Special issue from the launching meeting of networks and heterogeneous media

1.   Introduction

This paper is concerned with the finite element approximation of system of J = 2 quasi-variational inequalities QVIs with term sources and obstacles depending on solution: Find a vector $U = ({u}^{1}, {u}^{2})\in {\left({H}_{0}^{1}\right({\mathit{\Omega}} \left)\right)}^{2}$ satisfying

Where Ω is a bounded smooth domain of ${\mathbb{R}}^{N}$ with N$\ge$ 1, each ${a}^{i}\left(., .\right)$ is a continuous elliptic bilinear form, $(., .)$ is the inner product in ${L}^{2}\left({\mathit{\Omega}} \right)$ and each ${f}^{i}$ is a regular, nonlinear functional depending on solutions. The obstacle M provide the coupling between the unknowns u¹; u²

k is a positive number. We point out that in the case where ${f}^{i}$ are independent of the solution, the system (1.1) coincides with that introduced by Bensoussan and Lions in ^[1] which arises in the management of energy production problems.

It is easy to note that the structure of system (1.1) is analogous to that of the classical obstacle problem ^[2] where the term source and obstacle are depending upon the solution sought. The terminology QVI being chosen is a result of this remark.

Numerical analysis of system of quasi-variational inequalities where term sources not depending on solutions were achieved in several works, we refer to ^{[3,4,5,6,7,8]} for system of quasi-variational inequalities with coercive or noncoercive operators.

For results on systems related to evolutionary Hamilton-Jacobi-Bellman equation we refer to ^[9,10,11].

The main objective of this paper is to show that problem (1.1) can be properly approximated by a finite element method and an optimal ${L}^{\infty }$-error estimates is derived, which coincides with the optimal convergence order of elliptic variational inequalities of an obstacle type problem ^[12].

The approximation is carried out by first introducing a modified Bensoussan-Lions type iterative scheme depending on parameters which is shown to converge geometrically to the continuous solution. By a symmetrical approach, using the standard finite element method and a discrete maximum principle (DMP), the geometric convergence of the discrete modified Bensoussan-Lions type iterative scheme depending upon parameters is given as well. An ${L}^{\infty }$-error estimates is then established combining the geometric convergence of both the continuous and discrete iterative schemes and the known uniform error estimates in elliptic VIs.

It is worth mentioning that even the guiding idea of this paper rests on the algorithmic approach followed in many papers cited above, the treatment of the geometric convergence of both continuous and discrete schemes is totally different because of the nonlinear nature of terms sources. Also, it is used for the first time for a system of QVIs.

An outline of this paper is as follows: In section 2, we lay down some definitions and classical results related to variational inequalities and prove a Lipschitz continuous and discrete dependency with respect to the source term, the boundary condition and the obstacle. Section 3 discusses the continuous Bensoussan-Lions type iterative scheme and proves its geometrical convergence. In Section 4, we establish the finite element counter parts of the continuous system and the continuous Bensoussan-Lions type iterative scheme respectively and the geometrical convergence of the discrete scheme. Section 5 is devoted the ${L}^{\infty }$-error analysis of the method.

2.   Preliminaries

We are given functions ${a}_{jk}^{i}\left(x\right), {a}_{k}^{i}\left(x\right), {a}_{0}^{i}\left(x\right), 1\le i\le 2$ sufficiently smooth functions such that $1\le j, k\le N$

where ${\beta }^{i}$ is a positive constant. We define the bilinear forms: For all $u, v\in {H}_{0}^{1}\left({\mathit{\Omega}} \right)$

We are given right-hand sides

a nonlinear functional and Lipschitz continuous on $\mathbb{R}$; that is

such that

where ${\beta }^{i}$ is a constant defined in (2.1). For $W = \left({w}^{1}, {w}^{2}\right)\in {\left({L}_{+}^{\infty }\left({\mathit{\Omega}} \right)\right)}^{2}$ we introduce the norm

2.1.   Elliptic variational inequalities

Let be Ω a bounded polyhedral domain of ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$ with sufficiently smooth boundary $\partial {\mathit{\Omega}}$. We consider the bilinear form of the same form of those defined in (2.2), the linear form

The right hand side

the obstacle

the boundary condition $g\in {L}^{\infty }(\partial {\mathit{\Omega}})$ and the nonempty convex set

We consider the variational inequality V.I.: Find $u\in {K}^{g}$ such that

2.2.   A monotonicity property

Proposition 1 Let $\left(f, g, \psi \right);$ $\left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$be a pair of data and $\zeta = \sigma \left(f, g, \psi \right)$; ${\widetilde {\zeta }} = \sigma \left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$ the corresponding solution to (2.8). If $f\le {\widetilde f}$ in Ω, $g\le {\widetilde g}$ on ∂Ω and $\psi \le {\widetilde {\psi }}$ then, $\zeta \le {\widetilde {\zeta }}$ in Ω.

Proof. The proof is an adaptation of the proof of the monotonicity property of the solution of Ⅵ with nonlinear source term (see ^[13]). According to ^[14], $\zeta = \underset{}{\mathrm{max}}\left\{{\underline \zeta }\right\}$ where $\left\{{\underline \zeta }\right\}$ is the set of all the subsolutions of $\zeta$. Hence, $\forall {\underline \zeta }\in \left\{{\underline \zeta }\right\}$, ${\underline \zeta }$ satisfies

By using the conditions $f\le {\widetilde f}$ in Ω, $g\le {\widetilde g}$ on ∂Ω and $\psi \le {\widetilde {\psi }}$, we get

with

Thus, $\zeta$ is a subsolution of ${\widetilde {\zeta }} = \sigma \left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$, that is $\zeta \le {\widetilde {\zeta }}$ in Ω.

2.3.   A Lipschitz continuous dependency with respect to the boundary condition, the source term and the obstacle

This subsection is devoted to the establishment of a Lipschitz continuous dependence property of the solution with respect to the source term, the boundary condition and the obstacle by which we first, set out and demonstrate.

Proposition 2 Let $\left(f, g, \psi \right);$ $\left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$be a pair of data and$\zeta = \sigma \left(f, g, \psi \right)$; ${\widetilde {\zeta }} = \sigma \left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$ the corresponding solution to (2.8). Then, we have

Proof. The proof is an adaptation of the proof of a Lipschitz property of the solution of Ⅵ with nonlinear source term (see ^[13]). First, set

Then,

So,

Thus, for all $0 < v,$

with

So, according to the property ${\widetilde {\zeta }}$ is a subsolution of$\sigma \left(f+{a}_{0}\left(x\right)\varphi, g+\varphi, \psi +\varphi \right) = \sigma (f, g, \psi)+\varphi$, that is

or

Similarly, interchanging the roles of the couples $\left(f, g, \psi \right);$ $\left({\widetilde f}, {\widetilde g}, {\widetilde {\psi }}\right)$, we obtain

which completes the proof.

Let ${\tau }_{h}$ be a triangulation of Ω with meshsize h, V_h be the space of finite elements consisting of continuous piecewise linear functions v vanishing on ∂Ω and ${\varphi }_{s}$; s = 1, 2, …, m(h) be the basis functions of V_h.

The discrete counterpart of (2.8) consists of finding ${u}_{h}\in {K}_{h}^{g}$ such that

Where

${\pi }_{h}$ is an interpolation operator on $\partial {\mathit{\Omega}}$ and r_his the usual finite element restriction operator on ${\mathit{\Omega}}$.

Theorem 3 (See ^[12] Under conditions (2.5) and (2.6), there exists a constant C independent of h such that

2.4.   A Lipschitz discrete dependency with respect to the boundary condition, the source term and the obstacle

Assuming that the DMP is satisfied, i.e. the matrix resulting from the finite element discretization is an M-matrix (see ^[15,16]), we prove the Lipschitz discrete dependence with respect to the boundary condition, the source term and the obstacle by a similar study to that undertaken previously for the Lipschitz continuous dependence property.

Proposition 4 Let $\left(f, g, {r}_{h}\psi \right);$ $\left({\widetilde f}, {\widetilde g}, {r}_{h}{\widetilde {\psi }}\right)$be a pair of data and ${\zeta }_{h} = {\sigma }_{h}\left(f, g, {r}_{h}\psi \right)$; ${{\widetilde {\zeta }}}_{h} = {\sigma }_{h}\left({\widetilde f}, {\widetilde g}, {r}_{h}{\widetilde {\psi }}\right)$ the corresponding solution to (2.14). If $f\le {\widetilde f}$ in Ω, $g\le {\widetilde g}$ on ∂Ω and ${r}_{h}\psi \le {r}_{h}{\widetilde {\psi }}$ then, ${\zeta }_{h}\le {{\widetilde {\zeta }}}_{h}$ in Ω.

Proof. The proof is similar to that of the continuous case.

The proposition below establishes a Lipschitz discrete dependence of the solution with respect to the data.

Proposition 5 Let the (d.m.p) holds. Then, we have

Proof. The proof is similar to that of the continuous case.

3.   The continuous problem

We define the following fixed-point mapping

Where ${\mathrm{\zeta }}^{\mathrm{i}}\in {H}_{0}^{1}\left({\mathit{\Omega}} \right)\cap {L}^{\infty }\left({\mathit{\Omega}} \right)$ is a solution to the following variational inequality

Thanks to ^[1,2], ${\mathrm{\zeta }}^{\mathrm{i}}$ is the unique solution to coercive variational inequality (3.1).

Remark 1 We remark that the solution $U = \left({u}^{1}, {u}^{2}\right)$ of the system (1.1) is the fixed point of the mapping T; that is$TU = U$.

3.1.   A continuous iterative scheme

Starting from ${U}^{0} = \left({u}^{\mathrm{1, 0}}, {u}^{\mathrm{2, 0}}\right)$ where ${u}^{i, 0}$; i = 1; 2 is solution of the variational equation

and for all $0 < {w}_{i} < 1$; $i = \mathrm{1, 2}$ we define the sequences $\left({u}^{1, n+1}\right)$ and $\left({u}^{2, n+1}\right)$ such that ${u}^{1, n+1}$ and ${u}^{2, n+1}$ the components of the vector ${U}^{n+1}$, solve the following elliptic variational inequalities respectively

3.2.   Convergence of the continuous iterative scheme

Theorem 2 The sequences $\left({u}^{1, n+1}\right)$ and $\left({u}^{2, n+1}\right)$ converge geometrically to the solution $U = \left({u}^{1}, {u}^{2}\right)$ of the system (1.1); there exist a positive real $\rho \in \left(\mathrm{0, 1}\right)$ which depends on ${\alpha }_{i}$ and ${w}_{i}$ such that for all $n\ge 0$

where

Proof. The proof will carry out by induction.

● We first deal with the case

● Indeed for n = 0; using (1.1), (3.2), (3.3) and (2.9), we have

So,

We distinguish two cases

or

(3.9) in conjunction with case (3.10) implies

with

which implies

By replacing (3.14) in (3.13), we get

which coincides with (3.8).

(3.9) in conjunction with (3.11) implies

with

${‖{u}^{2}-{u}^{\mathrm{2, 0}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$ is bounded below by both ${‖{u}^{1}-{u}^{\mathrm{1, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$

and

So,

or

Then,

or

(3.15), (3.17) and (3.18) generate the following three possibilities

or

or

All possibilities are true in the same time because they coincide with (3.8). So, there is either a contradiction and thus case (3.11) is impossible or case (3.11) is possible if and only if

Hence, both cases (3.10) and (3.11) imply (3.14).

● Let us now discuss the second case

(3.9) in conjunction with (3.10) implies (3.14) with

which contradicts (3.19) which means that (3.10) is impossible. (3.9) in conjunction with (3.11) we get (3.17) and (3.18). So,

or

We remark that both alternatives are true in same time because both coincide with (3.19) which implies that in case (3.11), we must have

Hence, in both cases (3.8) and (3.19), we obtain (3.14). Hence,

● As

we need to deal also with ${‖{u}^{2}-{u}^{\mathrm{2, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)},$ by following the same reasoning as that adopted for u¹ and u^1,1, we get

Again we distinguish two possibilities

or

(3.21) and (3.22) imply

with

By substituting (3.24) in (3.25), we get

which coincides with (3.20). (3.21) and (3.23) imply

with

It is clear that ${‖{u}^{1}-{u}^{\mathrm{1, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$ is bounded below by both

and

which leads us to distinguish the following possibilities

or

Then,

or

Thus, (3.26)-(3.28) imply that the three following alternatives are required

or

or

It is clear that all alternatives coincide with (3.20). So, we must have

Thus, in both cases (3.22) and (3.23) we obtain (3.24). Hence,

(3.20) and (3.29) imply

● Let us assume that, for $n\ge 0$

● We prove

By adopting the same arguments for (1.1), (3.2), (3.3) and (2.9) as that applied for the previous iterates, we get

So,

Also we distinguish two cases:

or

(3.32) in conjunction with (3.33) implies

with

By replacing (3.35) in (3.36) we get, according to (3.30); i = 1

which matches with (3.30); i = 2. (3.32) in conjunction with (3.34) implies

with

${‖{u}^{2}-{u}^{2, n}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$ is bounded below by both ${‖{u}^{1}-{u}^{1, n+1}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$

and

So,

or

Thus,

or

By taking into account (3.37), we get

or

or

Three possibilities are true because all coincide with (3.30). So, we necessarily get

Thus, both cases (3.33) and (3.34) imply (3.35). Hence, by using (3.30) we get (3.31) for i = 1. The proof for (3.31); i = 2 is obtain in similar way by using (3.31); i = 1 and (3.35) so, it will be omitted. The desired result (3.6) follows naturally from (3.31).

4.   Statement of discrete problem

This section, we will handle the discrete problem by a perfect symmetry in the treatment of that the continuous one. Indeed, we define the discrete system of QVIs: Find a vector ${U}_{h} = \left({u}_{h}^{1}, {u}_{h}^{2}\right)\in {\left({V}_{h}\right)}^{2}$ such that

The related discrete fixed-point mapping

where ${\zeta }_{h}^{i}\in {V}_{h}$ is the unique solution to the following discrete variational inequality

Remark 1 We remark that the solution ${U}_{h} = \left({u}_{h}^{1}, {u}_{h}^{2}\right)$ of the system (4.1) is the fixed point of the mapping ${T}_{h}$; that is ${T}_{h}{U}_{h} = {U}_{h}.$

4.1.   A discrete iterative scheme

Starting from ${U}_{h}^{0} = \left({u}_{h}^{\mathrm{1, 0}}, {u}_{h}^{\mathrm{2, 0}}\right)$ where ${u}_{h}^{i, 0} = {r}_{h}{u}^{i, 0};i = \mathrm{1, 2}$ is the discrete analog of ${u}^{i, 0}$ then,

For all $0 < {w}_{i} < 1;i = \mathrm{1, 2}$ we define the discrete sequences $\left({u}_{h}^{1, n+1}\right)$ and $\left({u}_{h}^{2, n+1}\right)$ such that ${u}_{h}^{1, n+1}$ and ${u}_{h}^{2, n+1}$ components of the vector ${U}_{h}^{n+1}$ solve discrete elliptic variational inequalities

4.2.   Convergence of the discrete iterative scheme

Theorem 2 The discrete sequences $\left({u}_{h}^{1, n+1}\right)$ and $\left({u}_{h}^{2, n+1}\right)$ converge geometrically to the discrete solution ${U}_{h} = \left({u}_{h}^{1}, {u}_{h}^{2}\right)$ of the system (4.1); there exist a positive real $\rho \in \left(\mathrm{0, 1}\right)$ defined in (3.7) such that for all $n\ge 0$

Proof. The proof is similar to that of the continuous case.

5.   ${\mathit{L}}^{\mathit{\infty }}$-error analysis

This section is devoted to the proof of the main result of this paper. For that purpose we need to introduce an auxiliary system.

5.1.   Auxiliary system

Let ${w}_{h}^{i, 0} = {u}_{h}^{i, 0};i = \mathrm{1, 2}$ be an initialization. For all $0 < {w}_{i} < 1;i = \mathrm{1, 2}$ we define the discrete sequences $\left({w}_{h}^{1, n+1}\right)$ and $\left({w}_{h}^{2, n+1}\right)$ such that ${w}_{h}^{1, n+1}$ and ${w}_{h}^{2, n+1}$ solve coercive variational inequalities

It is clear that ${w}_{h}^{i, n+1};i = \mathrm{1, 2}$ components of the vector ${W}_{h}^{n+1}$ are finite element approximation of ${u}^{i, n+1}$ defined in (3.2)–(3.4). Thus, making use of (2.16); we get

The algorithmic approach used in the present paper rests on the following crucial lemma, where the error estimate between the nth iterate ${U}^{n}$ and its discrete counter parts ${U}_{h}^{n+1}$ is established.

Lemma 1 Let $\left({U}^{n+1}\right)$ and $\left({U}_{h}^{n+1}\right)$ be the vectors whose components are sequences defined in (3.2)–(3.5) and (4.4)–(4.7) respectively. Then,

Where

Proof. The proof of the lemma rests on the discrete Lipschitz continuous dependency with respect to source term and obstacle and will carry out by induction.

● For n = 0, we have

(5.1), (5.2), (4.4), (4.5) and (2.17) imply

So,

Therefore,

We distinguish two cases

or

(5.8) in conjunction with (5.9) imply

with

So,

with (5.11). Then,

By replacing (5.12) in (5.11) we obtain

According to (5.5) and (4.3) we get,

which coincides with (4.3).

(5.8) and (5.10) imply

with

Then, multiplying (5.13) by ${\alpha }_{1}{w}_{1}$ and adding ${\alpha }_{1}\left(1-{w}_{1}\right){‖{u}^{\mathrm{1, 0}}-{u}_{h}^{\mathrm{1, 0}}‖}_{{L}^{\mathrm{\infty }}\left({\mathit{\Omega}} \right)},$ we obtain

We note that

is bounded by both

and

So,

or

Therefore, according to (5.5) and (4.3), we get

or

So, the last two alternatives are true at the same time because both coincide with (4.3). We necessarily deduce that

By replacing (5.14) in (5.13); we get (5.12). Hence, in both cases (5.9) and (5.10); we can write

Thus,

● In a similar way, that is by following the same steps as for ${u}^{\mathrm{1, 1}}$ and ${u}_{h}^{\mathrm{1, 1}}$, ${u}^{\mathrm{2, 1}}$ and ${u}_{h}^{\mathrm{2, 1}}$ satisfy

So,

We distinguish also two cases

or

(5.16) in conjunction with case (5.17); we get

${‖{u}^{\mathrm{2, 1}}-{u}_{h}^{\mathrm{2, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}\le {‖{u}^{\mathrm{2, 1}}-{w}_{h}^{\mathrm{2, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$+${\alpha }_{2}{w}_{2}{‖{u}^{\mathrm{2, 1}}-{u}_{h}^{\mathrm{2, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$

with

So,

with, according to (5.20)

Then,

Therefore,

which coincides with (5.15). The conjunction of (5.16) with case (5.18), implies

with

Then, by multiplying (5.21) by ${\alpha }_{2}{w}_{2}$ and adding${\alpha }_{2}\left(1-{w}_{2}\right){‖{u}^{\mathrm{2, 0}}-{u}_{h}^{\mathrm{2, 0}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$, we obtain that the term ${\alpha }_{2}{w}_{2}{‖{u}^{\mathrm{2, 1}}-{u}_{h}^{\mathrm{2, 1}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}+{\alpha }_{2}\left(1-{w}_{2}\right){‖{u}^{\mathrm{2, 0}}-{u}_{h}^{\mathrm{2, 0}}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)}$is bounded by both

and

So, we distinguish again, the two following alternatives

or

We remark that both alternatives coincide with (5.15), which implies that case (5.18) is possible if and only if

By substituting (5.22) in (5.21), we get (5.20). Hence, in both cases (5.17) and (5.18), we get

Thus,

(5.15) and (5.23) imply

● Let us assume that for $n\ge 0$ and i = 1, 2

● And prove for i = 1, 2

We operate in the same way as in iterate n = 0. Let us begin with case i = 1 in (5.25)

So, by applying (2.17), we get

We distinguish again two cases

or

(5.26) in conjunction with case (5.27) implies

and

Then,

with, according to (5.29)

(5.24) implies

with

Thus,

and as ${\alpha }_{1}{w}_{1} <1$

Hence,

and

which corresponds with (5.24) for i = 2: Inequality (5.26) with (5.28) imply

and

By multiplying (5.30) by ${\alpha }_{1}{w}_{1}$ and adding the term ${\alpha }_{1}\left(1-{w}_{1}\right){‖{u}^{1, n}-{u}_{h}^{1, n}‖}_{{L}^{\infty }\left({\mathit{\Omega}} \right)},$ we get that the term

is bounded by the following two terms

and

So, we need to distinguish the followings possibilities

or

which implies

or

By using (5.24), we can write

or

Only the last alternative is true because it matches with (5.24) for i = 2. So, in (5.28) we get

By replacing (5.31) in (5.30); we get (5.29). Hence, in both cases (5.27) and (5.28), we obtain

So,

Therefore,

By using the last inequality (5.32) and by adopting the same reasoning we prove (5.25); i = 2, therefore, we get (5.6).

5.2.   The main result

Theorem 2 Let $U$ and ${U}_{h}$ be the solution of systems (1.1) and (4.8), respectively. Then, there exists a constant C independent of h such that

Proof. Making use of (3.6), (5.6) and (4.8), we have

As $n\to +\infty$ and by using (5.5) we get (5.33).

6.   Conclusions

In this work an optimal convergence order is derived for a class of system of two elliptic quasi-variational inequalities where terms sources and obstacles depend upon the solution, where the continuous and discrete Lipschitz dependence with respect to the terms sources, boundary condition and obstacles' played a leading role in obtaining the main result of this paper. As (1.1) plays a key role in solving Hamilton-Jacobi-Bellman equation the results obtained in this paper can give an optimal error estimate for HJB equation also even for $J\ge 2$. The approach used and the results obtained in this paper (optimal convergence order) remain valid when we deal with systems of $J\ge 2$ quasi-variational inequalities with terms sources depends on solution and the obstacles ⁱ independent of the solution, that is systems of the form; Find a vector $U = ({u}^{1}, \dots, {u}^{J})\in {\left({H}_{0}^{1}\right({\mathit{\Omega}} \left)\right)}^{J}$ satisfying

Acknowledgments

The author states that no funding source or sponsor has participated in the realization of this work.

Conflict of interest

All authors declare no conflicts of interest in this paper.