Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic PDES

Abstract

We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic PDE on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a Lipschitz asssumption on the nonlinearity, we establish, on each subdomain, an optimal 𝐿^∞ error estimate between the discrete Schwarz sequence and the exact solution of the PDE.

1. Introduction

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains.

Extensive analysis of Schwarz alternating method for nonlinear elliptic boundary value problems can be found in ([1–4] and the references therein). Also the effectiveness of Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. See proceedings of the annual domain decomposition conference beginning with [5].

In this paper, we are interested in the error analysis in the maximum norm for a class of nonlinear elliptic problems in the context of overlapping nonmatching grids: we consider a domain which is the union of two overlapping subdomains where each subdomain has its own triangulation. This kind of discretizations are very interesting as they can be applied to solve many practical problems which cannot be handled by global discretizations. They are earning particular attention of computational experts and engineers as they allow the choice of different mesh sizes and different orders of approximate polynomials in different subdomains according to the different properties of the solution and different requirements of the practical problems.

Quite a few works on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems are known in the literature (cf., e.g., [6–9]).

To prove the main result of this paper, we proceed as in [7]. More precisely, we develop an approach which combines a geometrical convergence result due to Lions [2] and a lemma which consists of estimating the error in the maximum norm between the continuous and discrete Schwarz iterates. The optimal convergence order is then derived making use of standard finite element 𝐿^∞-error estimate for linear elliptic equations.

In the present paper, the proof of this lemma stands on a Lipschitz continuous dependency with respect to both the boundary condition and the source term for linear elliptic equations (see Proposition 2.7) while in [7] the proof stands on a Lipschitz continuous dependency only with respect to the boundary condition for the elliptic obstacle problem.

To the best of our knowledge, this paper provides the first 𝐿^∞-error analysis for overlapping nonmatching grids for nonlinear elliptic PDEs. We also believe that our convergence result will have important implications in the computation of the solution of this type of problems on composite grids.

Now, we give an outline of the paper. In Section 2 we state a continuous alternating Schwarz sequences and define their respective finite element counterparts in the context of nonmatching overlapping grids. Section 3 is devoted to the 𝐿^∞-error analysis of the method.

2. Preliminaries

We begin by laying down some definitions and classical results related to linear elliptic equations.

2.1. Linear Elliptic Equations

Let Ω be a bounded polyhedral domain of ℝ²  or ℝ³ with sufficiently smooth boundary 𝜕Ω. We consider the bilinear form𝑎(𝑢,𝑣)=_Ω(∇𝑢⋅∇𝑣)𝑑𝑥, (2.1) the linear form𝑓,𝑣=_Ω𝑓(𝑥)⋅𝑣(𝑥)𝑑𝑥, (2.2) the right hand side 𝑓,aregularfunction, (2.3) the space𝑉^(𝑔)=𝑣∈𝐻¹(Ω)suchthat𝑣=𝑔on𝜕Ω, (2.4) where 𝑔 is a regular function defined on 𝜕Ω.

We consider the linear elliptic equation: find 𝜉∈𝑉^(𝑔) such that𝑎(𝜁,𝑣)+𝑐(𝜉,𝑣)=𝑓,𝑣,∀𝑣∈𝑉^(𝑔), (2.5) where 𝑐 is a positive constant such that𝑐≥𝛽>0. (2.6)

Let 𝑉_ℎ be the space of finite elements consisting of continuous piecewise linear functions 𝑣 vanishing on 𝜕Ω and 𝜑_𝑠,𝑠=1,2,…,𝑚(ℎ) be the basis functions of 𝑉_ℎ.

The discrete counterpart of (2.5) consists of finding 𝜁_ℎ∈𝑉_ℎ^(𝑔)  such that𝑎𝜁_ℎ,𝑣+𝑐𝜁_ℎ,𝑣=𝑓,𝑣,∀𝑣∈𝑉_ℎ^(𝑔), (2.7) where𝑉_ℎ^(𝑔)=𝑣∈𝑉_ℎ∶𝑣=𝜋_ℎ𝑔on𝜕Ω (2.8) and 𝜋_ℎ is an interpolation operator on 𝜕Ω.

Theorem 2.1 (see (cf. [13])). Under suitable regularity of the solution of problem (2.5), there exists a constant 𝐶 independent of ℎ such that 𝜁−𝜁_ℎ≤𝐶ℎ²logℎ. (2.9)

Lemma 2.2 (see (cf. [4])). Let 𝑤∈𝐻₁(Ω)∩𝐶(Ω) satisfy 𝑎(𝑤,𝜙)+𝑐(𝑤,𝜙)≥0forall nonnegative 𝜙∈𝐻¹₀(Ω), and 𝑤≥0 on 𝜕Ω. Then 𝑤≥0 on Ω.

The proposition below establishes a Lipschitz continuous dependency of the solution with respect to the data.

Notation 2.3. Let (𝑓;𝑔); (𝑓,𝑔) be a pair of data and 𝜁=𝜕(𝑓,𝑔); 𝜁̃=𝜕(𝑓,𝑔) the corresponding solutions to (2.5).

Proposition 2.4. Under conditions of the preceding Lemma 2.2, we have 𝜁−𝜁̃_𝐿∞(Ω)≤max1𝛽𝑓−𝑓_𝐿∞(Ω),𝑔−𝑔_{𝐿∞(𝜕Ω)}. (2.10)

Proof. First, set 1Φ=max𝛽𝑓−𝑓_𝐿∞(Ω),𝑔−𝑔_{𝐿∞(𝜕Ω)}. (2.11) Then 𝑓≤𝑓+𝑓−𝑓_𝐿∞(Ω)𝑐≤𝑓+𝛽𝑓−𝑓_𝐿∞(Ω)1≤𝑓+𝑐max𝛽𝑓−𝑓_𝐿∞(Ω),𝑔−𝑔_{𝐿∞(𝜕Ω)}≤𝑓+𝑐Φ. (2.12) So 𝑎𝜁̃,𝜙+𝑐𝜁̃,𝜙≤𝑎𝜁,𝜙+𝑐𝜁,𝜙+𝑐Φ,𝜙,∀𝜙≥0,𝜙∈𝐻¹₀(Ω)≤𝑎𝜁+Φ,𝜙+𝑐𝜁+Φ,𝜙=𝑓+𝑐Φ,𝜙. (2.13) On the other hand, we have 𝜁+Φ−𝜁̃≥0on𝜕Ω. (2.14) So 𝑎̃̃̃𝜁+Φ−𝜁,𝜙+𝑐𝜁+Φ−𝜁,𝜙≥0𝜁+Φ−𝜁≥0on𝜕Ω. (2.15) Thus, making use of Lemma 2.2, we get 𝜁+Φ−𝜁̃≥0onΩ. (2.16) Similarly, interchanging the roles of the couples (𝑓,𝑔) and (𝑓,𝑔), we obtain 𝜁̃+Φ−𝜁≥0onΩ, (2.17) which completes the proof.

Remark 2.5. Lemma 2.2 stays true in the discrete case.

Indeed, assume that the discrete maximum principle (d.m.p) holds; that is, the matrix resulting from the finite element discretization is an M-Matrix (cf. [10, 11]). Then we have the following

Lemma 2.6. Let 𝑤∈𝑉_ℎ satisfy 𝑎(𝑤,𝑠)+𝑐(𝑤,𝑠)≥0,𝑠=1,2,…,𝑚(ℎ) and 𝑤≥0 on 𝜕Ω. Then 𝑤≥0 on Ω.

Proof. The proof is a direct consequence of the discrete maximum principle.

Let (𝑓,𝑔);(𝑓,𝑔) be a pair of data and 𝜁_ℎ=𝜕_ℎ(𝑓,𝑔);𝜁̃_ℎ=𝜕_ℎ(𝑓,𝑔) the corresponding solutions to (2.7).

Proposition 2.7. Let the d.m.p hold. Then, under conditions of Lemma 2.6, we have 𝜁_ℎ−𝜁̃_{ℎ𝐿∞(Ω)}1≤max𝛽𝑓−𝑓_𝐿∞(Ω),𝑔−𝑔_{𝐿∞(𝜕Ω)}. (2.18)

Proof. The proof is similar to that of the continuous case. Indeed, as the basis functions of the space 𝑉_ℎ are positive, it suffices to use the discrete maximum principle.

2.2. Schwarz Alternating Methods for Nonlinear PDEs

Consider the nonlinear PDE−Δ𝑢+𝑐𝑢=𝑓(𝑢)inΩ,𝑢=0on𝜕Ω, (2.19) or in its weak form𝑎(𝑢,𝑣)+𝑐(𝑢,𝑣)=𝑓(𝑢),𝑣,∀𝑣∈𝐻¹₀(Ω), (2.20) where 𝑓(⋅) is a nondecreasing nonlinearity. Thanks to [12], problem (2.19) has a unique solution.

Let us also assume that 𝑓(⋅) is a Lipschitz continuous on ℝ; that is, 𝑓(𝑥)−𝑓𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈ℝ (2.21) such that 12<𝑘𝛽<1, (2.22) where 𝛽 is the constant defined in (2.6).

We decompose Ω into two overlapping smooth subdomains Ω₁ and Ω₂ such thatΩ=Ω₁∪Ω₂. (2.23)

We denote by 𝜕Ω_𝑖 the boundary of Ω_𝑖 and Γ_𝑖=𝜕Ω_𝑖∩Ω_𝑗 and assume that the intersection of Γ_𝑖 and Γ_𝑗, 𝑖≠𝑗 is empty. Let 𝑉^(𝑤𝑗)_𝑖=𝑣∈𝐻¹Ω_𝑖suchthat𝑣=𝑤_𝑗onΓ_𝑖. (2.24)

We associate with problem (2.20) the following system: find (𝑢₁,𝑢₂)∈𝑉^(𝑢2)₁×𝑉^(𝑢1)₂ solution to 𝑎₁𝑢₁,𝑣+𝑐𝑢₁,𝑣=𝑓𝑢₁,𝑣,∀𝑣∈𝐻¹₀Ω₁,𝑎₂𝑢₂,𝑣+𝑐𝑢₂,𝑣=𝑓𝑢₂,𝑣,∀𝑣∈𝐻¹₀Ω₂, (2.25) where𝑎_𝑖(𝑢,𝑣)=_Ω𝑖()𝑢∇𝑢⋅∇𝑣𝑑𝑥,_𝑖=𝑢/Ω_𝑖,𝑖=1,2. (2.26)

2.3. The Continuous Schwarz Sequences

Let 𝑢₀ be an initialization in 𝐶₀(Ω) (i.e., continuous functions vanishing on 𝜕Ω) such that 𝑎𝑢₀,𝑣+𝑐𝑢₀,𝑣=𝑓,𝑣,∀𝑣∈𝐻¹₀(Ω). (2.27) Starting from 𝑢⁰₂=𝑢⁰/Ω₂, we respectively define the alternating Schwarz sequences (𝑢₁^𝑛+1) on Ω₁ such that 𝑢₁^𝑛+1∈𝑉^(𝑢𝑛2) solves 𝑎₁𝑢₁^𝑛+1,𝑣+𝑐𝑢₁^𝑛+1,𝑣=𝑓𝑢₁^𝑛+1,𝑣,∀𝑣∈𝐻¹₀Ω₁;𝑛≥0, (2.28) and (𝑢₂^𝑛+1) on Ω₂ such that 𝑢₂^𝑛+1∈𝑉^{(𝑢1𝑛+1)} solves𝑎₂𝑢₂^𝑛+1,𝑣+𝑐𝑢₂^𝑛+1,𝑣=𝑓𝑢₂^𝑛+1,𝑣,∀𝑣∈𝐻¹₀Ω₂;𝑛≥0. (2.29)

Theorem 2.8 (see (cf. [2, pages 51–63])). The sequences (𝑢₁^𝑛+1);(𝑢₂^𝑛+1); 𝑛≥0 produced by the Schwarz alternating method converge geometrically to the solution (𝑢₁,𝑢₂) of the system (2.25). More precisely, there exist two constants 𝑘₁,𝑘₂∈(0;1) which depend on (Ω₁,Γ₂) and (Ω₂,Γ₁), respectively, such that for all 𝑛≥0; 𝑢₁−𝑢₁^𝑛+1_𝐿∞(Ω1)≤𝑘^𝑛₁𝑘^𝑛₂𝑢⁰−𝑢_𝐿∞(Γ1),𝑢₂−𝑢₂^𝑛+1_𝐿∞(Ω2)≤𝑘₁^𝑛+1𝑘^𝑛₂𝑢⁰−𝑢_𝐿∞(Γ2). (2.30)

2.4. The Discretization

For 𝑖=1,2, let 𝜏^ℎ𝑖 be a standard regular and quasiuniform finite element triangulation in Ω_𝑖;ℎ_𝑖, being the meshsize. The two meshes being mutually independent Ω₁∩Ω₂, a triangle belonging to one triangulation does not necessarily belong to the other. We consider the following discrete spaces: 𝑉_ℎ𝑖=𝑣∈𝐶Ω_𝑖∩𝐻¹₀Ω_𝑖suchthat𝑣/𝐾∈𝑃₁,∀𝐾∈𝜏^ℎ𝑖 (2.31) and for every 𝑤∈𝐶(Ω_𝑖), we set 𝑉_ℎ^(𝑤)_𝑖=𝑣∈𝑉_ℎ𝑖∶𝑣=0on𝜕Ω_𝑖∩𝜕Ω;𝑣=𝜋_ℎ𝑖(𝑤)onΓ_𝑖, (2.32) where 𝜋_ℎ𝑖 denote an interpolation operator on Γ_𝑖.

The Discrete Maximum Principle (see [10, 11])
We assume that the respective matrices resulting from the discretizations of problems (2.28) and (2.29) are M-matrices.
Note that as the two meshes ℎ₁ and ℎ₂ are independent over the overlapping subdomains, it is impossible to formulate a global approximate problem which would be the direct discrete counterpart of problem (2.20).

2.5. The Discrete Schwarz Sequences

Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (2.28) and (2.29).

Indeed, let 𝑢_0ℎ be the discrete analog of 𝑢₀, defined in (2.27); we, respectively, define by 𝑢^𝑛+1_1ℎ∈𝑉^{(𝑢𝑛2ℎ)}_ℎ1 such that 𝑎₁𝑢^𝑛+1_1ℎ,𝑣+𝑐𝑢^𝑛+1_1ℎ,𝑣=𝑓𝑢^𝑛+1_1ℎ,𝑣∀𝑣∈𝑉_ℎ1;𝑛≥0 (2.33) and 𝑢^𝑛+1_2ℎ∈𝑉^{(𝑢𝑛+11ℎ)}_ℎ2 such that 𝑎₂𝑢^𝑛+1_2ℎ,𝑣+𝑐𝑢^𝑛+1_2ℎ,𝑣=𝑓𝑢^𝑛+1_2ℎ,𝑣∀𝑣∈𝑉_ℎ2;𝑛≥0. (2.34)

3. 𝐿^∞-Error Analysis

This section is devoted to the proof of the main result of the present paper. To that end we begin by introducing two discrete auxiliary sequences and prove a fundamental lemma.

3.1. Two Auxiliary Schwarz Sequences

For 𝑤⁰_2ℎ=𝑢⁰_2ℎ, we define the sequences (𝑤^𝑛+1_1ℎ) and (𝑤^𝑛+1_2ℎ) such that 𝑤^𝑛+1_1ℎ∈𝑉^(𝑢𝑛2)_ℎ1 solves 𝑎₁𝑤^𝑛+1_1ℎ,𝑣+𝑐𝑤^𝑛+1_1ℎ,𝑣=𝑓𝑢₁^𝑛+1,𝑣,∀𝑣∈𝑉_ℎ1;𝑛≥0 (3.1) and 𝑤^𝑛+1_2ℎ∈𝑉^{(𝑢1𝑛+1)}_ℎ2 solves 𝑎₂𝑤^𝑛+1_2ℎ,𝑣+𝑐𝑤^𝑛+1_2ℎ,𝑣=𝑓𝑢₂^𝑛+1,𝑣,∀𝑣∈𝑉_ℎ2;𝑛≥0, (3.2) respectively.

It is then clear that 𝑤^𝑛+1_1ℎ and 𝑤^𝑛+1_2ℎ are the finite element approximation of 𝑢₁^𝑛+1 and 𝑢₂^𝑛+1 defined in (2.28), (2.29), respectively. Then, as 𝑓(⋅) is continuous, ‖𝑓(𝑢^𝑛_𝑖)‖_∞≤𝐶 (independent of 𝑛), and, therefore, making use of standard maximum norm estimates for linear elliptic problems, we have 𝑢^𝑛_𝑖−𝑤^𝑛_{𝑖ℎ𝐿∞(Ω𝑖)}≤𝐶ℎ²logℎ, (3.3) where 𝐶 is a constant independent of both ℎ and 𝑛.

Notation 3.1. From now on, we shall adopt the following notations: |⋅|₁=‖⋅‖_𝐿^∞_(Γ1);|⋅|₂=‖⋅‖_𝐿^∞_(Γ2),‖⋅‖₁=‖⋅‖_𝐿^∞_(Ω1);‖⋅‖₂=‖⋅‖_𝐿^∞_(Ω2),𝜋_ℎ1=𝜋_ℎ2=𝜋_ℎ. (3.4)

3.2. The Main Results

The following lemma will play a key role in proving the main result of this paper.

Lemma 3.2. Let 𝜌=𝑘/𝛽. Then, under assumption (2.22), there exists a constant 𝐶 independent of both ℎ and 𝑛 such that 𝑢_𝑖^𝑛+1−𝑢^𝑛+1_𝑖ℎ𝑖≤𝐶ℎ²logℎ1−𝜌;𝑖=1,2. (3.5)

Proof. We know from standard 𝐿^∞-error estimate for linear problem (see [13]) that there exists a constant 𝐶 independent of ℎ such that 𝑢⁰−𝑢⁰_{ℎ𝐿∞(Ω)}≤𝐶ℎ²logℎ. (3.6)
Now, since 1/2<𝜌<1, then 1<𝜌/(1−𝜌), and therefore 𝑢⁰₂−𝑢⁰_2ℎ2≤𝐶ℎ²logℎ≤𝜌𝐶ℎ²logℎ1−𝜌. (3.7)
Let us now prove (3.5) by induction. Indeed for 𝑛=1, using the Proposition 2.7, we have in domain 1 𝑢¹₁−𝑢¹_1ℎ1≤𝑢¹₁−𝑤¹_1ℎ1+𝑤¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+𝑤¹₁−𝑢¹_1ℎ1≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢¹₁−𝑓𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ1≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢¹₁−𝑓𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ2≤𝐶ℎ²logℎ+max𝜌𝑢¹₁−𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ2. (3.8)
We then have to distinguish between two cases: (1)  max𝜌𝑢¹₁−𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ2=𝜌𝑢¹₁−𝑢¹_1ℎ1 (3.9) or(2)  max𝜌𝑢¹₁−𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ2=𝑢⁰₂−𝑢⁰_2ℎ2. (3.10)
Case (1)implies 𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+𝜌𝑢¹₁−𝑢¹_1ℎ1,𝑢⁰₂−𝑢⁰_2ℎ2≤𝜌𝑢¹₁−𝑢¹_1ℎ1. (3.11)
Then 𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ,𝑢1−𝜌₀₂−𝑢⁰_2ℎ2≤𝜌𝑢¹₁−𝑢¹_1ℎ1≤𝜌𝐶ℎ²logℎ.1−𝜌 (3.12) Case (2) implies 𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+𝑢⁰₂−𝑢⁰_2ℎ2, (3.13)𝜌𝑢¹₁−𝑢¹_1ℎ1≤𝑢⁰₂−𝑢⁰_2ℎ2. (3.14) So, by multiplying (3.13) by 𝜌 we get 𝜌𝑢¹₁−𝑢¹_1ℎ1≤𝜌𝐶ℎ²logℎ+𝜌𝑢⁰₂−𝑢⁰_2ℎ2. (3.15) So, 𝜌‖𝑢¹₁−𝑢¹_1ℎ‖₁ is bounded by both 𝜌𝐶ℎ²|logℎ|+𝜌‖𝑢⁰₂−𝑢⁰_2ℎ‖₂ and ‖𝑢⁰₂−𝑢⁰_2ℎ‖₂. This implies that (a)  𝑢⁰₂−𝑢⁰_2ℎ2≤𝜌𝐶ℎ²logℎ+𝜌𝑢⁰₂−𝑢⁰_2ℎ2 (3.16) or (b)𝜌𝐶ℎ²logℎ+𝜌𝑢⁰₂−𝑢⁰_2ℎ2≤𝑢⁰₂−𝑢⁰_2ℎ2. (3.17)
That is, (a)𝑢⁰₂−𝑢⁰_2ℎ2≤𝜌𝐶ℎ²logℎ1−𝜌 (3.18) or(b)𝑢⁰₂−𝑢⁰_2ℎ2≥𝜌𝐶ℎ²logℎ1−𝜌. (3.19) It follows that only the case (a) is true, that is, 𝑢⁰₂−𝑢⁰_2ℎ2≤𝜌𝐶ℎ²logℎ1−𝜌. (3.20)
Thus 𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+𝑢⁰₂−𝑢⁰_2ℎ2≤𝐶ℎ²logℎ+𝜌𝐶ℎ²logℎ≤1−𝜌𝐶ℎ²logℎ.1−𝜌 (3.21)
So, in both cases (1) and (2), we have 𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ1−𝜌. (3.22)
Similarly, we have in domain 2 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ+𝑤¹₂−𝑢¹_2ℎ2≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢¹₂−𝑓𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ2≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢¹₂−𝑓𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+max𝜌𝑢¹₂−𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ1. (3.23)
So (1)  max𝜌𝑢¹₂−𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ1=𝜌𝑢¹₂−𝑢¹_2ℎ2 (3.24) or(2)max𝜌𝑢¹₂−𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ1=𝑢¹₁−𝑢¹_1ℎ1. (3.25)Case (1) implies 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ+𝜌𝑢¹₂−𝑢¹_2ℎ2,𝑢¹₁−𝑢¹_1ℎ1≤𝜌𝑢¹₂−𝑢¹_2ℎ2. (3.26)
So 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ,𝑢1−𝜌₁₁−𝑢¹_1ℎ1≤𝜌𝑢¹₂−𝑢¹_2ℎ2≤𝜌𝐶ℎ²logℎ≤1−𝜌𝐶ℎ²logℎ,1−𝜌 (3.27) while case (2) implies 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ+𝑢¹₁−𝑢¹_1ℎ1, (3.28)𝜌𝑢¹₂−𝑢¹_2ℎ1≤𝑢¹₁−𝑢¹_1ℎ1. (3.29)
So, by multiplying (3.28) by 𝜌 we get 𝜌𝑢¹₂−𝑢¹_2ℎ2≤𝜌𝐶ℎ²logℎ+𝜌𝑢¹₁−𝑢¹_1ℎ1, (3.30) and hence 𝜌‖𝑢¹₂−𝑢¹_2ℎ‖₂ is bounded by both 𝜌𝐶ℎ²|logℎ|+𝜌‖𝑢¹₁−𝑢¹_1ℎ‖₁ and ‖𝑢¹₁−𝑢¹_1ℎ‖₁. Then(a)  𝑢¹₁−𝑢¹_1ℎ1≤𝜌𝐶ℎ²logℎ+𝜌𝑢¹₁−𝑢¹_1ℎ1 (3.31) or(b)𝐶ℎ²logℎ+𝜌𝑢¹₁−𝑢¹_1ℎ1≤𝑢¹₁−𝑢¹_1ℎ1, (3.32) which implies (a)𝑢¹₁−𝑢¹_1ℎ1≤𝜌𝐶ℎ²logℎ<1−𝜌𝐶ℎ²logℎ1−𝜌 (3.33) or(b)𝜌𝐶ℎ²logℎ≤𝑢1−𝜌¹¹−𝑢¹_1ℎ1<𝐶ℎ²logℎ1−𝜌. (3.34)
Hence, (a) and (b) are true because they both coincide with (3.22). So, there is either a contradiction and thus case (2) is impossible or case (2) is possible only if 𝑢¹₁−𝑢¹_1ℎ1=𝜌𝐶ℎ²logℎ+𝜌𝑢¹₁−𝑢¹_1ℎ1, (3.35) that is, 𝑢¹₁−𝑢¹_1ℎ1=𝜌𝐶ℎ²logℎ1−𝜌. (3.36)
Thus 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ+𝑢¹₁−𝑢¹_1ℎ1≤𝐶ℎ²logℎ+𝜌𝐶ℎ²logℎ≤1−𝜌𝐶ℎ²logℎ.1−𝜌 (3.37)
That is, both cases (1) and (2) imply 𝑢¹₂−𝑢¹_2ℎ2≤𝐶ℎ²logℎ1−𝜌. (3.38)
Now, let us assume that 𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝐶ℎ²logℎ1−𝜌 (3.39) and prove that 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤11−𝜌𝐶ℎ²,𝑢logℎ_2𝑛+1−𝑢^𝑛+1_2ℎ2≤𝐶ℎ²logℎ.1−𝜌 (3.40)
Indeed, we have in domain 1 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ+𝑤₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢₁^𝑛+1−𝑓𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ1≤𝐶ℎ²1logℎ+max𝛽𝑓𝑢₁^𝑛+1−𝑓𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝐶ℎ²logℎ+max𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ2. (3.41)
We have again to distinguish between two cases (1)  max𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ2=𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1 (3.42) or(2)max𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ2=𝑢^𝑛₂−𝑢^𝑛_2ℎ2. (3.43)
Case (1) implies 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ+𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1,𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1. (3.44)
Then 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ,𝑢1−𝜌_𝑛2−𝑢^𝑛_2ℎ2≤𝐶ℎ²logℎ.1−𝜌 (3.45)
Case (2) implies 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ+𝑢^𝑛₂−𝑢^𝑛_2ℎ2, (3.46)𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝑢^𝑛₂−𝑢^𝑛_2ℎ2. (3.47)
So, by multiplying (3.46) by  𝜌, we get 𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝜌𝐶ℎ²logℎ+𝜌𝑢^𝑛₂−𝑢^𝑛_2ℎ2,𝜌𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝑢^𝑛₂−𝑢^𝑛_2ℎ2. (3.48)
Hence, we can see that 𝜌‖𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ‖₁ is bounded by both 𝜌𝐶ℎ²|logℎ|+𝜌‖𝑢^𝑛₂−𝑢^𝑛_2ℎ‖₂ and ‖𝑢^𝑛₂−𝑢^𝑛_2ℎ‖₂. So, we have 𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝜌𝐶ℎ²logℎ+𝜌𝑢^𝑛₂−𝑢^𝑛_2ℎ2 (3.49) or 𝜌𝐶ℎ²logℎ+𝜌𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝑢^𝑛₂−𝑢^𝑛_2ℎ2, (3.50) which implies (a)  𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝜌𝐶ℎ²logℎ<1−𝜌𝐶ℎ²logℎ1−𝜌 (3.51) or(b)𝜌𝐶ℎ²logℎ≤𝑢1−𝜌^𝑛2−𝑢^𝑛_2ℎ2<𝐶ℎ²logℎ1−𝜌. (3.52)
So, (a) and (b) are true because they both coincide with (3.39). This means that there is either contradiction and then case (2) is impossible, or case (2) is possible and then we must have 𝜌𝐶ℎ²logℎ+𝜌𝑢^𝑛₂−𝑢^𝑛_2ℎ2=𝑢^𝑛₂−𝑢^𝑛_2ℎ2, (3.53) that is 𝑢^𝑛₂−𝑢^𝑛_2ℎ2=𝜌𝐶ℎ²logℎ1−𝜌. (3.54)
So 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ+𝑢^𝑛₂−𝑢^𝑛_2ℎ2≤𝐶ℎ²logℎ+𝜌𝐶ℎ²logℎ=1−𝜌𝐶ℎ²logℎ.1−𝜌 (3.55)
Thus, both cases (1) and (2) imply 𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²logℎ1−𝜌. (3.56)
Estimate in domain 2 can be proved similarly using estimate (3.56).

Theorem 3.3. Let ℎ=max(ℎ₁,ℎ₂). Then, for 𝑛 large enough, there exists a constant 𝐶 independent of both ℎ and 𝑛 such that 𝑢_𝑖−𝑢^𝑛+1_𝑖ℎ1≤𝐶ℎ²logℎ,∀𝑖=1,2. (3.57)

Proof. Let us give the proof for 𝑖=1. The one for 𝑖=2 is similar and so will be omitted.
Indeed, Let 𝑘=𝑘₁𝑘₂. Then making use of Theorem 2.8 and Lemma 3.2, we get 𝑢₁−𝑢^𝑛+1_1ℎ1≤𝑢₁−𝑢₁^𝑛+1₁+𝑢₁^𝑛+1−𝑢^𝑛+1_1ℎ1≤𝑘^𝑛₁𝑘^𝑛₂𝑢⁰−𝑢₁+𝐶ℎ²logℎ1−𝜌≤𝑘^2𝑛𝑢⁰−𝑢₁+𝐶ℎ²logℎ.1−𝜌 (3.58) So, for 𝑛 large enough, we have 𝑘^2𝑛≤ℎ² (3.59) and thus 𝑢₁−𝑢^𝑛+1_1ℎ1≤𝐶ℎ²+𝐶ℎ²logℎ≤𝐶ℎ²logℎ, (3.60) which is the desired result.

Conclusion
We have established an error estimate for the finite element Schwarz alternating method for a nonlinear elliptic PDE on two subdomains with nonmatching grids combining a geometrical convergence result due to Lions and a standard finite element 𝐿^∞-error analysis for linear elliptic equations.The same approach may be extended to other types of problems such as linear parabolic PDEs (see [2]) and singularly perturbed advection-diffusion equations (see [14]).