An optimal error estimate of finite element method for parabolic quasi-variational inequalities with non linear source terms

Abstract

The main purpose of this paper is to analyze the convergence and regularity of the proposed algorithm (J. Nonlinear Sci. Appl. 9 (2016), 568–583) of the discontinuous Galerkin methods coupled with Euler time discretization scheme for parabolic quasi-variational inequalities with nonlinear source terms and an obstacle defined as impulse control problem.

Keywords

Finite elements Euler scheme fixed point PQVIs EQVIs geometric convergence

1. Introduction

In this paper, we extend our works [2,7] and continue to analyze the convergence of the proposed algorithm of the discontinuous Galerkin methods coupled with a theta time discretization scheme. Also an optimal error estimate with an asymptotic behavior in uniform norm are given for the parabolic quai-variational inequalities (PQVIs) with respect to the nonlinear source terms and an obstacle defined as an impulse control problem. Namely, we consider the following PQVIs: find $u \in L^{2} (0, T, H^{1})$ such that $\begin{matrix} (1.1) & \{\begin{matrix} u_{t} + A u ⩽ f (u) in Σ, \\ u ⩽ M u, \\ (\frac{\partial u}{\partial t} + A u - f (u)) (u - M u) = 0, \\ u (0, x) = u_{0} in Ω, u = 0 on \partial Ω, \end{matrix} \end{matrix}$ where Σ is a set in $R \times R^{n}$ defined as $Σ = Ω \times [0, T]$ with $T < + \infty$ and Ω is a smooth bounded domain of $R^{n}$ with sufficiently smooth boundary Γ, A is an operator defined on $H^{1} (Ω)$ as $\begin{matrix} (1.2) & A u = - \sum_{i, j = 1}^{n} \frac{\partial}{\partial x_{i}} (a_{i j} (x) \frac{\partial u}{\partial x_{j}}) + \sum_{j = 1}^{n} b_{j} (x) \frac{\partial u}{\partial x_{j}} + a_{0} (x) u \end{matrix}$ and $a (\cdot, \cdot)$ is the bilinear form associated with operator A defined in (1.2) $\begin{matrix} (1.3) & a (u, v) = \int_{Ω} (\sum_{i, j = 1}^{n} a_{i j} (x) \frac{\partial u}{\partial x_{i}} \frac{\partial v}{\partial x_{j}} + \sum_{j = 1}^{n} b_{j} (x) \frac{\partial u}{\partial x_{j}} v + a_{0} (x) u v) d x \end{matrix}$ and assumed to be noncoercive and whose coefficients $a_{i, j} (x)$ , $b_{j} (x)$ , $a_{0} (x) \in L^{\infty} (Ω) \cap C^{2} (\bar{Ω})$ , $x \in \bar{Ω}$ , $1 ⩽ i, j ⩽ n$ are sufficiently smooth functions and satisfy the following conditions $\begin{array}{l} (1.4) & a_{i j} (x) = a_{j i} (x); a_{0} (x) ⩾ β > 0, \\ (1.5) & \sum_{i, j = 1}^{n} a_{i j} (x) ξ_{i} ξ_{j} ⩾ γ | ξ |^{2}; ξ \in R^{2}, γ > 0, x \in \bar{Ω}, \end{array}$ where β and γ are constants.

Here, $f (\cdot)$ is a Lipschitz increasing nonlinear source term $\begin{matrix} (1.6) & f \in L^{2} (0, T, L^{\infty} (Ω)) \cap C^{1} (0, T, H^{- 1} (Ω)), f ⩾ 0, \end{matrix}$ with rate of Lipschitz c satisfying $\begin{matrix} (1.7) & c ⩽ β \end{matrix}$ and M is an operator given by $\begin{matrix} (1.8) & M u = k + inf_{ξ ⩾ 0, x + ξ \in \bar{Ω}} u (x + ξ), \end{matrix}$ where $k > 0$ and $ξ ⩾ 0$ and $\begin{matrix} (1.9) & M u \in L^{2} (0, T, W^{2, \infty} (Ω)) . \end{matrix}$

As in [18], we assume that M satisfies the following properties:

M is concave that is to say, for $u, v \in C (Ω)$ $\begin{matrix} (1.10) & M (δ u + (1 - δ) v) ⩾ δ M (u) + (1 - δ) M (v) \end{matrix}$ and $\begin{matrix} (1.11) & \forall η \in R, M (u + η) = M (u) + η . \end{matrix}$

The symbol

{(\cdot, \cdot)}_{Ω}

stands for the inner product in

L^{2} (Ω)

The stationary free boundary problems are accomplished in some applications; for example, in stochastic control, the solution of (1.1) characterizes the infimum of the cost function associated to an optimally controlled stochastic switching process without costs for switching and for the calculus of quasi stationary state for the simulation of petroleum or gaseous deposit (cf., e.g., [1]). From the mathematical analysis point of view, elliptic case of the problem (1.1) has been intensively studied in the late 1980s (cf. [8,9,11,12] and [10,15–18]). Moreover, in [13] and [14], the existence and uniqueness of strong as well as weak solutions of parabolic variational inequalities with linear source terms have been established based on a Lagrange multiplier treatment where the existence has been obtained as the unique asymptotic limit of solutions to a family of appropriately regularized nonlinear parabolic equations. On the numerical and computational side (cf. [3–6,8,9] and [10,15–17]). However, as far as Galerkin method is concerned, only few papers are known in the literature (cf. [5,6,10,15–17]).

In [5], we applied a new time space discretization using the Euler time scheme combined with a finite element spatial approximation, we found that (1.1) can be transformed into the full-discrete system of elliptic quasi-variational inequalities (EQVIs) and we proposed a new iterative discrete algorithm to show the existence and uniqueness of the discrete solution, and we gave a proof of asymptotic behavior in uniform norm using the theta scheme combined with Galerkin method.

Also, in [4] we analyzed the stability in uniform norm of theta-scheme with respect to the t-variable combined with a Galerkin spatial approximation for the evolutionary inequalities and quasi variational inequalities with an obstacle defined as an impulse control problem. In addition in [3], a quasi-optimal $L^{\infty}$ -error estimate has been established of a system of PQVI related to the management of energy production with mixed boundary condition with linear source terms using our discrete algorithm based on a theta time scheme combined with a finite element spatial approximation

In [7], we have given a new proof for the existence and uniqueness of the solution of HJB equation which is consisted of 4 steps, where, it stand on main properties of our discrete iterative algorithm using the theta scheme combined with Galerkin method and which has been introduced for proving the asymptotic behavior in $L^{\infty}$ -norm in the previous works (cf., e.g., [4,6]). Then in [2], the same study has been extended to the parabolic quasi-variational inequalities with impulse control and nonlinear source terms using Euler time scheme combined with a finite element spatial approximation.

In this completed work of [2], an $L^{\infty}$ -error estimate is established combining the geometric convergence of discrete iterative schemes using the known $L^{\infty}$ -error estimate for a stationary and evolutionary free boundary problems (cf., e.g., [2,7]) which plays a major role in the finite element error analysis section. Finally an asymptotic behavior in uniform norm is deduced which investigated the evolutionary free boundary problem similar to that in [4] and [5].

The structure of this paper is as follows. In Sections 2 and 3, we consider the discrete parabolic quasi-variational inequalities (PQVIs), discretize the iterative scheme by the standard finite element method combined with Euler scheme and an algorithm iterative discrete scheme is introduced. Then its geometric convergence is proved with respect to $L^{\infty}$ -stability of the solution and right-hand side and its characterization as the least upper bound of the subsolutions set (see [8]). It is worth mentioning that this approach is entirely different from the one developed for the evolutionary problem. Also, it is used for the first time for system of stationary QVIs. In Section 4, a fundamental lemma and an optimal error estimates on uniform norm with an asymptotic behavior in uniform norm are proved, for presented problem. Finally, we make some comments on the approach and the results presented in this paper.

2. Parabolic quasi-variational inequalities

The problem (1.1) can be transformed into the following continuous parabolic quasi-variational inequalities: find $u \in (L^{2} (0, T, H^{1} (Ω)))$ solution to $\begin{matrix} (2.1) & \{\begin{matrix} (\frac{\partial u}{\partial t}, v - u) + a (u, v - u) ⩾ (f (u), v - u), \\ u ⩽ M u, v ⩽ M u, \\ u (0, x) = u_{0} in Ω, \end{matrix} \end{matrix}$ where $a (\cdot, \cdot)$ is the bilinear form associated with operator A defined in (1.2).

2.1. The temporal discretization

We discretize the problem (2.1) temporally by using the semi-implicit scheme. Therefore, we search a sequence of elements $u^{k} \in H_{0}^{1} (Ω)$ which approaches $u (t_{k})$ , $t_{k} = k Δ t$ , with initial data $u^{0} = u_{0}$ .

Thus, we have for $k = 1, \dots, n$ $\begin{matrix} (2.2) & \{\begin{matrix} \frac{u^{k} - u^{k - 1}}{Δ t} + A u^{k} ⩽ f^{k} (u^{k}) in Σ, \\ u ⩽ M u, v ⩽ M u, \\ u^{0} (x) = u_{0} in Ω, u = 0 on \partial Ω . \end{matrix} \end{matrix}$

Firstly we define the mapping $\begin{matrix} (2.3) & \begin{matrix} T : L_{+}^{\infty} (Ω) & ⟶ L^{\infty} (Ω) \\ W & ⟶ T W = ξ^{k} = \partial (F^{k} (w), M u^{k}), \end{matrix} \end{matrix}$ where $L_{+}^{\infty} (Ω)$ denotes the positive cone of $L^{\infty} (Ω)$ , such that $ξ^{k}$ is the solution of the following problem $\begin{matrix} (2.4) & \{\begin{matrix} \frac{ξ^{k} - ξ^{k - 1}}{Δ t} + A ξ^{k} ⩽ f (ξ^{k}) & in Σ, \\ ξ^{k} ⩽ M u . \end{matrix} \end{matrix}$

2.2. An iterative semi-discrete algorithm

We choose $u^{0} = u_{0}$ the solution of the following semi-discrete equation $\begin{matrix} (2.5) & A^{0} u = g^{0} \end{matrix}$ and $g^{0}$ is an M regular function.

Now we give the following semi-discrete algorithm $\begin{matrix} (2.6) & u^{k} = T u^{k - 1}, k = 1, \dots, n, \end{matrix}$ where $u^{k}$ is the solution of the problem (2.2).

Remark 1.
We denote by $\begin{matrix} (2.7) & Q = {w \in L_{+}^{\infty}, such that 0 ⩽ w ⩽ u^{0}}, \end{matrix}$ where $u^{0}$ is solution of (2.5).

Since $f^{k} (\cdot) ⩾ 0$ , and $u_{h}^{0} = u_{h 0} ⩾ 0$ , combining comparison results in variational inequalities with a simple induction, it follows that $u^{k} ⩾ 0$ , i.e., $u^{k} ⩾ 0$ , $\forall k = 1, \dots, n$ and $T w ⩾ 0$ .

Furthermore, by (2.6) and (2.7) we have $\begin{matrix} u^{1} = T u^{0} ⩽ u^{0} . \end{matrix}$ Similar to that in previous works [5] and [2], the mapping T is a monotone increasing function for the stationary free boundary problem with non linear source term. Then it can be easily verified that $\begin{matrix} u^{2} = T u^{1} ⩽ T u^{0} = u^{1} ⩽ u^{0}, \end{matrix}$ thus, inductively $\begin{matrix} u^{k + 1} = T u^{k} ⩽ u^{k} ⩽ \dots ⩽ u^{0}, \forall k = 1, \dots, n, \end{matrix}$ also it can be seen that the sequence ${(u^{k})}_{k}$ stays in Q.

According to the assumption (1.6), we have $f (\cdot)$ is increasing and under Remark 1, we have for $k = 1, \dots, n$ $\begin{matrix} f (u^{k}) ⩽ f (u^{k - 1}), \end{matrix}$ then we can rewrite (2.2) as follows: $\begin{matrix} (2.8) & \{\begin{matrix} \frac{u^{k} - u^{k - 1}}{Δ t} + A u^{k} ⩽ f (u^{k - 1}) in Σ, \\ ξ^{k} ⩽ M u, \\ ξ^{0} (x) = ξ_{0} in Ω, ξ = 0 on \partial Ω, \end{matrix} \end{matrix}$ where $ξ^{k}$ is a solution of (2.4).

Also, the (2.8) can be transformed into the following system of the semi discrete parabolic quasi-variational inequalities (PQVIs) $\begin{matrix} (2.9) & \{\begin{matrix} (\frac{u^{k} - u^{k - 1}}{Δ t}, v - u^{k}) + a (u^{k}, v - u^{k}) ⩾ (f (u^{k - 1}), v - u^{k}), \\ u ⩽ M u, \\ u (0, x) = u_{0} in Ω . \end{matrix} \end{matrix}$
2.3. The spatial discretization

Let Ω be decomposed into triangles and $τ_{h}$ denotes the set of all those elements $h > 0$ is the mesh size. We assume that the family $τ_{h}$ is regular and quasi-uniform. We consider the usual basis of affine functions $φ_{l}$ , $l = {1, \dots, m (h)}$ defined by $φ_{l} (M_{s}) = δ_{l s}$ where $M_{s}$ is a vertex of the considered triangulation. We introduce the following discrete spaces $V^{h}$ of finite element $\begin{matrix} (2.10) & V^{h} = \{\begin{matrix} v \in L^{2} (0, T, H_{0}^{1} (Ω)) \cap C (0, T, H_{0}^{1} (\bar{Ω})), such that \\ v ∣_{K} \in P_{1}, K \in τ_{h}, \\ u (\cdot, 0) = u_{0} in Ω, u = 0 on \partial Ω, \end{matrix}\} \end{matrix}$ where $r_{h}$ is the usual interpolation operator defined by $\begin{matrix} (2.11) & v \in L^{2} (0, T, H_{0}^{1} (Ω)) \cap C (0, T, H_{0}^{1} (\bar{Ω})), r_{h} v = \sum_{i = 1}^{m (h)} v (M_{i}) φ_{i} (x) \end{matrix}$ and $P_{1}$ denotes the space of polynomials with degree at most 1.

In the sequel of the paper, we shall make use of the discrete maximum principle assumption (DMP). In other words, we shall assume that the matrices ${(A)}_{p s} = a (φ_{p}, φ_{s})$ is M-matrices (cf. [9]).

We discretize in space the problem (2.9), i.e. that we approach the space $H_{0}^{1}$ by a space discretization of finite dimensional $V_{h} \subset H_{0}^{1}$ , we get the following discrete PQVIs: $\begin{matrix} (2.12) & \{\begin{matrix} (\frac{u_{h}^{k} - u_{h}^{k - 1}}{Δ t}, v_{h} - u_{h}^{k}) + a (u_{h}^{k}, v_{h} - u_{h}^{k}) ⩾ (f (u_{h}^{k - 1}), v_{h} - u_{h}^{k}), \\ u_{h}^{k} ⩽ r_{h} M u_{h}^{k}, \\ u^{0} (x) = u_{0} in Ω, \end{matrix} \end{matrix}$ which implies $\begin{matrix} (2.13) & \{\begin{matrix} (\frac{u_{h}^{k}}{Δ t}, v_{h} - u_{h}^{k}) + a (u_{h}^{k}, v_{h} - u_{h}^{k}) ⩾ (f (u_{h}^{k - 1}) + \frac{u_{h}^{k - 1}}{Δ t}, v_{h} - u_{h}^{k}), \\ u_{h}^{k} ⩽ r_{h} M u_{h}^{k}, \\ u_{h}^{k} (0) = u_{0 h}^{k} in Ω . \end{matrix} \end{matrix}$

Then, the problem (2.13) can be reformulated into the following coercive discrete system of elliptic quasi-variational inequalities (EQVIs) $\begin{matrix} (2.14) & \{\begin{matrix} b (u_{h}^{k}, v_{h} - u_{h}^{k}) ⩾ (f (u_{h}^{k - 1}) + λ u_{h}^{k - 1}, v_{h} - u_{h}^{k}), u_{h}^{k} \in V^{h}, \\ u_{h}^{k} ⩽ r_{h} M u_{h}^{k}, \\ u_{h}^{k} (0) = u_{0 h}^{k} in Ω, \end{matrix} \end{matrix}$ such that $\begin{matrix} (2.15) & \{\begin{matrix} b (u_{h}^{k}, v_{h} - u_{h}^{k}) = λ (u_{h}^{k}, v_{h} - u_{h}^{k}) + a (u_{h}^{k}, v_{h} - u_{h}^{k}), & u_{h}^{k} \in V^{h}, \\ λ = \frac{1}{Δ t} = \frac{1}{k} = \frac{T}{n}, & k = 1, \dots, n . \end{matrix} \end{matrix}$

We shall first recall some results related to coercive quasi variational inequalities that are necessarily in proving some useful qualitative properties.

Definition 1.
$ζ_{h}^{k}$ is said to be a subsolution for the system of EQVIs (2.14) if $\begin{matrix} (2.16) & \{\begin{matrix} b (ζ_{h}^{k}, φ_{s}) ⩽ (f (w) + λ ζ_{h}^{k - 1}, φ_{s}), \forall φ_{s}, s = 1, \dots, m (h), \\ ζ_{h}^{k} ⩽ r_{h} M ζ_{h}^{k} . \end{matrix} \end{matrix}$
Theorem 1 ([6]).

Under the discrete maximum principle assumption [ 9 ], there exists a constant $α > 0$ such that: $\begin{matrix} (2.17) & b (u_{h}^{k}, u_{h}^{k}) = a (u_{h}^{k}, u_{h}^{k}) + λ (u_{h}^{k}, u_{h}^{k}) ⩾ α {‖ u_{h}^{k} ‖}_{H^{1} (Ω)}, \end{matrix}$ where $\begin{matrix} λ = (\frac{‖ b_{j} ‖_{\infty}^{2}}{2 γ} + \frac{γ}{2} + ‖ a_{0} ‖_{\infty}), α = \frac{γ}{2} . \end{matrix}$

Notation 1.
Let $X_{h}$ be the set of discrete subsolutions. Then, we have the following theorem.
Theorem 2 ([2]).

Under the discrete maximum principle, the solution of the system of EQVI ( 2.14 ) is the maximum element of $X_{h}$ .

Remark 2.
In this situation, the existence of a unique continuous solution to the stationary system can be handled in the spirit of (2.5) or by adapting the algorithmic approach developed for the coercive and noncoercive problems using the Bensoussan’s algorithm, cf. (1.5) just a brief description of it and skip over the proofs.

3. Existence and uniqueness for discrete PQVIs

In [2], we have proved the existence and uniqueness of the discrete QVIs (2.14) using the algorithm based on Euler time scheme combined with a finite element method which has already been used our previous researches regarding the evolutionary free boundary problems (see [5]).

3.1. A fixed-point mapping associated with the system of QVIs

For that, let us first introduce the initial iteration ${\bar{u}}_{h}^{0}$ is solution of $\begin{matrix} (3.1) & b ({\bar{u}}_{h}^{0}, v_{h}) = (f ({\bar{u}}_{h}^{0}), v_{h}), \forall v_{h} \in V_{h}^{i}, \end{matrix}$ where $\begin{matrix} b ({\bar{u}}_{h}^{0}, v_{h}) = a ({\bar{u}}_{h}^{0}, v_{h}) + μ ({\bar{u}}_{h}^{0}, v_{h}) . \end{matrix}$

Now, we consider the mapping $\begin{matrix} (3.2) & \begin{matrix} T_{h} : L_{+}^{\infty} (Ω) & ⟶ V_{h} \\ W & ⟶ T W = ξ_{h}^{k} = \partial_{h} (f ({\bar{u}}_{h}^{k - 1}) + λ {\bar{u}}_{h}^{k - 1}, r_{h} M w^{k - 1}), \end{matrix} \end{matrix}$ where $L_{+}^{\infty} (Ω)$ denotes the positive cone of $L^{\infty} (Ω)$ and $ξ_{h}^{k}$ is solution of the following problem $\begin{matrix} (3.3) & \{\begin{matrix} b (ξ_{h}^{k}, v_{h} - ξ_{h}^{k}) ⩾ (F^{k - 1}, v_{h} - ξ_{h}^{k}), \\ ξ_{h}^{k} ⩽ r_{h} M w^{k - 1}, k = 1, \dots, n, \end{matrix} \end{matrix}$ where $\begin{matrix} F^{k - 1} = f (u_{h}^{k - 1}) + λ u_{h}^{k - 1} . \end{matrix}$

3.2. An iterative discrete algorithm

As we have chosen before in the iterative semi-discrete algorithm ${\bar{u}}_{h}^{0} = {\bar{u}}_{h 0}$ the solution of the following full-discrete equation $\begin{matrix} (3.4) & b ({\bar{u}}_{h}^{0}, v_{h}) = (g^{0}, v_{h}), v_{h} \in V^{h}, \end{matrix}$ where $g^{0}$ is a linear and a regular function.

Now we give the full following discrete algorithm $\begin{matrix} (3.5) & {\bar{u}}_{h}^{k} = T_{h} {\bar{u}}^{k - 1}, k = 1, \dots, n, \end{matrix}$ where ${\bar{u}}_{h}^{k}$ is the solution of the problem (2.14).

Let $F^{k - 1} (w) = f (w) + λ w$ , ${\tilde{F}}^{k - 1} (\tilde{w}) = f (\tilde{w}) + λ \tilde{w} \in L^{\infty} (Ω)$ be the corresponding right-hand sides to the EQVIs.

Lemma 1 ([9]).

Under the previous assumption and the DMP we have, if $\begin{matrix} F^{k - 1} (w) ≧ F^{k - 1} (\tilde{w}), \end{matrix}$ then $\begin{matrix} u_{h}^{k} = \partial (F^{k - 1} (w)) ≧ {\tilde{u}}_{h}^{k} = \partial (F^{k - 1} (\tilde{w})) . \end{matrix}$

Theorem 3 ([2]).

The sequences $({\bar{u}}_{h}^{k})$ well defined in $k$ and converge to the unique solution of system of inequalities ( 2.14 ), where $\begin{matrix} (3.6) & k = {w \in L_{+}^{\infty} (Ω) such that 0 ⩽ w ⩽ {\bar{u}}_{h}^{0}} . \end{matrix}$

3.3. Regularity of sequences of (3.5)

Theorem 4 (Lewy Stampacchia inequality [1]).

Let A be an elliptic operator defined in ( 1.2 ) and u the solution of an elliptic variational inequalities (EVIs) with a simple obstacle $ψ \in H^{1} (Ω)$ , $ψ > 0$ in $\partial Ω$ and the right hand side $f \in L^{\infty} (Ω)$ such that $A u ⩾ g$ in the sense of $H^{- 1}$ , where $g \in L^{2} (Ω)$ . Then $\begin{array}{l} (3.7) & f ⩾ A ψ ⩾ f \land g, \\ (3.8) & A u \in L^{\infty} (Ω) \end{array}$ and $\begin{matrix} (3.9) & u \in W^{2, p} (Ω) . \end{matrix}$

Lemma 2.
For $i = 1, \dots, M$ , we have $\begin{matrix} (3.10) & {‖ {\bar{u}}_{h}^{k} ‖}_{W^{2, p} (Ω)} ⩽ C, 2 ⩽ p < \infty, \end{matrix}$ where ${\bar{u}}_{h}^{k}$ is a subsolution of the problem ( 2.14 ).
Proof.
It is clear that $\begin{matrix} {\bar{u}}_{h}^{1} = \partial_{h} (F^{0}, r_{h} M w^{0}) \end{matrix}$ is the solution of (2.14) with the obstacle $r_{h} M w^{0} = r_{h} ψ$ and the right hand side $F^{0}$ and ${\bar{u}}_{h}^{0} \in W^{2, p} (Ω)$ .

Since $\begin{matrix} ‖ ψ ‖_{W^{2, p} (Ω)} ⩽ C_{1}, \end{matrix}$ then, we have $\begin{matrix} B ψ ⩾ - C_{1} + λ ψ, \end{matrix}$ where $\begin{matrix} B = A + λ I . \end{matrix}$

Using Lewy Stampacchia inequality, we get $\begin{matrix} {‖ {\bar{u}}_{h}^{1} ‖}_{W^{2, p} (Ω)} ⩽ C_{2}, \end{matrix}$ where $C_{2} = max (- C_{1}, λ C_{3})$ , with $‖ ψ ‖_{W^{2, p} (Ω)} ⩽ C_{3}$ .

Now, we assume that $\begin{matrix} {‖ {\bar{u}}_{h}^{k - 1} ‖}_{W^{2, p} (Ω)} ⩽ C_{4}, \end{matrix}$ then $ψ = r_{h} M w^{k}$ verify $\begin{matrix} ‖ ψ ‖_{W^{2, p} (Ω)} ⩽ C_{4} \end{matrix}$ and we have $B ψ ⩾ - C_{4}$ in $H^{- 1} (Ω)$ , then $\begin{matrix} {‖ {\bar{u}}_{h}^{k} ‖}_{W^{2, p} (Ω)} ⩽ C_{3} . \end{matrix}$ □
4. Geometrical convergence of the discrete algorithm

Lemma 3 ([2]).

For $0 ⩽ λ ⩽ inf (\frac{k}{‖ {\bar{u}}^{0} ‖_{\infty}}, 1)$ , where k is a positive constant defined in ( 1.8 ), then we have $\begin{matrix} (4.1) & T_{h} (0) ⩾ λ {‖ {\bar{u}}_{h}^{0} ‖}_{\infty} . \end{matrix}$

Proposition 1 ([2]).

Let $ω \in [0, 1]$ such that $\begin{matrix} (4.2) & w - v ⩽ ω w, \forall v, w \in k, \end{matrix}$ then, we have $\begin{matrix} (4.3) & T_{h} v - T_{h} w ⩽ ω (1 - λ) T_{h} v . \end{matrix}$

Proposition 2.
Under the assumptions and previous notations, we have $\begin{matrix} (4.4) & {‖ {\bar{u}}_{h}^{k} - u_{h}^{\infty} ‖}_{\infty} ⩽ {(1 - λ)}^{k} ‖ {\bar{u}}_{h}^{0} ‖, \end{matrix}$ where ${\bar{u}}_{h}^{k}$ is the subsolution of ( 2.14 ), and $u_{h}^{\infty}$ is an asymptotic semi-discrete solution of ( 1.1 ) in space using the standard finite element method.
Proof.
Using Theorem 3, we have $\begin{matrix} 0 ⩽ u_{h}^{\infty} ⩽ {\bar{u}}_{h}^{0}, \end{matrix}$ then $\begin{matrix} 0 ⩽ {\bar{u}}_{h}^{0} - u_{h}^{\infty} ⩽ {\bar{u}}_{h}^{0} . \end{matrix}$

Using Proposition 1 with $ω = 1$ , we get $\begin{matrix} 0 ⩽ T_{h} {\bar{u}}_{h}^{0} - T_{h} u_{h}^{\infty} ⩽ (1 - λ) T {\hat{U}}_{h}^{0}, \end{matrix}$ or $\begin{matrix} 0 ⩽ {\bar{u}}_{h}^{1} - u_{h}^{\infty} ⩽ (1 - λ) {\bar{u}}_{h}^{1} . \end{matrix}$

Using Proposition 1 again with $ω = 1 - λ$ , we get $\begin{matrix} 0 ⩽ T_{h} {\bar{u}}_{h}^{1} - T_{h} u_{h}^{\infty} ⩽ (1 - λ) (1 - λ) T {\bar{u}}_{h}^{1}, \end{matrix}$ i.e. $\begin{matrix} 0 ⩽ {\bar{u}}_{h}^{2} - u_{h}^{\infty} ⩽ {(1 - λ)}^{2} {\bar{u}}_{h}^{2} \end{matrix}$ and by induction $\begin{matrix} 0 ⩽ {\bar{u}}_{h}^{k} - u_{h}^{\infty} ⩽ {(1 - λ)}^{k} {\bar{u}}_{h}^{k} . \end{matrix}$

Under the third step of the proof of Theorem 3 in [2], we deduce that $\begin{matrix} 0 ⩽ {\bar{u}}_{h}^{k} - u_{h}^{\infty} ⩽ {(1 - λ)}^{k} {\bar{u}}_{h}^{0} . \end{matrix}$ □
5. Optimal error estimates and asymptotic behavior

Before discussing the results, it is interesting to introduce the result of the following problems $\begin{matrix} (5.1) & {\tilde{u}}_{h}^{k} = T_{h} {\bar{u}}_{h}^{k - 1}, k = 2, 3, \dots, \end{matrix}$ where ${\bar{u}}_{h}^{0}$ solution of (3.4) and ${\bar{u}}_{h}^{k - 1}$ is the subsolution of (2.14).

Theorem 5.
For all $k = 1, \dots, n$ and C is a constant independent with n, we have the following estimate $\begin{matrix} (5.2) & {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} ⩽ C h^{2} | log h |^{2}, \end{matrix}$ where ${\bar{u}}^{k}$ is the subsolution of semi-discrete problem in time using the Euler scheme.
Proof.
The proof is similar to that in [8] which has been treated the finite element approximation of elliptic quasi variational inequalities with nonlinear operators. □

The following lemma will play a crucial role in obtaining the approximation error: Lemma 4.
For all $k = 1, \dots, n$ and C independent of n,we have the following estimate $\begin{matrix} (5.3) & {‖ {\bar{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} ⩽ \sum_{p = 0}^{k} {‖ {\bar{u}}^{p} - {\tilde{u}}_{h}^{p} ‖}_{\infty} . \end{matrix}$
Proof.
By induction, we have $\begin{matrix} {\bar{u}}^{1} = T {\bar{u}}^{0}, {\bar{u}}_{h}^{1} = T_{h} {\bar{u}}_{h}^{0} and {\tilde{u}}_{h}^{1} = T_{h} {\bar{u}}^{0}, \end{matrix}$ then $\begin{array}{rcl} {‖ {\bar{u}}^{1} - {\bar{u}}_{h}^{1} ‖}_{\infty} & ⩽ & {‖ {\bar{u}}^{1} - {\tilde{u}}_{h}^{1} ‖}_{\infty} + {‖ {\tilde{u}}_{h}^{1} - {\bar{u}}_{h}^{1} ‖}_{\infty} \\ (5.4) & ⩽ & {‖ {\bar{u}}^{1} - {\tilde{u}}_{h}^{1} ‖}_{\infty} + {‖ T_{h} {\bar{u}}^{0} - T_{h} {\bar{u}}_{h}^{0} ‖}_{\infty} . \end{array}$

Since $T_{h}$ is Lipschitz, thus $\begin{array}{rcl} {‖ {\bar{u}}^{1} - {\bar{u}}_{h}^{1} ‖}_{\infty} & ⩽ & {‖ {\bar{u}}^{1} - {\tilde{u}}_{h}^{1} ‖}_{\infty} + {‖ {\bar{u}}^{0} - {\bar{u}}_{h}^{0} ‖}_{\infty} \\ ⩽ & \sum_{p = 0}^{1} {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} . \end{array}$

Assume that $\begin{matrix} {‖ {\bar{u}}^{k - 1} - {\bar{u}}_{h}^{k - 1} ‖}_{\infty} ⩽ \sum_{p = 0}^{k - 1} {‖ {\bar{u}}^{p - 1} - {\tilde{u}}_{h}^{p - 1} ‖}_{\infty}, \end{matrix}$ then, we have $\begin{array}{rcl} {‖ {\bar{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} & ⩽ & {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} + {‖ {\tilde{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} \\ ⩽ & {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} + {‖ T_{h}^{k - 1} \bar{u} - T_{h}^{k - 1} {\bar{u}}_{h} ‖}_{\infty} \\ ⩽ & {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} + {‖ {\bar{u}}^{k - 1} - {\bar{u}}_{h}^{k - 1} ‖}_{\infty} . \end{array}$

Using the induction assumption, we get $\begin{array}{rcl} {‖ {\bar{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} & ⩽ & {‖ {\bar{u}}^{k} - {\tilde{u}}_{h}^{k} ‖}_{\infty} + \sum_{p = 0}^{k - 1} {‖ {\bar{u}}^{p} - {\tilde{u}}_{h}^{p} ‖}_{\infty} \\ ⩽ & \sum_{p = 0}^{k} {‖ {\bar{u}}^{p} - {\tilde{u}}_{h}^{p} ‖}_{\infty} . \end{array}$ □

5.1. $L^{\infty}$ -Optimal error estimate

Theorem 6.
For all $k = 1, \dots, n$ and C independent by n, we have the following estimate $\begin{matrix} (5.5) & {‖ u^{k} - u_{h}^{k} ‖}_{\infty} ⩽ C h^{2} | log h |^{3} . \end{matrix}$
Proof.
We have $\begin{array}{rcl} {‖ u^{k} - u_{h}^{k} ‖}_{\infty} & ⩽ & {‖ u^{k} - {\bar{u}}^{k} ‖}_{\infty} + {‖ {\bar{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} \\ ⩽ & {‖ u^{k} - {\bar{u}}^{k} ‖}_{\infty} + {‖ {\bar{u}}^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} + {‖ {\bar{u}}_{h}^{k} - u_{h}^{k} ‖}_{\infty} \\ ⩽ & {‖ u^{k} - {\bar{u}}^{k} ‖}_{\infty} + {‖ {\bar{u}}_{h}^{k} - u_{h}^{k} ‖}_{\infty} + {‖ {\bar{u}}^{0} - {\bar{u}}_{h}^{0} ‖}_{\infty} + \sum_{p = 1}^{k} {‖ {\bar{u}}^{p} - {\tilde{u}}_{h}^{p} ‖}_{\infty} . \end{array}$

From the initial data in (3.4), we have ${\bar{u}}^{0} = u (x, 0) = u_{0}$ and ${\bar{u}}_{h}^{0} = u_{0 h}$ , then, it can be used the following standard error estimate [8] and [9] which investigated the stationary case $\begin{matrix} (5.6) & {‖ {\bar{u}}^{0} - {\bar{u}}_{h}^{0} ‖}_{\infty} ⩽ C h^{2} | log h |^{2} . \end{matrix}$

Using the estimates (5.2), (5.3) and (5.5), thus $\begin{array}{l} {‖ u^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} ⩽ & C h^{2} | log h |^{2} + k C h^{2} | log h |^{2} \\ + {(1 - λ)}^{k} ‖ {\bar{u}}^{0} ‖ + {(1 - λ)}^{k} ‖ {\bar{u}}_{h}^{0} ‖ . \end{array}$

Setting $\begin{matrix} {(1 - λ)}^{k} = h^{2}, \end{matrix}$ then $\begin{matrix} k = \frac{2 | log h |}{| log (1 - λ) |} . \end{matrix}$

Therefore, it can be deduced $\begin{matrix} {‖ u^{k} - {\bar{u}}_{h}^{k} ‖}_{\infty} ⩽ C h^{2} | log h |^{3} . \end{matrix}$ □
Proposition 3 ([5]).

Under the assumption ( 1.7 ), we have for all $k = 1, \dots, n$ the following estimates: $\begin{matrix} {‖ u_{h}^{n} - u_{h}^{\infty} ‖}_{\infty} ⩽ {(\frac{1 + (Δ t) c}{1 + (Δ t) β})}^{n} {‖ u_{h}^{0} - u_{h}^{\infty} ‖}_{\infty}, k = 0, \dots, n, \end{matrix}$ $u_{h}^{\infty}$ is an asymptotic continuous solution of ( 1.1 ) and $u_{h_{0}}$ solution of ( 2.14 ).

Now we evaluate the variation in $L^{\infty}$ -norm between $u (T, x)$ , the discrete solution calculated at the moment $T = n Δ t$ and $U^{\infty}$ , the asymptotic continuous solution of (1.1).

Theorem 7.
Under the results of Proposition 3 and Theorem 6 , we have $\begin{matrix} {‖ u_{h}^{n} - u^{\infty} ‖}_{\infty} ⩽ C [h^{2} | log h |^{3} + {(\frac{1 + (Δ t) c}{1 + (Δ t) β})}^{n}] . \end{matrix}$
Proof.
Using Theorem 5 and Proposition 3, it can be easily obtained: $\begin{array}{rcl} {‖ u_{h}^{n} - u^{\infty} ‖}_{\infty} & ⩽ & {‖ u_{h}^{n} - u_{h}^{\infty} ‖}_{\infty} + {‖ u_{h}^{\infty} - u^{\infty} ‖}_{\infty} \\ ⩽ & C [h^{2} | log h |^{3} + {(\frac{1 + (Δ t) c}{1 + (Δ t) β})}^{n}], \end{array}$ which completes the proof. □

6. Simple numerical example

We use the following example:

Consider the following evolutionary HJB equation $\begin{matrix} (6.1) & \{\begin{matrix} {max}_{1 ⩽ i ⩽ 2} (\frac{\partial u^{i}}{\partial t} + A^{i} u^{i} - f^{i}) = 0, in [0, 1] \times [0, 2], \\ u (x, 0) = φ (x) = x, \\ u (0, t) in Ω = 0, \end{matrix} \end{matrix}$ and $\begin{matrix} (6.2) & A^{1} u = \frac{\partial^{2} u}{\partial x^{2}}, A^{2} u = \frac{\partial^{2} u}{\partial x^{2}} + u, f^{1} (u) = f^{2} (u) = max (A^{1} u, A^{2} u) . \end{matrix}$

The exact solution of the problem is $\begin{matrix} u (x, t) = (x^{4} - x^{5}) sin (10 x) cos (20 π t) . \end{matrix}$

Setting $t = 0.1$ , $h = 0.2$ and $d = 0.5$ , we get the following results at the last iteration $t_{final}$ $\begin{matrix} (6.3) & \begin{matrix} x_{i} & 0.2 & 0.4 & 0.6 & 0.8 \\ e = ‖ u^{\infty} - u_{h} ‖_{\infty} & 3.15723 (- 5) & 8.20085 (- 5) & 3.11281 (- 5) & 2.12364 (- 5) \end{matrix} \end{matrix}$

Also, if we set $t = 0.15$ , $h = 0.2$ and $d = 0.5$ , we get the following results: $\begin{matrix} (6.4) & \begin{matrix} x_{i} & 0.2 & 0.4 & 0.6 & 0.8 \\ e = ‖ u^{\infty} - u_{h} ‖_{\infty} & 7.058191 (- 5) & 4.25441 (- 5) & 2.32197 (- 5) & 5.08795 (- 5) \end{matrix} \end{matrix}$

We can calculate the following estimate: $h^{2} | log h |^{3} = 0.119213 \times 10^{- 1}$ .

Table (6.3) and (6.4) show the uniform norm of the asymptotic behavior $\begin{matrix} (6.5) & e = {‖ u^{\infty} - u_{h} (x_{i}, t_{final}) ‖}_{\infty} \end{matrix}$ when iteration terminated according to the value $(x_{i}, t_{final})$ , $i = 1, \dots, 4$ for the two different steps of the time discretization $t = 0.2$ and 0.15. Thus, we can see that $e < h^{2} | log h |^{3}$ , necessarily e less than $\begin{matrix} (6.6) & h^{2} | log h |^{3} + {(\frac{1 + (Δ t) c}{1 + (Δ t) β})}^{n} . \end{matrix}$

7. Conclusion

In this paper, the regularity and convergence of the presented algorithm sequences of the discontinuous Galerkin methods coupled with Euler time discretization scheme of PQVIs are analyzed. Also an optimal error estimate with an asymptotic behavior in uniform norm are given with respect to the same proposed boundary conditions in [2]. A future work will propose a relaxation scheme for solving these qualitative problems which are an extension of Lions and Mercier’s scheme [16] for solving the elliptic case. The convergence of the new scheme will be established and the numerical example will be shown to prove that the new presented scheme is efficient.

Footnotes

Acknowledgements

The author gratefully acknowledge Qassim University in Kingdom of Saudi Arabia and He would like to thank the anonymous referees for their careful reading and for relevant remarks/suggestions which helped him to improve the paper.

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An optimal error estimate of finite element method for parabolic quasi-variational inequalities with non linear source terms

Abstract

Keywords

1. Introduction

2. Parabolic quasi-variational inequalities

2.1. The temporal discretization

2.2. An iterative semi-discrete algorithm

Definition 1. ζ h k is said to be a subsolution for the system of EQVIs (2.14) if (2.16) b ( ζ h k , φ s ) ⩽ ( f ( w ) + λ ζ h k − 1 , φ s ) , ∀ φ s , s = 1 , … , m ( h ) , ζ h k ⩽ r h M ζ h k . Theorem 1 ([6]).

Notation 1. Let X h be the set of discrete subsolutions. Then, we have the following theorem. Theorem 2 ([2]).

3.1. A fixed-point mapping associated with the system of QVIs

3.2. An iterative discrete algorithm

Lemma 1 ([9]).

Theorem 3 ([2]).

3.3. Regularity of sequences of (3.5)

Theorem 4 (Lewy Stampacchia inequality [1]).

Lemma 3 ([2]).

Proposition 1 ([2]).

7. Conclusion

Footnotes

Acknowledgements

References

Notation 1.
Let $X_{h}$ be the set of discrete subsolutions. Then, we have the following theorem.
Theorem 2 ([2]).