We study a coercive quasi-variational inequality (QVI) system and propose a generalized Schwarz method using finite element approximations. The discrete solution is iteratively constructed through monotone upper and lower sequences, and its convergence is rigorously established in the \(\mathfrak{L}^\infty\) norm. This framework ensures stability, geometric convergence, and efficient computation on overlapping subdomains.
Quasi-variational inequalities (QVIs) provide a powerful mathematical framework for modeling equilibrium and control problems in which the constraints depend on the unknown solution itself. They naturally arise in various applied fields such as optimal switching, impulse control, and energy management, where several coupled processes interact through obstacle-type conditions. The theoretical foundations of QVIs were first established in [1, 2], where the classical theory of variational inequalities was extended to the case in which the admissible set depends on the solution. This generalization introduced strong nonlinear and interdependent features that make QVIs particularly challenging to analyze.
From a numerical perspective, the accurate and stable approximation of QVIs remains a difficult problem. Early studies investigated both finite element and finite difference schemes for variational and quasi-variational inequalities [3– 6]. The work in [7] considered noncoercive QVIs arising in impulse control, while [8] established one of the first \(\mathfrak{L}^\infty\)-error estimates for coercive elliptic QVI systems, providing an important theoretical foundation for later numerical developments.
More recently, domain decomposition and Schwarz-type iterative techniques have been successfully applied to elliptic QVIs. The studies in [9– 11] demonstrated that overlapping subdomain strategies can efficiently handle large and coupled elliptic QVIs while ensuring convergence and stability.
Motivated by these developments, the present paper proposes a generalized overlapping domain decomposition method for systems of coercive elliptic QVIs. The approach combines a finite element discretization with an iterative Schwarz scheme applied over overlapping subdomains. The convergence of the method is established in the \(\mathfrak{L}^\infty\) norm by constructing monotone upper and lower sequences, which ensure stability, geometric convergence, and improved accuracy near subdomain interfaces.
The paper is organized as follows. §2 presents the continuous and discrete formulations of the coercive QVI system and discusses its main analytical properties. §3 introduces the Schwarz domain decomposition algorithm and describes the iterative construction of the monotone sequences. §4 proves the monotone convergence of these sequences toward the discrete solution in the \(\mathfrak{L}^\infty\) norm.
This section describes the coupled system of quasi-variational inequalities (QVIs) that forms the basis of our analysis. We begin with the continuous formulation that models nonlinear relations between components through solution-dependent constraints. Then, we provide the discrete approximation and discuss the main assumptions ensuring the well-posedness of the problem.
Let \(\Omega \subset \mathbb{R}^{\mathtt{d}}\) (\(\mathtt{d} \ge 1\)) be a bounded domain with a smooth boundary \(\partial \Omega\). We consider a system of \(\mathfrak{m}\) elliptic QVIs whose unknown is the vector function \[\vartheta = (\vartheta^{1}, \ldots, \vartheta^{\mathfrak{m}}) \in (\mathcal{H}_0^1(\Omega))^{\mathfrak{m}}.\]
Each component \(\vartheta^{\mathfrak{j}}\) satisfies, for \(1 \le \mathfrak{j} \le \mathfrak{m}\), the following inequality: \[\label{iqv} \left\{ \begin{array}{l} \mathfrak{a}^{\mathfrak{j}}\!\left(\vartheta^{\mathfrak{j}},\, \mathtt{u} – \vartheta^{\mathfrak{j}}\right) \ge \big(G^{\mathfrak{j}},\, \mathtt{u} – \vartheta^{\mathfrak{j}}\big), \quad \forall \mathtt{u} \in \mathcal{H}_0^1(\Omega), \\[6pt] \vartheta^{\mathfrak{j}} \le \mathcal{M}\vartheta^{\mathfrak{j}}, \qquad \vartheta^{\mathfrak{j}} \ge 0, \qquad \mathtt{u} \le \mathcal{M}\vartheta^{\mathfrak{j}}. \end{array} \right. \tag{1}\]
The system in (1) involves the following elements:
\(\bullet\) Bilinear form. The term \(\mathfrak{a}^{\mathfrak{j}}(\cdot,\cdot)\) corresponds to an elliptic differential operator. It is continuous and coercive on \(\mathcal{H}^1(\Omega)\), ensuring stability of the formulation.
\(\bullet\) Source term. Each equation includes a function \(G^{\mathfrak{j}} \in \mathfrak{L}^{\infty}(\Omega)\) with \(G^{\mathfrak{j}} \ge 0\), representing an external input or forcing term.
\(\bullet\) Coupling operator. The nonlinear operator \(\mathcal{M}\) links the components by a constraint that depends on the solution itself. A common expression is \[\mathcal{M}\vartheta^{\mathfrak{i}}(\mathtt{x}) = \mathtt{k} + \inf_{\eta \neq \mathfrak{i}} \vartheta^{\eta}(\mathtt{x}), \quad \mathtt{k} > 0,\] where \(\mathtt{k}\) denotes the switching cost between different modes or states.
The elliptic operator \(N^{\mathfrak{j}}\) related to \(\mathfrak{a}^{\mathfrak{j}}(\cdot,\cdot)\) takes the general form: \[N^{\mathfrak{j}} = \sum_{\mathfrak{t,s}=1}^{\mathtt{d}} \mathfrak{a}_{\mathfrak{ts}}^{\mathfrak{j}}(\mathtt{x}) \frac{\partial^2}{\partial \mathtt{x}_{\mathfrak{t}}\partial \mathtt{x}_{\mathfrak{s}}} + \sum_{\mathfrak{s}=1}^{\mathtt{d}} \mathfrak{c}_{\mathfrak{s}}^{\mathfrak{j}}(\mathtt{x}) \frac{\partial}{\partial \mathtt{x}_{\mathfrak{s}}} + \mathfrak{a}_0^{\mathfrak{j}}(\mathtt{x}),\] where all coefficients are smooth, symmetric, and satisfy the ellipticity condition: \[\sum_{\mathfrak{t,s}=1}^{\mathtt{d}} \mathfrak{a}_{\mathfrak{ts}}^{\mathfrak{j}}(\mathtt{x}) \zeta_{\mathfrak{t}}\zeta_{\mathfrak{s}} \ge \mu |\zeta|^2, \qquad \mathfrak{a}_0^{\mathfrak{j}}(\mathtt{x}) \ge \delta > 0, \quad \mu,\delta > 0.\]
The bilinear form \(\mathfrak{a}^{\mathfrak{j}}\) is expressed as: \[\mathfrak{a}^{\mathfrak{j}}(\mathtt{u},\mathtt{v}) = \int_{\Omega} \Big( \sum_{\mathfrak{t,s}=1}^{\mathtt{d}} \mathfrak{a}_{\mathfrak{ts}}^{\mathfrak{j}} \frac{\partial \mathtt{u}}{\partial \mathtt{x}_{\mathfrak{t}}} \frac{\partial \mathtt{v}}{\partial \mathtt{x}_{\mathfrak{s}}} + \sum_{\mathfrak{s}=1}^{\mathtt{d}} \mathfrak{b}_{\mathfrak{s}}^{\mathfrak{j}} \frac{\partial \mathtt{u}}{\partial \mathtt{x}_{\mathfrak{s}}}\mathtt{v} + \mathfrak{a}_0^{\mathfrak{j}} \mathtt{uv} \Big)\, d\mathtt{x}.\]
It satisfies the coercivity inequality: \[\mathfrak{a}^{\mathfrak{j}}(\mathtt{u},\mathtt{u}) \ge \beta \|\mathtt{u}\|_{\mathcal{H}^1(\Omega)}^2, \quad \beta > 0.\]
Under the above regularity and coercivity assumptions, the coupled QVI system (1) admits a unique solution \[(\vartheta^1,\ldots,\vartheta^{\mathfrak{m}}) \in (\mathcal{W}^{2,\mathfrak{p}}(\Omega))^{\mathfrak{m}}, \qquad 2 \le \mathfrak{p} < \infty.\]
This result guarantees both well-posedness and regularity of the continuous formulation, which will serve as the foundation for its discrete approximation.
This section presents the discrete version of the continuous system (1). The approximation is developed using conforming finite element spaces defined on overlapping subdomains. We first describe the finite element setting, then give the discrete quasi-variational inequality (QVI) system, and finally recall the main error estimate.
The computational domain \(\Omega\)
is divided into \(\mathfrak{q}\)
overlapping subregions \[\Omega =
\bigcup_{\mathfrak{i}=1}^{\mathfrak{q}}\Omega_{\mathfrak{i}},\]
each equipped with a regular triangulation \(\tau^{\mathfrak{h}_{\mathfrak{i}}}\)
characterized by the local mesh size \(\mathfrak{h}_{\mathfrak{i}} > 0\). The
triangulations are independent from one subdomain to another, which
allows local refinement and flexibility in the overlapping
structure.
For every subdomain \(\Omega_{\mathfrak{i}}\), we introduce the
local finite element space \[V_{\mathfrak{h}}
= \Big\{
v_{\mathfrak{h}} \in C(\overline{\Omega_{\mathfrak{i}}}) :
v_{\mathfrak{h}} \text{ is affine on each element of }
\tau^{\mathfrak{h}_{\mathfrak{i}}},
\;
v_{\mathfrak{h}} = 0 \text{ on }
\partial\Omega_{\mathfrak{i}}\!\cap\!\Omega_{\mathfrak{l}},
\;
\forall\,\mathfrak{l}\neq\mathfrak{i}
\Big\}.\]
Given a boundary trace \(w \in C(\overline{\Sigma_{\mathfrak{i}}})\), the constrained subspace is defined by \[V_{\mathfrak{h}_{\mathfrak{i}}}^{(w)} = \Big\{ v_{\mathfrak{h}} \in V_{\mathfrak{h}} : v_{\mathfrak{h}} = 0 \text{ on } \partial\Omega\!\cap\!\Omega_{\mathfrak{i}}, \; v_{\mathfrak{h}} = \pi_{\mathfrak{h}_{\mathfrak{i}}}(w) \text{ on } \Sigma_{\mathfrak{i}} \Big\},\] where \(\pi_{\mathfrak{h}_{\mathfrak{i}}}\) is the nodal interpolation operator on \(\Sigma_{\mathfrak{i}}\). These constructions guarantee a conforming finite element discretization over the overlapping configuration.
The discrete version of system (1) is written as \[\label{iqv_d} \left\{ \begin{array}{l} \mathfrak{a}^{\mathfrak{j}} \!\left( \vartheta_{\mathfrak{h}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \vartheta_{\mathfrak{h}}^{\mathfrak{j}} \right) \ge \left( \mathfrak{G}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \vartheta_{\mathfrak{h}}^{\mathfrak{j}} \right), \quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}, \\[6pt] \vartheta_{\mathfrak{h}}^{\mathfrak{j}} \le \mathtt{r}_{\mathfrak{h}} \mathcal{M} \vartheta_{\mathfrak{h}}^{\mathfrak{j}}, \qquad \vartheta_{\mathfrak{h}}^{\mathfrak{j}} \ge 0. \end{array} \right. \tag{2}\]
Here \(\mathtt{r}_{\mathfrak{h}}\) denotes the standard finite element interpolation operator. Using the canonical basis \(\{\varphi_{\mathfrak{t}}\}\) of \(V_{\mathfrak{h}}\), the discrete operators take the matrix representation \[(\mathfrak{G}^{\mathfrak{j}}(\vartheta_{\mathfrak{h}}))_{\mathfrak{t}} = (\mathfrak{G}^{\mathfrak{j}}, \varphi_{\mathfrak{t}}), \qquad (\mathcal{N}^{\mathfrak{j}})_{\mathfrak{ts}} = \mathfrak{a}^{\mathfrak{j}} (\varphi_{\mathfrak{t}}, \varphi_{\mathfrak{s}}).\]
If the stiffness matrices \(\mathcal{N}^{\mathfrak{j}}\) are \(M\)-matrices, the discrete system (2) admits a unique componentwise solution \(\vartheta_{\mathfrak{h}}^{\mathfrak{j}} \in V_{\mathfrak{h}}\). This property ensures stability and monotonicity of the numerical approximation.
The accuracy of the discrete approximation is characterized by the following estimate.
Theorem 1. [12] Let \(\vartheta\) and \(\vartheta_{\mathfrak{h}}\) be the exact and discrete solutions of (1) and (2), respectively. There exists a constant \(\mathfrak{C} > 0\), independent of the mesh size \(\mathfrak{h}\), such that \[\| \vartheta – \vartheta_{\mathfrak{h}} \|_{\mathfrak{L}^\infty(\Omega)} \le \mathfrak{C}\, \mathfrak{h}^2 |\log \mathfrak{h}|^3.\]
This result provides a uniform \(\mathfrak{L}^\infty\) convergence rate. It serves as the theoretical foundation for the numerical experiments and iterative algorithms introduced in the following sections.
This section develops a practical iterative Schwarz framework designed to solve systems of quasi-variational inequalities (QVIs) over overlapping subdomains. The main idea is to decompose the global computational region into smaller parts that can be processed in parallel, with iterative exchange of boundary information to ensure global consistency.
Consider a smooth bounded domain \(\Omega \subset \mathbb{R}^2\), decomposed into \(\mathfrak{q}\) overlapping subregions: \[\Omega = \bigcup_{\mathfrak{i}=1}^{\mathfrak{q}} \Omega_{\mathfrak{i}}, \qquad \Omega_{\mathfrak{i}} \cap \Omega_{\mathfrak{l}} \neq \emptyset \text{ for } \mathfrak{i} \neq \mathfrak{l}.\]
Each local domain \(\Omega_{\mathfrak{i}}\) has a smooth boundary \(\Gamma_{\mathfrak{i}} = \partial \Omega_{\mathfrak{i}}\). The interface between two neighboring subdomains is denoted by \[\Sigma_{\mathfrak{i}} = \partial \Omega_{\mathfrak{i}} \cap \Omega_{\mathfrak{l}}, \qquad \mathfrak{l} \neq \mathfrak{i}.\]
It is assumed that these interfaces do not overlap: \[\overline{\Sigma}_{\mathfrak{i}} \cap \overline{\Sigma}_{\mathfrak{l}} = \emptyset, \qquad \mathfrak{i} \neq \mathfrak{l}.\]
The restriction of a function \(\vartheta\) to any local domain is taken to belong to \[\vartheta_{|\Omega_{\mathfrak{i}}} \in \mathcal{W}^{2,\mathfrak{p}}(\Omega_{\mathfrak{i}}), \quad \mathfrak{i} = 1, \dots, \mathfrak{q}.\]
Such a geometric partition enables parallel computations while maintaining smooth data transfer across the overlapping interfaces.
For the continuous QVI system (1), each subdomain problem is defined as an independent variational inequality, subject to coupling conditions on the overlaps. The local iterates \(\vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1}\) are computed within admissible sets \(\mathcal{K}_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} \subset \mathcal{H}_0^1(\Omega)\) satisfying: \[\label{Sch_C} \begin{cases} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1}, \mathtt{u} – \vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} \right) \geq \left(G_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u} – \vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} \right), \quad \forall \mathtt{u} \in \mathcal{H}_0^1(\Omega), \\[6pt] \vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} \leq \mathcal{M}\vartheta_{\mathfrak{i}}^{\mathfrak{j}+1,\mathfrak{n}+1}, \qquad \mathtt{u} \leq \mathcal{M}\vartheta_{\mathfrak{i}}^{\mathfrak{j}+1,\mathfrak{n}+1}, \\[6pt] \vartheta_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} = \vartheta_{\mathfrak{l}}^{\mathfrak{j},\mathfrak{n}+1_{\mathfrak{il}}} \text{ on } \Gamma_{\mathfrak{i}}. \end{cases} \tag{3}\]
The associated admissible set reads \[\mathcal{K}_{\mathfrak{i}}^{\mathfrak{j},\mathfrak{n}+1} = \left\{ \mathtt{u} \in \mathcal{H}_0^1(\Omega) ~\middle|~ \mathtt{u} \leq \mathcal{M}\vartheta_{\mathfrak{i}}^{\mathfrak{j}+1,\mathfrak{n}+1},~ \mathtt{u} = \vartheta_{\mathfrak{l}}^{\mathfrak{j},\mathfrak{n}+1_{\mathfrak{il}}} \text{ on } \Gamma_{\mathfrak{i}} \right\}.\]
To manage the ordering of updates, we introduce the binary indicator \[1_{\mathfrak{il}} = \begin{cases} 1, & \text{if } \mathfrak{i} > \mathfrak{l}, \\[4pt] 0, & \text{if } \mathfrak{i} < \mathfrak{l}. \end{cases}\]
The local source term and restriction of the global solution are defined as \[G_{\mathfrak{i}}^{\mathfrak{j}} = \left(G^{\mathfrak{j}}\right)_{|\Omega_{\mathfrak{i}}}, \qquad \vartheta_{\mathfrak{i}}^{\mathfrak{j}} = \vartheta^{\mathfrak{j}}_{|\Omega_{\mathfrak{i}}}.\]
This iterative construction ensures that each local solution continuously refines the global approximation by exchanging updated traces on overlapping interfaces.
In the discrete finite element framework, the iteration (2) is reformulated in terms of local discrete iterates \(\vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \in \mathcal{K}_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \subset V_{\mathfrak{h}}\), which satisfy \[\label{sdp} \begin{cases} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1}, \mathtt{u}_{\mathfrak{h}} – \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \right) \geq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \right), \\[6pt] \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j}+1,\mathfrak{n}+1}, \quad \mathtt{u}_{\mathfrak{h}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j}+1,\mathfrak{n}+1}, \\[6pt] \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} = \vartheta_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1_{\mathfrak{il}}} \text{ on } \Gamma_{\mathfrak{i}}. \end{cases} \tag{4}\]
The corresponding discrete admissible set is defined by \[\mathcal{K}_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} = \left\{ \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}} ~\middle|~ \mathtt{u}_{\mathfrak{h}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M} \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j}+1,\mathfrak{n}+1},~ \mathtt{u}_{\mathfrak{h}} = \vartheta_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1_{\mathfrak{il}}} \text{ on } \Gamma_{\mathfrak{i}} \right\}.\]
The initial approximation is given by \[\check{\vartheta}_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},0} = 0, \quad \mathfrak{i} = 1,\dots,\mathfrak{q}, \quad \mathfrak{j} = 1,\dots,\mathfrak{m},\] and the discrete supersolution satisfies \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},0}, \mathtt{u}_{\mathfrak{h}} \right) = \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} \right), \quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}.\]
This iterative discrete scheme establishes a consistent update rule across subdomains and forms the foundation for proving monotone convergence of the discrete solution sequence.
Theorem 2. Let \(\left( \vartheta_{\mathfrak{i}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}+1} \right)_{\mathfrak{i}=\overline{1,\mathfrak{q}}}^{\mathfrak{j}=\overline{1,\mathfrak{m}};\mathfrak{n}\in\mathbb{N}}\) denote the sequence generated by the discrete Schwarz iteration. If the initialization is taken as a subsolution or a supersolution, then the iterative process converges monotonically to the unique discrete solution of the QVI system (2).
Proof. The proof is divided into three main parts in order to clearly establish the monotone convergence property.
Increasing property of the subsolution sequence:
We aim to prove that the sequence of discrete subsolutions constructed by the Schwarz process increases with respect to the iteration counter. The reasoning proceeds by verifying the first iteration (base case) and then showing the general result by induction.
Base Case: For any subdomain \(\Omega_{\mathfrak{i}}\) with index \(\mathfrak{i} < \mathfrak{l}\), the initialization step is given by \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0} = 0, \qquad \mathfrak{n}=0.\]
At the first iteration, the function \(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}\)
is computed as the solution of \[\left\{
\begin{array}{l}
\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1},
\mathtt{u}_{\mathfrak{h}} –
\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}\right)
\geq
(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} –
\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}),
\quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}, \\[4pt]
\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \leq
\mathtt{r}_{\mathfrak{h}}\mathcal{M}\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,1},
\quad \mathtt{u}_{\mathfrak{h}} \leq
\mathtt{r}_{\mathfrak{h}}\mathcal{M}\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,1},
\\[4pt]
\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} =
\check{\vartheta}_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},0} \text{ on
} \Gamma_{\mathfrak{i}},
\quad \mathtt{u}_{\mathfrak{h}} =
\check{\vartheta}_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},0} \text{ on
} \Gamma_{\mathfrak{i}}.
\end{array}
\right. \tag{5}\] Because the local right-hand side satisfies \(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}} \geq
0\), the weak formulation immediately implies \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}
\geq 0 = \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}.\]
Consequently, we can derive the following inequality: \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}
\leq \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \quad \text{and}
\quad \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \geq 0 \quad
\text{together imply that} \quad
\check{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},0} \geq 0. \tag{6}\]
Hence, the monotone behavior is verified for the first iteration.
Inductive Step: Assume that for some integer \(\mathfrak{n} \geq 0\), the discrete sequence satisfies the following inequalities: \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}-1} \leq \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}, \qquad \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} = \check{\vartheta}_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}-1} \text{ on } \Gamma_{\mathfrak{i}}.\] We now prove that the same ordering remains valid at level \(\mathfrak{n}+1\).
The function \(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}\) satisfies the local variational inequality: \[\label{eq:inductive_revised} \left\{ \begin{array}{l} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\! \left( \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}, \mathtt{u}_{\mathfrak{h}} – \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} \right) \geq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} \right), \quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}, \\[4pt] \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,\mathfrak{n}}, \\[4pt] \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} = \check{\vartheta}_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}-1} \text{ on } \Gamma_{\mathfrak{i}}. \end{array} \right. \tag{7}\]
To compare successive iterations, we substitute \(\mathtt{u}_{\mathfrak{h}} = \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1} – \mathtt{v}_{\mathfrak{h}}\), where \(\mathtt{v}_{\mathfrak{h}} \geq 0\), and substitute this expression into the variational inequality (4): \[\label{eq:v1} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}} \!\left( \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1}, \mathtt{v}_{\mathfrak{h}} \right) \leq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_{\mathfrak{h}} \right). \tag{8}\] Likewise, by inserting \(\mathtt{u}_{\mathfrak{h}} = \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} – \mathtt{v}_{\mathfrak{h}}\) into (7), we obtain \[\label{eq:v2} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}} \!\left( \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}, \mathtt{v}_{\mathfrak{h}} \right) \leq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_{\mathfrak{h}} \right). \tag{9}\] Subtracting (9) from (8) gives \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}} \!\left( \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} – \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1}, \mathtt{v}_{\mathfrak{h}} \right) \leq 0.\] Since the bilinear form \(\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}(\cdot,\cdot)\) is positive definite, it follows that \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1} \geq \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}.\] Hence, by mathematical induction, the entire sequence satisfies \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1} \geq \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} \geq 0, \qquad \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1} = \check{\vartheta}_{\mathfrak{l}\mathfrak{h}}^{\mathfrak{j},\mathfrak{n}} \text{ on } \Gamma_{\mathfrak{i}},\quad \text{then it follows that:} \quad \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \geq 0 \quad \text{implies} \quad \check{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j,n}} \geq 0.\] This proves that the discrete subsolution sequence is monotone increasing.
Decreasing property of the supersolution sequence
We now demonstrate that the discrete sequence \(\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}+1} \right)_{\mathfrak{i}=\overline{1,\mathfrak{q}}}^{\mathfrak{j}=\overline{1,\mathfrak{m}};\,\mathfrak{n}\in\mathbb{N}}\) is monotonically decreasing, assuming that the initial function \(\hat{\vartheta}_{\mathfrak{h}}\) is chosen as a discrete supersolution.
Base Case: We start with the first iteration, corresponding to \(\mathfrak{n}=0\). For each subdomain \(\Omega_{\mathfrak{i}}\) with index \(\mathfrak{i}<\mathfrak{l}\), the first local iterate \(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}\) is defined as the solution of the discrete quasi-variational system: \[\label{eq:sup1} \left\{ \begin{array}{l} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}, \mathtt{u}_{\mathfrak{h}} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \right) \geq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \right), \quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}, \\[6pt] \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,1}, \quad \mathtt{u}_{\mathfrak{h}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,1}, \\[6pt] \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} = \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},0} \text{ on } \Gamma_{\mathfrak{i}}, \quad \mathtt{u}_{\mathfrak{h}} = \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},0} \text{ on } \Gamma_{\mathfrak{i}}. \end{array} \right. \tag{10}\]
Let \(\varphi \in \mathcal{H}_0^1(\Omega)\) and denote by \(\varphi_* = \max(\varphi,0)\) its nonnegative part. We introduce the auxiliary test function \[\mathtt{u}_{\mathfrak{h}} = \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_*,\] and by substituting it into (10), and after some algebraic manipulation, we obtain: \[\label{eq:sup2} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}, \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_* \right) \leq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_* \right). \tag{11}\]
Next, using \(\mathtt{u}_{\mathfrak{h}} = \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_*\) as a test function in the variational inequality that defines the initial supersolution, we obtain: \[\label{eq:sup3} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}, \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_* \right) = \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_* \right). \tag{12}\]
Subtracting (12) from (11) gives
the inequality \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left(
\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} –
\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0},
\left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} –
\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_*
\right)
\leq 0 , \quad \text{where } \quad
\left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} –
\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\right)_{*} \geq
0.\] This directly implies that: \[\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}
\leq \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0},
\quad
\text{and on } \Gamma_{\mathfrak{i}}, \quad
\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1}
= \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},0}, \quad \text{then}
\quad \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},1} \geq 0 \quad
\text{implies} \quad
\hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},0}\geq 0.\]
Therefore, the first iteration step preserves the decreasing nature of
the sequence.
Inductive Step: Assume that for a certain iteration index \(\mathfrak{n}\geq1\), the following inequalities hold: \[\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} \leq \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}-1}, \qquad \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} = \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j},\mathfrak{n}-1} \text{ on } \Gamma_{\mathfrak{i}}.\] We now show that the same ordering property remains valid for the next iteration, \(\mathfrak{n}+1\).
The function \(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}}\) satisfies the local inequality: \[\label{eq:sup4} \left\{ \begin{array}{l} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}}, \mathtt{u}_{\mathfrak{h}} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \right) \geq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_{\mathfrak{h}} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \right), \quad \forall \mathtt{u}_{\mathfrak{h}} \in V_{\mathfrak{h}}, \\[6pt] \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,\mathfrak{n}}, \quad \mathtt{u}_{\mathfrak{h}} \leq \mathtt{r}_{\mathfrak{h}}\mathcal{M}\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j}+1,\mathfrak{n}}, \\[6pt] \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} = \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j,n}-1} \text{ on } \Gamma_{\mathfrak{i}}. \end{array} \right. \tag{13}\]
We now replace \(\mathtt{u}_{\mathfrak{h}}\) by \(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \mathtt{v}_{\mathfrak{h}}\), with \(\mathtt{v}_{\mathfrak{h}}\geq0\), in the problem (4). This gives: \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}, \mathtt{v}_{\mathfrak{h}} \right) \leq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_{\mathfrak{h}} \right).\] Similarly, substituting \(\mathtt{u}_{\mathfrak{h}} = \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} – \mathtt{v}_{\mathfrak{h}}\) in (13) yields \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}}, \mathtt{v}_{\mathfrak{h}} \right) \leq \left( \mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_{\mathfrak{h}} \right).\] Subtracting these two inequalities leads to: \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}}, \mathtt{v}_{\mathfrak{h}} \right) \leq 0.\] As a result of this inequality, we obtain: \[\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \leq 0. \tag{14}\] We immediately deduce that: \[\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \leq \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}}, \quad \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} = \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j,n}} \text{ on } \Gamma_{\mathfrak{i}}, \quad \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \geq 0 \Rightarrow \hat{\vartheta}_{\mathfrak{lh}}^{\mathfrak{j,n}} \geq 0.\] Thus, by the principle of mathematical induction, the sequence \(\left( \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \right)\) is monotonically decreasing.
Comparison between subsolution and supersolution iterations:
This step aims to demonstrate that the two iterative sequences \(\left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}\right)_{\mathfrak{i}=\overline{1,\mathfrak{q}}}^{\mathfrak{j}=\overline{1,\mathfrak{m}};\mathfrak{n}\in\mathbb{N}}\) and \(\left(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}\right)_{\mathfrak{i}=\overline{1,\mathfrak{q}}}^{\mathfrak{j}=\overline{1,\mathfrak{m}};\mathfrak{n}\in\mathbb{N}}\) are ordered at each iteration, that is, \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \leq \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}, \quad \forall\, \mathfrak{i},\mathfrak{j},\mathfrak{n}.\]
Base Case: For \(\mathfrak{n}=0\) and every subdomain index \(\mathfrak{i}\in\{1,\dots,\mathfrak{q}\}\), we start with the initialization \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0} = 0.\] The function \(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}\) is obtained by solving the linear system \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}, \mathtt{u}_\mathfrak{h}\right) = \left(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{u}_\mathfrak{h}\right), \tag{15}\] where \(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}} \ge 0\). Consequently, \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0} = 0 \quad \text{and} \quad \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0} \ge 0,\] which implies the initial ordering \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0} \le \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},0}.\]
Inductive Step. Suppose that at iteration \(\mathfrak{n}\), the inequality \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \le \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}} \quad \text{holds on } \Gamma_\mathfrak{i}.\] We now prove that it remains true for the next iteration \(\mathfrak{n}+1\).
From the discrete system (4), consider the test function \(\mathtt{u}_\mathfrak{h} = \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \mathtt{v}_\mathfrak{h}\) with \(\mathtt{v}_\mathfrak{h} \ge 0\). Substituting into the variational inequality yields \[\label{eq:ineq_hat_mod} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left(\hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}, \mathtt{v}_\mathfrak{h}\right) \le \left(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_\mathfrak{h}\right). \tag{16}\] Repeating the same reasoning for \(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}\) from (8), we get \[\label{eq:ineq_check_mod} \mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left(\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}, \mathtt{v}_\mathfrak{h}\right) \le \left(\mathfrak{G}_{\mathfrak{i}}^{\mathfrak{j}}, \mathtt{v}_\mathfrak{h}\right). \tag{17}\] Subtracting (16) from (17) gives \[\mathfrak{a}_{\mathfrak{i}}^{\mathfrak{j}}\!\left( \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}, \mathtt{v}_\mathfrak{h}\right) \le 0.\] Using the coercivity of the bilinear form, we deduce that \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} – \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \le 0,\] confirming the desired inequality for all iterations: \[\check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1} \le \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j,n}+1}. \tag{18}\]
Extension to Indices \(\mathfrak{i}>\mathfrak{l}\).
For subdomains indexed by \(\mathfrak{i}>\mathfrak{l}\), the same
argument applies directly. Repeating the reasoning above shows that the
ordering property is preserved on every subdomain \(\Omega_\mathfrak{l}\).
By letting \(\mathfrak{n}\) tend to infinity, we derive : \[\lim_{\mathfrak{n}\to\infty} \check{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} = \lim_{\mathfrak{n}\to\infty} \hat{\vartheta}_{\mathfrak{ih}}^{\mathfrak{j},\mathfrak{n}} = \vartheta_{\mathfrak{h}}^{\mathfrak{j}}, \qquad \mathfrak{j}=\overline{1,\mathfrak{m}}.\] This result shows that the iterative sequences eventually coincide and converge monotonically toward the unique discrete solution of the problem. ◻
In this section, we derive an upper bound for the approximation error in the \(\mathfrak{L}^\infty\)-norm and discuss the convergence rate of the proposed Schwarz iteration. To simplify the notation, we set \[|\cdot|_{\mathfrak{i}} = \|\cdot\|_{\mathfrak{L}^\infty(\Gamma_{\mathfrak{i}})}, \qquad \|\cdot\|_{\mathfrak{i}} = \|\cdot\|_{\mathfrak{L}^\infty(\Omega_{\mathfrak{i}})}, \quad \mathfrak{i} = \overline{1,\mathfrak{q}}.\]
Theorem 3. [10] Let \(\left( \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right)\) and \(\left( \vartheta_{\mathfrak{lh}}^{\mathfrak{n}+1} \right)\) be the discrete sequences produced by the Schwarz iterative process for \(\mathfrak{n} \ge 0\). Then, these sequences converge geometrically to the unique discrete solution \((\vartheta^1, \ldots, \vartheta^{\mathfrak{m}})\) of the QVI system. More precisely, there exist constants \(\kappa_1, \ldots, \kappa_{\mathfrak{q}} \in (0,1)\), depending only on the overlapping pairs \((\Omega_{\mathfrak{i}}, \Sigma_{\mathfrak{i}})\) and \((\Omega_{\mathfrak{l}}, \Sigma_{\mathfrak{l}})\), such that for every \(\mathfrak{n} \ge 0\): \[\sup_{\bar{\Omega}_{\mathfrak{i}}} \left| \vartheta_{\mathfrak{ih}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right|_{\mathfrak{i}} \le \kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \kappa_{\mathfrak{l}}^{\mathfrak{n}+1} \sup_{\Sigma_{\mathfrak{i}}} \left| \vartheta_{\mathfrak{ih}} – \vartheta_{\mathfrak{ih}}^0 \right|_{\mathfrak{i}},\] and \[\sup_{\bar{\Omega}_{\mathfrak{l}}} \left| \vartheta_{\mathfrak{lh}} – \vartheta_{\mathfrak{lh}}^{\mathfrak{n}+1} \right|_{\mathfrak{l}} \le \kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \kappa_{\mathfrak{l}}^{\mathfrak{n}} \sup_{\Sigma_{\mathfrak{l}}} \left| \vartheta_{\mathfrak{lh}} – \vartheta_{\mathfrak{lh}}^0 \right|_{\mathfrak{l}}.\]
Theorem 4. Let \(\vartheta\) denote the exact solution of the continuous problem (1). For any pair of indices \(\mathfrak{i} = \overline{1,\mathfrak{q}-1}\) and \(\mathfrak{l} = \overline{2,\mathfrak{q}}\) with \(\mathfrak{i} < \mathfrak{l}\), there exists a constant \(\mathfrak{C}\), independent of both the mesh size \(\mathfrak{h}\) and the iteration number \(\mathfrak{n}\), such that: \[\left\| \vartheta_\mathcal{J} – \vartheta_{\mathcal{J}\mathfrak{h}}^{\mathfrak{n}+1} \right\|_\infty \le \mathfrak{C}\,\mathfrak{h}^2 |\log \mathfrak{h}|^4, \qquad \mathcal{J} = \mathfrak{i}, \mathfrak{l}.\]
Proof. We start with the case \(\mathcal{J} = \mathfrak{i}\). The total error can be decomposed into two components: \[\left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right\|_\infty \le \left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}} \right\|_\infty + \left\| \vartheta_{\mathfrak{ih}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right\|_\infty.\]
Applying Theorems 1 and 3, we have: \[\begin{aligned} \left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right\|_\infty &\le \mathfrak{C}_1 \mathfrak{h}^2 |\log \mathfrak{h}|^3 + \kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \left\| \vartheta_{\mathfrak{ih}} – \vartheta_{\mathfrak{ih}}^0 \right\|_{\mathfrak{i}} \\[2mm] &\le \mathfrak{C}_1 \mathfrak{h}^2 |\log \mathfrak{h}|^3 + \kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \left( \left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}} \right\|_{\mathfrak{i}} + \left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}}^0 \right\|_{\mathfrak{i}} \right). \end{aligned}\]
Combining these bounds yields: \[\left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right\|_\infty \le \mathfrak{C}_1 \mathfrak{h}^2 |\log \mathfrak{h}|^3 + \kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \mathfrak{C}_2 \mathfrak{h}^2 |\log \mathfrak{h}|^3.\]
If we assume \(\kappa_{\mathfrak{i}}^{\mathfrak{n}+1} \le |\log \mathfrak{h}|\), the final estimate becomes: \[\left\| \vartheta_{\mathfrak{i}} – \vartheta_{\mathfrak{ih}}^{\mathfrak{n}+1} \right\|_\infty \le \mathfrak{C}\,\mathfrak{h}^2 |\log \mathfrak{h}|^4.\]
The same argument applies for \(\mathcal{J} = \mathfrak{l}\), which concludes the proof. ◻
In this paper, we have presented an efficient and accurate overlapping domain decomposition strategy for coupled quasi-variational inequality systems. The method extends the classical Schwarz approach to coercive frameworks while ensuring monotone geometric convergence under the \(\mathfrak{L}^\infty\) norm. Emphasis was placed on improving the rate of convergence and stability through enhanced overlap communication. Future work will focus on nonlinear and high-dimensional Hamilton–Jacobi–Bellman systems, combining the present approach with advanced multigrid and mixed finite element techniques to achieve greater efficiency .
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