v\in L^{2} ( \Omega ) ; v \backslash T\in H^{1} ( \Omega ) , \forall T\in\Upsilon_{jh}, \int_{e}v\backslash T=\int_{e}v \backslash K, \forall e\in \zeta _{T}\cap\zeta_{K}\cap \zeta_{jh}^{\mathrm{int}}, \\& \hphantom{Y_{jh}^{0}={}}T, K\in\Upsilon_{jh}, \int_{e}v \backslash T=0, \forall e\in \zeta_{T}\cap\Gamma, T\in \Upsilon_{jh} \biggr\} . \end{aligned}$$

We recall that for a node \(x\in N_{jh}\), we denote by \(\lambda _{x}\) the standard hat function such that \(\lambda_{x} ( y ) =\delta_{xy}\), \(\forall y\in N_{jh}\), where \(N_{jh}\) is the set of nodes of \(\Upsilon_{jh}\) and \(N_{jh}^{\mathrm{int}}\) denotes the set of interior nodes of \(\Upsilon_{jh}\).

Definition 3.1

For \(v\in Y_{jh}\) and \(w\in Y_{jh}^{0}\), Clément interpolation is defined as follows:
$$\begin{aligned}& I_{C}v=\sum_{x\in N_{jh}}\vert w_{x}\vert ^{-1} \biggl( \int_{w_{x}}v \biggr) \lambda_{x}, \end{aligned}$$
(12)
$$\begin{aligned}& I_{C}^{0}w=\sum_{x\in N_{jh}^{\mathrm{int}}}\vert w_{x}\vert ^{-1} \biggl( \int_{w_{x}}w \biggr) \lambda_{x}. \end{aligned}$$
(13)
We define the gradient jump of \(u_{h}^{j}\) in normal and tangential direction as follows:
$$ J_{e,\eta}^{j}=\left \{ \textstyle\begin{array}{l@{\quad}l} [ \nabla u_{h}^{j}\cdot \eta_{e} ]& \text{if }e\in \zeta _{jh}^{\mathrm{int}}, \\ 0&\text{if }e\in\zeta_{jh}\backslash\zeta_{jh}^{\mathrm{int}}.\end{array}\displaystyle \right . $$
(14)
If \(d=2\), then
$$ J_{e,t}^{j}=\left \{ \textstyle\begin{array}{l@{\quad}l} [ \nabla u_{h}^{j}\cdot t_{e} ] &\text{if }e\in\zeta_{jh}^{\mathrm{int}}, \\ -\nabla u_{h}^{j}\cdot t_{e}&\text{if }e\in\zeta _{jh}\backslash \zeta_{jh}^{\mathrm{int}}. \end{array}\displaystyle \right . $$
(15)
If \(d=3\), then
$$ J_{e,t}^{j}=\left \{ \textstyle\begin{array}{l@{\quad}l} [ \nabla u_{h}^{j}\times n_{e} ]& \text{if }e\in\zeta _{jh}^{\mathrm{int}}, \\ - \nabla u_{h}^{j}\times n_{e}&\text{if }e\in\zeta _{jh}\backslash \zeta_{jh}^{\mathrm{int}}. \end{array}\displaystyle \right . $$

Lemma 3.1

[7]

For all \(v\in Y_{jh}\) and \(w\in Y_{jh}^{0}\), we have
$$\begin{aligned}& \Vert v-I_{C}v\Vert _{T}\lesssim h_{T}\Vert \nabla _{h}v\Vert _{\widetilde{w}_{T}},\quad \forall T\in \Upsilon_{jh}, \end{aligned}$$
(16)
$$\begin{aligned}& \Vert v-I_{C}v\Vert _{e}\lesssim h_{e}^{\frac{1}{2}}\Vert \nabla_{h}v\Vert _{\widetilde{w}_{e}},\quad \forall e\in\zeta_{jh}, \end{aligned}$$
(17)
$$\begin{aligned}& \bigl\Vert w-I_{C}^{0}w\bigr\Vert _{T}\lesssim h_{T}\Vert \nabla_{h}w\Vert _{\widetilde{w}_{T}},\quad \forall T\in \Upsilon_{jh}, \end{aligned}$$
(18)
$$\begin{aligned}& \bigl\Vert w-I_{C}^{0}w\bigr\Vert _{e}\lesssim h_{e}^{\frac{1}{2} }\Vert \nabla_{h}w\Vert _{\widetilde{w}_{e}}, \quad \forall e\in \zeta_{jh}^{\mathrm{int}}, \end{aligned}$$
(19)
$$\begin{aligned}& \bigl\Vert \nabla I_{C}^{0}w\bigr\Vert \lesssim \Vert \nabla _{h}w\Vert _{\widetilde{w}_{T}}, \quad \forall T\in \Upsilon_{jh}. \end{aligned}$$
(20)

Next, we need the following Green’s formulas.

If D is open bounded of \(\mathbb{R}^{2}\) and \(v, w\in H^{1} ( D ) \), then
$$ \int_{D}\nabla v\operatorname{curl}w=\int _{\partial D}v\operatorname{curl}w\cdot \eta =\int _{\partial D}\nabla v\cdot tw, $$
(21)
where t is the unit tangent vector along ∂D and \(\operatorname{curl} w=\bigl( {\scriptsize\begin{matrix}{} \partial_{2}w \cr -\partial_{1}w \end{matrix}} \bigr)\).
Similarly if D is open bounded of \(\mathbb{R}^{3}\) and \(v\in H^{1} ( D ) \), \(w\in H^{1} ( D ) ^{3}\), then we have
$$ \int_{D}\nabla v\operatorname{curl}w=\int _{\partial D}v\operatorname{curl}w\cdot \eta =\int _{\partial D} ( \nabla v\times\eta ) \cdot w. $$
(22)

4 A posteriori analysis of time discretization

For each j, \(j=1,\ldots,n\), the refinement indicator is defined by
$$ \eta_{t}^{j}=\tau\bigl\Vert \nabla_{h} \bigl( u_{h}^{j+1}-u_{h}^{j} \bigr) \bigr\Vert +\tau\biggl\Vert \frac {u_{h}^{j+1}-2u_{h}^{j}+u_{h}^{j-1}}{\tau}\biggr\Vert , $$
(23)
\(e^{\tau}=u-u^{n}\) indicate the error with respect to the discretization time.

Proposition 4.1

(Upper and lower bounds of the error in time)

The following a posteriori error estimate holds for all \(t_{j+1}\), \(j=1,\ldots,n-1\):
$$\begin{aligned}& \biggl\Vert ( \partial_{t}u ) ( t_{j+1} ) - \frac{ u^{j+1}-u^{j}}{\tau}\biggr\Vert _{H^{-1} ( \Omega ) }+\bigl\Vert u ( t_{j+1} ) -u^{j+1}\bigr\Vert \\& \quad \lesssim \sum_{m=1}^{j} \eta_{t}^{m}+\tau \Vert \nabla U_{0}\Vert + \tau^{2}\Vert \nabla U_{1}\Vert +\sum _{m=1}^{j}\tau\sum_{k=0}^{1} \bigl\Vert \nabla \bigl( u^{m+1-k}-u_{h}^{m+1-k} \bigr) \bigr\Vert \\& \qquad {}+\sum_{m=1}^{j}\tau\sum _{k=0}^{1}\bigl\Vert \delta u^{m+1-k}-\delta_{h}^{m+1-k}\bigr\Vert , \end{aligned}$$
(24)
$$\begin{aligned}& \eta_{t}^{j} \leq\sum_{k=0}^{1} \bigl\Vert ( \partial _{t}u ) ( t_{j+1-k} ) -\delta u^{j+1-k}\bigr\Vert _{H^{-1} ( \Omega ) }+\bigl\Vert u ( t_{j+1} ) -\overline{u^{j+1-k}}\bigr\Vert +\biggl\Vert \int_{t_{j}}^{tj+1} \nabla \bigl( u-u^{n} \bigr) ( s ) \, ds\biggr\Vert \\& \hphantom{\eta_{t}^{j} \leq{}}{}+\biggl\Vert \int_{t_{j}}^{tj+1} \bigl( \partial_{t}u-\delta u^{n} \bigr) ( s )\, ds\biggr\Vert +\tau \sum_{k=0}^{1}\bigl\Vert \nabla \bigl( u^{j+1-k}-u_{h}^{j+1-k} \bigr) \bigr\Vert \\& \hphantom{\eta_{t}^{j} \leq{}}{}+\tau \sum_{k=0}^{1}\bigl\Vert \delta u^{j+1-k}-\delta _{h}u_{h}^{j+1-k} \bigr\Vert . \end{aligned}$$
(25)

Proof

See [3]. □

5 A posteriori analysis of space discretization

The error indicator is defined by
$$ \eta_{T}^{j}=h_{T}\biggl\Vert f_{h}^{j}-\frac{ u_{h}^{j}-2u_{h}^{j-1}+u_{h}^{j-2}}{\tau^{2}}\biggr\Vert +\sum _{e\in\zeta_{jh}}h_{e}^{\frac{1}{2}} \bigl( \bigl\Vert J_{e,\eta}^{j}\bigr\Vert _{e}+\bigl\Vert J_{e,t}^{j}\bigr\Vert _{e} \bigr), $$
the global error estimator \(\eta^{j}\) is given by
$$ \eta^{j}=\sqrt{\sum_{T\in\Upsilon_{jh}} \bigl( \eta _{T}^{j} \bigr) ^{2}}, $$
the higher order term depending on the datum f is defined as
$$ \operatorname{osc} ( f,w_{T} ) ^{2}=\sum _{T\in\Upsilon _{jh}}h_{T}^{2}\bigl\Vert f-f_{h}^{j}\bigr\Vert _{w_{T}}^{2},\quad \text{where } \bigl( f_{h}^{j} \bigr) \backslash T:= \frac{1}{\vert T\vert }\int_{T}f_{h}^{j}, T \in\Upsilon_{jh}. $$

Our main result is the following theorem.

Theorem 5.1

(Upper bound)

The following inequality holds:
$$\begin{aligned} \bigl\Vert e^{n}\bigr\Vert ^{2}+\sum _{j=1}^{n}\tau^{2}\bigl\Vert \nabla_{h}e^{j}\bigr\Vert ^{2} \lesssim&\bigl\Vert e^{0}\bigr\Vert ^{2}+\bigl\Vert e^{1} \bigr\Vert ^{2}+\tau ^{2}\sum_{j=1}^{n} \operatorname{osc} ( f,w_{T} ) ^{2} \\ &{}+\sum _{j=1}^{n}\max \bigl\{ h_{j}^{2}, \tau^{2} \bigr\} \bigl( \eta^{j} \bigr) ^{2}. \end{aligned}$$

To prove this theorem we need some lemmas. As our approximated scheme is a nonconforming one, we need to use an appropriate Helmholtz decomposition of the error.

Lemma 5.1

(Helmholtz decomposition of the error)

Let \(e^{j}=u^{j}-u_{h}^{j}\), then we have the following decomposition of error \(e^{j}\):
$$ \nabla_{h}e^{j}=\nabla\varphi^{j}+ \operatorname{curl}\chi^{j} $$
(26)
with \(\chi^{j}\in H^{1} ( \Omega ) \) and \(\varphi^{j}\in H_{0}^{1} ( \Omega ) \); furthermore, the following inequalities hold:
$$\begin{aligned}& \bigl\Vert \nabla\varphi^{j}\bigr\Vert \leq\bigl\Vert \nabla _{h}e^{j}\bigr\Vert , \end{aligned}$$
(27)
$$\begin{aligned}& \bigl\Vert \operatorname{curl}\chi^{j}\bigr\Vert \leq\bigl\Vert \nabla _{h}e^{j}\bigr\Vert . \end{aligned}$$
(28)

Proof

We consider the following problem: find \(\varphi ^{j}\in H_{0}^{1} ( \Omega ) \), a solution of
$$ \left \{ \textstyle\begin{array}{l} \operatorname{div} ( \nabla_{h}e^{j}-\nabla\varphi^{j} ) =0\quad \text{in }\Omega, \\ \varphi^{j}=0\quad \text{on }\Gamma. \end{array}\displaystyle \right . $$
(29)
The weak formulation of that problem (29) is
$$ \int_{\Omega}\nabla\varphi^{j}\nabla v= \int_{\Omega}\nabla _{h}e^{j}\nabla v ,\quad \forall v\in H_{0}^{1} ( \Omega ). $$
(30)
As the vector field \(( \nabla_{h}e^{j}-\nabla\varphi ^{j} ) \) is divergence-free in Ω, i.e., \(\operatorname{div} ( \nabla_{h}e^{j}-\nabla\varphi^{j} ) =0\) in Ω, by Theorem 1.3.1 of [8] if \(d=2\) or Theorem 1.3.4 of [8] if \(d=3\), there exists \(\chi^{j}\in H^{1} ( \Omega ) \) if \(d=2\) and \(\chi ^{j}\in H^{1} ( \Omega ) ^{3}\) if \(d=3\) such that
$$ \operatorname{curl}\chi^{j}=\nabla_{h}e^{j}- \nabla\varphi^{j}. $$
Estimate (27) directly follows by taking \(v=\varphi^{j}\) in (30). To prove the inequality, we use identity (26) and we get
$$ \int_{\Omega}\bigl\vert \operatorname{curl} \chi^{j}\bigr\vert ^{2}=\int_{\Omega}\operatorname{curl}\chi^{j} \bigl( \nabla_{h}e^{j}- \nabla\varphi ^{j} \bigr). $$
(31)
Using Green’s formula and taking into account that \(\varphi^{j}=0\) on Γ, we get
$$ \int_{\Omega}\bigl\vert \operatorname{curl}\chi^{j} \bigr\vert ^{2}=\int_{\Omega} \operatorname{curl} \chi^{j}\nabla_{h}e^{j}. $$
The Cauchy-Schwarz inequality implies
$$ \bigl\Vert \operatorname{curl}\chi^{j}\bigr\Vert \leq\bigl\Vert \nabla _{h}e^{j}\bigr\Vert . $$
 □

Lemma 5.2

The error satisfies the following identity:
$$ \sum_{T\in\Upsilon_{jh}}\int_{T} \nabla_{h}e^{j}\nabla v_{h}=\int _{\Omega}\frac{-e^{j}+2e^{j-1}-e^{j-2}}{\tau^{2}}v_{h},\quad \forall v_{h}\in V_{jh}^{0} ( \Omega ). $$
(32)

Proof

We only need to take \(v=v_{h}\) in (5), then taking the difference between (5) and (10) we get the result. □

Lemma 5.3

Let \(\varphi_{h}\in V_{jh}\) if \(d=2\) and \(\varphi_{h}\in ( V_{jh} ) ^{3}\) if \(d=3\), then the error verifies
$$ \sum_{T\in\Upsilon_{jh}}\int_{T} \nabla _{h}e^{j}\operatorname{curl}\varphi_{h}=0. $$
(33)

Proof

We integrate by parts the expression \(\int_{\Omega}\nabla_{h}e^{j}\operatorname{curl}\varphi_{h}\), using Green’s formula and taking into account that \(u^{j}\in H_{0}^{1} ( \Omega ) \), then we use the property of finite elements of Crouseix-Raviart (\(\int_{e} [ u_{h}^{j} ] =0 \)) and get (33). □

Lemma 5.4

Let \(\varphi\in H^{1} ( \Omega ) \) if \(d=2\) and \(\varphi\in ( H^{1} ( \Omega ) ) ^{3}\) if \(d=3\), then we have
$$ \int_{\Omega}\nabla_{h}e^{j} \operatorname{curl}\varphi=\sum_{e\in\zeta _{jh}}\int _{E}J_{E,t}^{j}\cdot\varphi. $$
(34)

Proof

The integration by parts and Green’s formula give us
$$ \int_{\Omega}\nabla_{h}e^{j} \operatorname{curl}\varphi=\int_{\Omega}\nabla u^{j}\operatorname{curl}\varphi-\sum _{T\in\Upsilon_{jh}}\int_{T}\nabla u_{h}^{j}\operatorname{curl}\varphi_{h}=-\sum_{T\in\Upsilon_{jh}} \int_{\partial T}\nabla u_{h}^{j}\cdot t_{T}\varphi, $$
because \(u^{j}\in H_{0}^{1} ( \Omega )\), and according to the definition of \(J_{E,t}^{j}\) we find (34). □

Lemma 5.5

Let \(\varphi\in H_{0}^{1} ( \Omega ) \), then \(e^{j}\) verifies
$$ \sum_{T\in\Upsilon_{jh}}\int_{T} \nabla_{h}e^{j}\nabla\varphi =\sum _{T\in \Upsilon_{jh}}\int_{T} \biggl( f^{j}- \frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau ^{2}} \biggr) \varphi+\sum_{e\in\zeta_{jh}} \int_{e}J_{e,\eta }^{j}\cdot\varphi. $$

Proof

We integrate by parts the expression \(\sum_{T\in \Upsilon_{jh}}\int_{T}\nabla_{h}e^{j}\nabla\varphi\) with \(\triangle u=0\) on each element \(T\in\Upsilon_{jh}\), and from the definition of \(J_{e,\eta}^{j}\) we conclude the proof. □

Remark 5.1

Lemmas 5.4, 5.5 imply that \(\forall \varphi\in H_{0}^{1} ( \Omega ) \) and \(\chi\in H^{1} ( \Omega ) \) if \(d=2\) and \(\chi \in ( H^{1} ( \Omega ) ) ^{3}\) if \(d=3\), we have
$$\begin{aligned} \sum_{T\in\Upsilon_{jh}}\int_{T} \nabla_{h}e^{j} ( \nabla\varphi + \operatorname{curl}\chi ) =&\sum_{T\in\Upsilon_{jh}}\int_{T} \biggl( f^{j}-\frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}} \biggr) \varphi \\ &{}+\sum_{e\in\zeta_{jh}}\int_{e} \bigl( J_{e,\eta}^{j}\cdot\varphi +J_{\eta,t}^{j}\cdot \chi \bigr). \end{aligned}$$

Note that the local error estimator \(\eta_{T}^{j}\) is inspired by the latter identity.

Proof

From what precedes (Lemmas 5.2 and 5.5), we can easily prove that
$$\begin{aligned} \tau^{2}\int_{\Omega} \nabla_{h}e^{j}\nabla\varphi^{j} =&\tau ^{2}\int_{\Omega} \biggl( f^{j}- \frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}} \biggr) \bigl( \varphi^{j}-I_{C}^{0} \varphi^{j} \bigr) \\ &{}+\tau^{2}\sum_{e\in\zeta_{jh}}\int _{E}J_{e,\eta}^{j} \bigl( \varphi^{j}-I_{C}^{0}\varphi^{j} \bigr) - \int_{\Omega} \bigl( e^{j}-2e^{j-1}+e^{j-2} \bigr) \varphi^{j}. \end{aligned}$$
(35)
From Lemmas 5.3 and 5.4 we get
$$ \int_{\Omega}\nabla_{h}e^{j} \operatorname{curl}\chi^{j}=\sum_{E\in \zeta_{jh}}\int _{E}J_{E,t}^{j} \bigl( \chi^{j}-I_{C}\chi ^{j} \bigr). $$
(36)
By using the Helmholtz decomposition of the error and identities (35)-(36), we obtain
$$\begin{aligned} \bigl\Vert e^{j}\bigr\Vert +\tau^{2}\int _{\Omega}\bigl\vert \nabla_{h}e^{j}\bigr\vert ^{2} =&\tau^{2}\int_{\Omega}\nabla _{h}e^{j}\nabla I_{C}^{0} \bigl( e^{j}-\varphi^{j} \bigr) + \bigl( -e^{j}+2e^{j-1}-e^{j-2},e^{j}- \varphi^{j}-I_{C}^{0} \bigl( e^{j}- \varphi ^{j} \bigr) \bigr) \\ &{}+\tau^{2}\int_{\Omega} \biggl( f^{j}- \frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}} \biggr) \bigl( \varphi^{j}-I_{C}^{0} \varphi^{j} \bigr) + \bigl( 2e^{j-1}-e^{j-2},e^{j} \bigr) \\ &{}+\tau^{2}\sum_{e\in\zeta_{jh}}\int _{e} \bigl( J_{e,\eta }^{j} \bigl( \varphi^{j}-I_{C}^{0}\varphi^{j} \bigr) +J_{e,t}^{j} \bigl( \chi^{j}-I_{C} \chi^{j} \bigr) \bigr). \end{aligned}$$
(37)
The Cauchy-Schwarz inequality and estimate (19) give
$$\begin{aligned} \sum_{e\in\zeta_{jh}}\int_{e}J_{e,\eta}^{j} \bigl( \varphi ^{j}-I_{C}\varphi^{j} \bigr) \leq& \sum_{e\in\zeta _{jh}}\bigl\Vert J_{e,\eta}^{j} \bigr\Vert \bigl\Vert \varphi ^{j}-I_{C} \varphi^{j}\bigr\Vert _{e}\lesssim\sum _{e\in\zeta _{jh}}\bigl\Vert J_{e,\eta}^{j}\bigr\Vert h_{e}^{\frac{1}{2}}\bigl\Vert \nabla\varphi ^{j}\bigr\Vert _{\widetilde{w}_{e}} \\ \lesssim&\sum_{e\in\zeta_{jh}}\eta_{T}^{j} \bigl\Vert \nabla \varphi^{j}\bigr\Vert _{\widetilde{w}_{e}}\lesssim\eta ^{j}\bigl\Vert \nabla\varphi^{j}\bigr\Vert . \end{aligned}$$
(38)
Similarly, using the Cauchy-Schwarz inequality and estimate (17), we get
$$ \sum_{e\in\zeta_{jh}}\int_{E}J_{E,t}^{j} \bigl( \chi ^{j}-I_{C}\chi^{j} \bigr) \lesssim \eta^{j}\bigl\Vert \nabla \chi^{j}\bigr\Vert . $$
(39)
By using the Helmholtz decomposition, Green’s formula and identity (36), we find
$$\begin{aligned} \int_{\Omega}\bigl\vert \operatorname{curl} \chi^{j}\bigr\vert ^{2} =&\int_{\Omega} \bigl( \nabla_{h}e^{j}-\nabla_{h} \varphi^{j} \bigr) \operatorname{curl}\chi^{j}=\int _{\Omega}\nabla_{h}e^{j}\operatorname{curl} \chi ^{j} \\ =&\sum_{e\in\zeta_{jh}}\int_{e}J_{e,t}^{j} \bigl( \chi ^{j}-I_{C}\chi^{j} \bigr) \lesssim \eta^{j}\bigl\Vert \nabla \chi^{j}\bigr\Vert . \end{aligned}$$
(40)
The Cauchy-Schwarz inequality and estimate (17) imply
$$ \int_{\Omega}\bigl\vert \operatorname{curl}\chi^{j} \bigr\vert ^{2}\lesssim\eta^{j}\bigl\Vert \nabla \chi^{j}\bigr\Vert . $$
Knowing that
$$ \bigl\Vert \operatorname{curl}\chi^{j}\bigr\Vert =\bigl\Vert \nabla\chi ^{j}\bigr\Vert , $$
we get
$$ \bigl\Vert \nabla\chi^{j}\bigr\Vert \lesssim\eta^{j}, $$
and consequently
$$ \sum_{e\in\zeta_{jh}}\int_{e}J_{e,t}^{j} \bigl( \chi ^{j}-I_{C}\chi^{j} \bigr) \lesssim \bigl( \eta^{j} \bigr) ^{2}. $$
(41)
On the other hand, the Cauchy-Schwarz inequality and estimate (18) give
$$\begin{aligned} \sum_{T\in\Upsilon_{jh}}\int_{T} \bigl( f^{j}-f_{h}^{j} \bigr) \bigl( \varphi^{j}-I_{C}^{0}\varphi^{j} \bigr) \lesssim& \sum_{T\in\Upsilon_{jh}}h_{T}\bigl\Vert f^{j}-f_{h}^{j}\bigr\Vert \bigl\Vert \nabla \varphi ^{j}\bigr\Vert _{\widetilde{w}_{T}} \\ \lesssim& \operatorname{osc} ( f,_{\widetilde {w}_{T}} ) \Vert \nabla\varphi \Vert . \end{aligned}$$
(42)
Similarly, we have
$$ \bigl\vert \bigl( -e^{j}+2e^{j-1}-e^{j-2},\varphi ^{j}-e^{j}-I_{C}^{0} \bigl( \varphi^{j}-e^{j} \bigr) \bigr) \bigr\vert \lesssim h_{T}\bigl\Vert e^{j}-2e^{j-1}+e^{j-2} \bigr\Vert \bigl\Vert \nabla \bigl( e^{j}-\varphi^{j} \bigr) \bigr\Vert . $$
Proceeding as in (40) we can prove that
$$ \bigl\Vert \nabla \bigl( e^{j}-\varphi^{j} \bigr) \bigr\Vert \lesssim\eta^{j}, $$
which implies that
$$ \bigl\vert \bigl( -e^{j}+2e^{j-1}-e^{j-2}, \varphi ^{j}-e^{j}-I_{C}^{0} \bigl( \varphi^{j}-e^{j} \bigr) \bigr) \bigr\vert \lesssim h_{T}\eta^{j}\bigl\Vert e^{j}-2e^{j-1}+e^{j-2} \bigr\Vert . $$
(43)
To estimate \(\Vert \nabla\varphi^{j}\Vert \), we have
$$ \bigl\Vert \nabla\varphi^{j}\bigr\Vert \leq\bigl\Vert \nabla _{h} \bigl( \varphi^{j}-e^{j} \bigr) \bigr\Vert +\bigl\Vert \nabla_{h}e^{j}\bigr\Vert \leq C \bigl( \eta^{j} \bigr) ^{2}+\bigl\Vert \nabla_{h}e^{j}\bigr\Vert ^{2}. $$
(44)
For the residual element, using the Cauchy-Schwarz inequality and estimation (18), we get
$$\begin{aligned} \sum_{T\in\Upsilon_{jh}}\int_{T} \biggl( f_{h}^{j}-\frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}} \biggr) \bigl( \varphi ^{j}-I_{C}^{0}\varphi^{j} \bigr) \lesssim&\sum_{T\in\Upsilon _{jh}}h_{T}\biggl\Vert f_{h}^{j}-\frac{u^{j}-2u^{j-1}+u^{j-2}}{\tau^{2}} \biggr\Vert \bigl\Vert \nabla\varphi^{j}\bigr\Vert _{\widetilde{w}_{T}} \\ \lesssim&\eta^{j}\bigl\Vert \nabla\varphi^{j}\bigr\Vert . \end{aligned}$$
(45)
Using the ϵ-inequality and replacing the previous estimates in (37), we find
$$ \bigl\Vert e^{j}\bigr\Vert ^{2}+\tau^{2}\int _{\Omega}\bigl\vert \nabla _{h}e^{j}\bigr\vert ^{2}\leq\frac{1}{2}\bigl\Vert e^{j-1}\bigr\Vert ^{2}+\frac{1}{2}\bigl\Vert e^{j-2} \bigr\Vert ^{2}+C \bigl( \max \bigl\{ h_{j}^{2}, \tau^{2} \bigr\} \bigl( \eta^{j} \bigr) ^{2}+\tau ^{2}\cdot\operatorname{osc} ( f,_{\widetilde{w}_{T}} ) ^{2} \bigr). $$
Summing from \(j=2\) until n results in
$$\begin{aligned} \bigl\Vert e^{n}\bigr\Vert ^{2}+\sum _{j=1}^{n}\tau ^{2}\int _{\Omega}\bigl\vert \nabla_{h}e^{j}\bigr\vert ^{2} \lesssim&\bigl\Vert e^{0}\bigr\Vert ^{2}+\bigl\Vert e^{1}\bigr\Vert ^{2}+\sum _{j=1}^{n}\max \bigl\{ h_{j}^{2}, \tau ^{2} \bigr\} \bigl( \eta^{j} \bigr) ^{2} \\ &{}+\sum _{j=1}^{n}\tau ^{2}\cdot \operatorname{osc} ( f,_{\widetilde{w}_{T}} ) ^{2}. \end{aligned}$$
 □

Theorem 5.2

(Lower bound of the error) [9, 10]

For each element \(T\in\Upsilon_{jh}\), \(j=2,\ldots,n\), the following estimate holds:
$$ \eta_{T}^{j}\lesssim h_{T}\biggl\Vert \frac {u_{h}^{j}-2u_{h}^{j-1}+u^{j-2}}{\tau^{2}}\biggr\Vert +\bigl\Vert \nabla_{h}e^{j} \bigr\Vert _{w_{T}}+h_{T}\bigl\Vert f^{j}-f_{h}^{j} \bigr\Vert _{w_{T}}. $$

Declarations

Acknowledgements

The authors are grateful for the referees’ careful reading and comments on this paper that led to the improvement of the original manuscript. This work is supported by the MESRS of Algeria (CNEPRU Project B01520130004).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University 8 Mai 1945 Guelma
(2)
Laboratory of Applied Mathematics and Modeling, University 8 Mai 1945 Guelma
(3)
Advanced Materials Laboratory, University of Badji Mokhtar Annaba

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© Chaoui et al. 2015