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Full discretization of wave equation
Boundary Value Problems volume 2015, Article number: 133 (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.
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,
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
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:
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
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
We define Rothe’s functions by a piecewise linear interpolation with respect to the time t,
the auxiliary functions are
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
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
We will use the following Crouseix-Raviart property:
where the jump of some function v across an edge/face at point x is defined by
\(\eta_{e}\) 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
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:
We further need
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:
We define the gradient jump of \(u_{h}^{j}\) in normal and tangential direction as follows:
If \(d=2\), then
If \(d=3\), then
Lemma 3.1
[7]
For all \(v\in Y_{jh}\) and \(w\in Y_{jh}^{0}\), we have
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
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
4 A posteriori analysis of time discretization
For each j, \(j=1,\ldots,n\), the refinement indicator is defined by
\(e^{\tau}=u-u^{n}\) 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 \(t_{j+1}\), \(j=1,\ldots,n-1\):
Proof
See [3]. □
5 A posteriori analysis of space discretization
The error indicator is defined by
the global error estimator \(\eta^{j}\) is given by
the higher order term depending on the datum f is defined as
Our main result is the following theorem.
Theorem 5.1
(Upper bound)
The following inequality holds:
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 \(e^{j}\):
with \(\chi^{j}\in H^{1} ( \Omega ) \) and \(\varphi^{j}\in H_{0}^{1} ( \Omega ) \); furthermore, the following inequalities hold:
Proof
We consider the following problem: find \(\varphi ^{j}\in H_{0}^{1} ( \Omega ) \), a solution of
The weak formulation of that problem (29) is
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
Estimate (27) directly follows by taking \(v=\varphi^{j}\) in (30). To prove the inequality, we use identity (26) and we get
Using Green’s formula and taking into account that \(\varphi^{j}=0\) on Γ, we get
The Cauchy-Schwarz inequality implies
□
Lemma 5.2
The error satisfies the following identity:
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
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
Proof
The integration by parts and Green’s formula give us
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
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
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
From Lemmas 5.3 and 5.4 we get
By using the Helmholtz decomposition of the error and identities (35)-(36), we obtain
The Cauchy-Schwarz inequality and estimate (19) give
Similarly, using the Cauchy-Schwarz inequality and estimate (17), we get
By using the Helmholtz decomposition, Green’s formula and identity (36), we find
The Cauchy-Schwarz inequality and estimate (17) imply
Knowing that
we get
and consequently
On the other hand, the Cauchy-Schwarz inequality and estimate (18) give
Similarly, we have
Proceeding as in (40) we can prove that
which implies that
To estimate \(\Vert \nabla\varphi^{j}\Vert \), we have
For the residual element, using the Cauchy-Schwarz inequality and estimation (18), we get
Using the ϵ-inequality and replacing the previous estimates in (37), we find
Summing from \(j=2\) until n results in
□
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:
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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).
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Chaoui, A., Ellaggoune, F. & Guezane-Lakoud, A. Full discretization of wave equation. Bound Value Probl 2015, 133 (2015). https://doi.org/10.1186/s13661-015-0396-3
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DOI: https://doi.org/10.1186/s13661-015-0396-3