Abstract
This paper deals with the numerical analysis of parabolic variational inequalities with nonlinear source terms, where the existence and uniqueness of the solution is provided by using Banach’s fixed point theorem. In addition, an optimally \(L^{\infty}\)-asymptotic behavior is proved using Euler time scheme combined with the finite element spatial approximation. The approach is based on Bensoussan–Lions algorithm for evolutionary free boundary problems using the concepts of subsolutions.
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1 Introduction
We consider the following parabolic variational inequality: Find \(u\in L^{2} ( 0,T;H_{0}^{1} ( \varOmega ) ) \) such that
where Ω is a bounded smooth and regular domain of \(\mathbb{R} ^{d}\), \(d\geq1\), with smooth boundary ∂Ω; the \(f (\cdot) \) and \(u_{0}=u_{0} ( x ) \) are given data; the ψ is a regular function in \(L^{2} ( 0,T;W^{2,\infty} ( \varOmega ) )\), and the \(\mathcal{L}\) is a second-order, uniformly elliptic operator of the form
Parabolic variational inequality (1.1) has arisen from many scientific, engineering, and economic problems such as heat control problem, Stefan problem, and American option problem (see [3, 5–13, 16, 18, 19, 21]).
In this paper, we give an \(L^{\infty}\)-error estimate for the numerical approximation of the solution of problem (1.1). From [2] (see also [8]), we know that (1.1) can be approximated by the following parabolic variational inequality with nonlinear source terms (PVI): Find \(u ( x,t ) \) such that \(u\in L^{2} ( 0,T;H_{0}^{1} ( \varOmega ) ) \), \(\frac{\partial u}{\partial t}\in L^{2} ( 0,T;L^{2} ( \varOmega ) ) \), and
where \(a ( \cdot,\cdot ) \) is a bilinear form continuous on \(H^{1} ( \varOmega ) \times H^{1} ( \varOmega ) \) corresponding to elliptic operator \(\mathcal{L}\) of second order defined as follows:
with \(a_{jk}(\cdot)\), \(b_{j}(\cdot)\), \(a_{0}(\cdot)\), smooth coefficients satisfying the following conditions:
and for each \(\xi\in \mathbb{R} ^{d}\) and for almost every \(x\in\varOmega\),
According to Theorem 2.3 in [8], there exists \(\gamma>0\) such that
The function \(f (\cdot) \) is a nondecreasing and Lipschitz continuous nonlinearity
with Lipschitz constant \(\alpha>0\), satisfying the following assumption:
where β is the constant defined in (1.3). The symbol \((\cdot,\cdot ) \) is the inner product in \(L^{2} ( \varOmega ) \).
Error estimates for piecewise linear finite element approximations of parabolic variational inequalities with linear source terms have been established in various papers: in [20] and [9] an \(L^{2}\)-error estimate is given by using a backward differencing in time. Also an \(L^{2}\)-error estimate is given in [23] by using a general finite difference discretization in time. Reference [4] gives an \(L^{2}\)-error estimate using the discretized truncation method. In [1] and [22] a posteriori error estimates have been proved. An \(L^{\infty}\)-error estimate has been proved in [15] and [17]. Recently an \(L^{\infty}\)-asymptotic behavior has been considered in [2] by using a semi-implicit time scheme combined with the finite element spatial approximation.
In this paper, we introduce a new approach to derive optimal \(L^{\infty}\)-asymptotic behavior for parabolic variational inequality with nonlinear source terms. This approach is based on Bensoussan–Lions algorithm for evolutionary free boundary problems using the concepts of subsolutions.
The paper is organized as follows. In Sect. 2, we state the continuous problem and study some qualitative properties. In Sect. 3, we consider the discrete problem and set up analogous discrete qualitative properties. In Sect. 4, we derive an \(L^{\infty}\)-error estimate of the approximation and we give the main result of the paper.
2 Semi continuous problem
2.1 Time discretization
In order to obtain a full discretization of (1.3), we consider a uniform mesh for the time variable t and define
\(\Delta t>0\) being the time-step, and \(\mathcal{N}= [ \frac{T}{\Delta t} ] \), the integral part of \(\frac {T}{\Delta t}\).
Next, we replace the time derivative by means of suitable difference quotients, thus constructing a sequence \(u^{n} ( x ) \in H_{0}^{1} ( \varOmega ) \) that approaches \(u ( t_{n},x ) \). For simplicity, we confine ourselves to the so-called semi-implicit scheme, which consists in replacing (1.3) by the following scheme: Find \(u^{n}\in H_{0}^{1} ( \varOmega ) \) such that
where
By adding \(( \frac{u^{n-1}}{\Delta t},v-u^{n} ) \) to both parties of inequalities (2.2), we get
As the bilinear form \(a ( \cdot,\cdot ) \) is noncoercive in \(H_{0}^{1} ( \varOmega ) \).
Set
Then the bilinear form \(b ( u,v ) \) is an elliptic, and therefore (2.4) can be written as the following coercive elliptic variational inequalities: Find \(u^{n}\in H_{0}^{1} ( \varOmega ) \) such that
where
Remark 1
Equation (2.6) is called the coercive continuous problem of elliptic variational inequalities (VI).
Notation 1
We denote by \(u^{n}=\partial ( f ( u^{n} ) ,\psi ) \) the solution of problem (2.6).
2.2 Existence and uniqueness
Next, using the preceding assumptions, we prove the existence of a unique solution for problem (2.6) by means of Banach’s fixed point theorem.
2.2.1 A fixed point mapping associated with continuous problem (2.6)
We consider the following mapping:
where \(\xi^{n}=\sigma ( f ( w ) ,\psi ) \) is the solution to the following variational inequalities:
Problem (2.10) being a coercive VI, thanks to [3] and [10], has one and only one solution.
Theorem 1
Under the preceding hypotheses and notation, the mapping\(\mathbb{T}\)is a contraction in\(L^{\infty} ( \varOmega ) \)with a contraction constant\(( \frac{\alpha\Delta t+1}{\beta\Delta t+1} ) \). Therefore, \(\mathbb{T}\)admits a unique fixed point which coincides with the solution of problem (2.6).
Proof
In [13], by taking \(\lambda=\frac{1}{\Delta t}\), we can easily get
□
The mapping \(\mathbb{T}\) clearly generates the following continuous algorithm.
2.3 A continuous iterative scheme
A continuous iterative scheme for the solution of problem (2.6) is given as follows.
Starting from \(u^{0}=u_{0}\) the solution of the following equation:
Now, we give the following algorithm:
where \(u^{n}\) is the solution to (2.6).
Making use of the propriety of mapping \(\mathbb{T}\), we have the following geometric convergence result.
Proposition 1
Let\(\rho=\frac{\alpha\Delta t+1}{\beta\Delta t+1}\), under conditions of Theorem 1, we have
where\(u^{\infty}\)is the asymptotic solution of the problem of variational inequalities: Find\(u^{\infty}\in H_{0}^{1} ( \varOmega ) \)such that
Proof
We adapt [2]. □
In what follows, we give some qualitative properties of the solution of problem (2.6).
2.4 Some qualitative properties of the solution of (2.6)
The solution \(u^{n}\) of (2.6) possesses the following properties.
2.4.1 A monotonicity property
Let \(u^{n}=\partial ( F ( u^{n} ) ,\psi ) \) (resp. \(\tilde{u}^{n}=\partial ( \tilde{F} ( \tilde{u} ^{n} ) ,\tilde{\psi} ) \)) be the solution of problem (2.6) with right-hand side \(F ( u^{n} ) =f ( u^{n} ) +\lambda u^{n-1}\) (resp. \(\tilde{F} ( \tilde{u} ^{n} ) =\tilde{f} ( \tilde{u}^{n} ) +\lambda\tilde{u} ^{n-1}\)). Then we have the following.
Lemma 1
If\(F ( u^{n} ) \geq\tilde{F} ( \tilde{u} ^{n} ) \)and\(\psi\geq\tilde{\psi}\), then
2.4.2 A continuous \(L^{\infty}\)-stability property
Proposition 2
Under conditions of Lemma1, we have
Proof
Let
Then, from (1.5), it is easy to see that
So, due to Lemma 1, it follows that
hence
Interchanging the role of \(F ( u^{n} ) \) and \(\tilde{F}^{n}\), we also get
Then, from (2.8), it is easy to see that
which completes the proof. □
2.4.3 The concept of continuous subsolution property
Definition 1
\(z^{n}\in H_{0}^{1} ( \varOmega ) \) is said to be a continuous subsolution for the problem of VI (2.6) if
Theorem 2
(cf. [6])
Let\(\mathbb{X}\)denote the set of such subsolutions, then the solution of (2.6) is the least upper bound of\(\mathbb{X}\).
3 The discrete problem
Let Ω be decomposed into triangles, and let \(\tau_{h}\) denote the set of all those elements; \(h>0\) is the mesh size. We assume that the family \(\tau_{h}\) is regular and quasi-uniform. We consider \(\phi_{l}\), \(l=1,2,\ldots,m ( h ) \), the usual basis of affine functions defined by \(\phi_{l} ( M_{s} ) =\delta_{l,s}\), where \(M_{s}\) is a vertex of the considered triangulation.
Let us \(\mathbb{V}_{h}\) denote the standard piecewise linear finite element space such that
The interpolation operator is applied to the function v continuous, defined by
and \(\mathbb{B}\) is the matrix with generic entries
In the sequel of the paper, we use the discrete maximum assumption (d.m.p.). In other words, we assume that the matrix \(\mathbb{B}\) is an M-matrix (cf. [14]).
Remark 2
Under the d.m.p., we achieve a similar study to that devoted to the continuous problem; therefore the qualitative properties and results stated in the continuous case are conserved in the discrete case.
As in the continuous situation, one can tackle the discrete problem by considering the equivalent formulation: Find \(u_{h}^{n}\in\mathbb{V}_{h}\) such that
Notation 2
We denote by \(u_{h}^{n}=\partial_{h} ( f^{n} ( u_{h}^{n} ) ,r_{h} \psi ) \) the solution of problem (3.4).
Existence and uniqueness of a solution of problem (3.4) can be shown similarly to that of the continuous case provided the discrete maximum principle is satisfied.
3.1 Existence and uniqueness
3.1.1 A fixed point mapping associated with discrete problem (3.4)
We consider the following mapping:
where \(\xi_{h}^{n}=\sigma_{h} ( f^{n} ( w ) ,r_{h}\psi ) \) is a solution of the following discrete coercive VI:
Theorem 3
Under the d.m.p. assumption and the preceding hypotheses and notation, the mapping\(\mathbb{T}_{h}\)is a contraction in\(L^{\infty} ( \varOmega ) \)with a contraction constant\(( \frac{\alpha\Delta t+1}{\beta\Delta t+1} ) \). Therefore, \(\mathbb {T}_{h}\)admits a unique fixed point which coincides with the solution of problem (3.4).
As in the continuous situation, one can define the following discrete iterative scheme.
3.2 A discrete iterative scheme
A discrete iterative scheme for the solution of problem (3.4) is given as follows.
Starting from \(u_{h}^{0}=u_{0,h}\), the solution of the following equation:
Now, we give the following algorithm:
where \(u_{h}^{n}\) is a solution of problem (3.4).
Using the above result, we are able to establish the following geometric convergence of sequence \(u_{h}^{n}\).
Proposition 3
Let\(\rho=\frac{\alpha\Delta t+1}{\beta\Delta t+1}\), under the d.m.p. assumption and Theorem3, we have
where\(u_{h}^{\infty}\)is the asymptotic solution of problem of variational inequalities: Find\(u_{h}^{\infty}\in\mathbb{V}_{h}\)such that
Proof
It is very similar to that of the continuous case. □
Under the d.m.p., the solution of discrete problem (3.4) possesses analogous properties to those of the continuous problem.
3.3 Some qualitative properties of the solution of (3.4)
As in the continuous situation, the solution \(u_{h}^{n}\) of system (3.4) possesses the following properties.
3.3.1 A monotonicity property
Let \(u_{h}^{n}=\partial_{h} ( F^{n},r_{h}\psi ) \) (resp. \(\tilde{u}_{h}^{n}=\partial_{h} ( \tilde{F}^{n},r_{h}\tilde{\psi} ) \)) the solution to (3.4) with right-hand side \(F^{n}\).
Lemma 2
If\(F^{n}\geq\tilde{F}^{n}\)and\(\psi\geq\tilde{\psi}\), then
3.3.2 A discrete \(L^{\infty}\)-stability
Proposition 4
Under the d.m.p. assumption and conditions of Lemma 2, we have
Proof
It is very similar to that of the continuous case. □
3.3.3 The concept of discrete subsolution
Definition 2
\(z_{h}^{n}\in\mathbb{V}_{h}\) is said to be a discrete subsolution for the system of quasi-variational inequalities (3.4) if
Theorem 4
Let\(\mathbb{X}_{h}\)be the set of such subsolutions, then under the d.m.p., the solution of (3.4) is the least upper bound of the set\(\mathbb{X}_{h}\).
4 Finite element error analysis
This section is devoted to deriving an error estimate, in the maximum norm, between the nth iterates \(u^{n}\) and their finite element counterpart \(u_{h}^{n}\). For that we first introduce two auxiliary sequences.
4.1 Two auxiliary sequences
4.1.1 A discrete sequence
We define the following discrete sequence \(\{ \bar{u}_{h}^{n} \} _{n\geq1}\), where \(\bar{u}_{h}^{n}\) is a solution to the following discrete problem of variational inequalities (VI):
where \(u^{n}\) is the solution to (2.6).
Lemma 3
(cf. [13])
There exists a constantCindependent ofh, n, and Δtsuch that
Proposition 5
There exists a sequence of discrete subsolutions\(\{ \alpha_{h}^{n} \} _{n\geq1}\)such that
where the constantCis independent ofh, Δt, andn.
Proof
For \(n=1\), we consider the discrete problem of VI:
Then as \(\bar{u}_{h}^{1}\) is a solution to a discrete VI, it is also a subsolution, i.e.,
or
Then
Using the Lipschitz continuity of \(f (\cdot) \), we have
On the other hand, due to [11]
Then
So, \(\bar{u}_{h}^{1}\) is a discrete subsolution for the VI whose solution is \(\bar{U}_{h}^{1}=\partial_{h} ( f ( u_{0,h} ) +C\,h^{2} \vert \log h \vert , r_{h}\psi ) \). Then \(u_{h}^{1}=\partial_{h} ( f ( u_{0,h} ) , r_{h}\psi ) \), and making use of Proposition 4, we have
Hence, making use of Theorem 4, we have
Putting
we get
and
Using Lemma 3, we get
For \(n+1\), let us now assume that
and we consider the discrete problem
Then
or
Then
Using the Lipschitz continuity of \(f (\cdot) \), we have
Using (4.2), we have
So, \(\bar{u}_{h}^{n+1}\) is a discrete subsolution for the VI whose solution is \(\bar{U}_{h}^{n+1}=\partial_{h} ( f ( \bar{u}_{h}^{n} ) +C\,h^{2} \vert \log h \vert ^{2},r_{h}\psi ) \). Then \(u_{h}^{n+1}=\partial_{h} ( f ( \bar{u}_{h}^{n} ) ,r_{h} \psi ) \), making use of Proposition 4, we have
Hence, applying Theorem 4, we get
Putting
we get
and
Using Lemma 3, we obtain
which completes the proof. □
4.1.2 A continuous sequence
We define the following continuous sequence \(\{ \bar{u}_{ ( h ) }^{n} \} _{n\geq1}\), where \(\bar{u}_{ ( h ) }^{n}\) is a solution to the following continuous problem of variational inequalities (VI):
where \(u_{h}^{n}\) is the solution of discrete problem (3.4).
Lemma 4
(cf. [13])
There exists a constantCindependent ofh, k, andnsuch that
where the constantCis independent ofh, n, and Δt.
Proposition 6
There exists a sequence of continuous subsolutions\(\{ \beta_{ ( h ) }^{n} \} _{n\geq1}\)such that
where the constantCis independent ofh, Δt, andn.
Proof
For \(n=1\), we consider the continuous problem of VI
Then, as \(\bar{u}_{ ( h ) }^{1}\) is a solution to a continuous VI, it is also a subsolution, i.e.,
or
Then
Using the Lipschitz continuity of \(f ( \cdot ) \), we have
On the other hand, due to [11]
Then
So, \(\bar{u}_{ ( h ) }^{1}\) is a continuous subsolution for the VI whose solution is \(\bar{U}_{ ( h ) }^{1}=\partial ( f ( u_{0} ) +C\,h^{2} \vert \log h \vert ,\psi ) \). Then \(u^{1}=\partial ( f ( u_{0} ) ,\psi ) \), and making use of Proposition 2, we have
Hence, making use of Theorem 2, we have
Putting
we get
and
Using Lemma 4, we obtain
For \(n+1\), let us now assume that
and consider the continuous problem
Then
or
Then
Using the Lipschitz continuity of \(f ( \cdot ) \), we have
Using (4.4), we have
So, \(\bar{u}_{ ( h ) }^{n+1}\) is a continuous subsolution for the VI whose solution is \(\bar{U}_{ ( h ) }^{n+1}=\partial ( f ( \bar{u}_{ ( h ) }^{n} ) +C\,h^{2} \vert \log h \vert ^{2},\psi ) \). Then \(u^{n+1}=\partial ( f ( \bar{u}_{ ( h ) }^{n} ) ,\psi ) \), and making use of Proposition 2, we have
and, making use of Theorem 2, we obtain
Now, taking
we have
and
Using Lemma 4, we obtain
which completes the proof. □
4.2 \(L^{\infty}\)-Error estimate
Now, guided by Propositions 5 and 6, we are in a position to prove the following.
Theorem 5
Under the conditions of Propositions5and6, we have
where the constantCis independent ofh, Δt, andn.
Proof
Using (4.3), we have
thus
and using (4.6), we have
Thus, we get
Therefore
which completes the proof. □
Corollary 1
In (4.7), passing to the limit, as\(n\rightarrow+\infty\), we get
4.3 \(L^{\infty}\)-Asymptotic behavior
Now we estimate the order of the difference between \(u_{h} ( T,\cdot)\), the discrete solution calculated at the moment \(T=n\Delta t\), and \(u^{\infty}\), the solution of problem (2.13).
Theorem 6
(The main result)
Under the conditions of Proposition3and Corollary1, the following inequality holds:
Proof
We have
thus
Indeed, applying the previous results of Proposition 3 and Corollary 1, we get
Then the following result can be deduced:
which completes the proof. □
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Boulaaras, S., Bencheikh Le Hocine, M.E.A. & Haiour, M. A new error estimate on uniform norm of a parabolic variational inequality with nonlinear source terms via the subsolution concepts. J Inequal Appl 2020, 78 (2020). https://doi.org/10.1186/s13660-020-02346-4
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DOI: https://doi.org/10.1186/s13660-020-02346-4