The Maximum Norm Analysis of a Nonmatching Grids Method for a Class of Parabolic p(x)-Laplacian Equation

Motivated by the work of Boulaaras and Haiour in [7], we provide a maximum norm analysis of Schwarz alternating method for parabolic p(x)-Laplacian equation, where an optimal error analysis each subdomain between the discrete Schwarz sequence and the continuous solution of the presented problem is established


Introduction
The problem: find u ∈ L 2 0, T ; W  Operator ∆ p(x) is called p(x)-Laplacian defined as: The constant α is assumed to be nonnegative satisfies f is a regular function such that f ∈ L 2 0, T, L p(x) (Ω) ∩ C 1 0, T, L q(x) (Ω) with 1 p(x) + 1 q(x) = 1.
In [7] Boulaaras and Haiour provided a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear parabolic partial differential equations on two overlapping subdomains with nonmatching grids. They considered 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,where a triangle belonging to one triangulation does not necessarily belong to the other one. Then according to Lipschitz assumption, they proved that for each subdomain an optimal error has been estimated by applying uniform norm between the discrete Schwarz sequence and the exact solution of a nonlinear parabolic partial differential equations. In this paper, the same approach can be extended to other types as a linear parabolic partial differential equations see [2] and singularly perturbed advection-diffusion equations (see [11]) using the overlapping domain decomposition method, where we applied it in a full discrete (see [7], [5] and [9]). In [7], the authors studied the overlapping domain decomposition method combined with a finite element approximation for Laplace equation, where an overlapping Schwarz method on nonmatching grids has been used on uniform norm of and they also proved the geometric convergence on every subdomain.
Aforementioned, in this paper, we can extend the study to p(x)-Laplacian equation, where we apply a maximum norm analysis of the finite element Schwarz alternating method of the presented problem on two overlapping subdomains with nonmatching grids. and we are following up the same procedures that have been mentioned above in [7] with respect to the stability analysis which has been given by our previous work in [7], we establish on each subdomain, an optimal error analysis between the discrete Schwarz sequence and the continuous solution of p(x)-Laplacian equation. In addition the geometric convergence is proved.

Nonlinear parabolic equation with function independent with solution
In this section we consider the parabolic problem and transform it into elliptic system and give some definitions and classical results related to nonlinear elliptic equations with the function f is a regular and independent of the solution u .

The semi-discrete of parabolic equation
The problem (1.1) can be reformulated into the following continuous parabolic variational equation: where A (., .) is the Nonlinear form defined as: The goal of this discretization is transform the parabolic equation into system of the elliptic equations, for this we apply the θ-schema in the equation(2.1). Thus we have, for any θ ∈ [0, 1] and k = 1, ..., N By multiplying and dividing by θ and by adding u k−1 θ∆t , v to both parties of the equalities (2.3), we get Then, the problem (2.4) can be reformulated into the following coercive discrete system of elliptic quasi-variational inequalities

Nonlinear elliptic equation
We consider the elliptic problem :find Let Ω be decomposed into triangles and τ h denotes the set of those elements, where 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 We discretize in space, i.e., we approach the space W , we get the following discrete system of elliptic equations Proof. The proof is easy, and the similar to that use in [18] the corresponding solutions to (2.5) .
Proposition 2.5. Under the previous notation and lemma 1, we have . (2.11) Proof. First, putting (Ω). On the other hand, we have By using the result of lemma 1, we get Similarly, interchanging the roles of the couples (F θ,k , ϕ θ,k ) and ( F θ,k , ϕ θ,k ), we get which completes the proof.

2.2.2.
The discrete maximum principle assumption (DMP). We assume the matrices resulting from the finite element discretization are M-matrix ( [12] and [13]). For convenience in all the sequels, C will be a generic constant independent on h. Then we have the following Proof. The proof is similar to that use in lemma 1.

(2.18)
Proof. The proof is similar to that of the continuous case.

Nonlinear parabolic equation with nonlinear function
Consider the nonlinear elliptic problem :find u θ,k ∈ W 1,p(x) 0 with Σ defined as blow and the function f is is a nondecreasing nonlinearity, assuming that f (·) is a Lipschitz continuous on R; that is Using the semi-discrete parabolic equation and variational formulation of (3.1) we find the following :find in this case F θ,k is Nonlinear Lipschitz continuous on R, because f that is.

Schwarz alternating methods for parabolic equation
We decompose (Ω) in two overlapping smooth subdomain Ω 1 and Ω 2 such that Ω = Ω 1 ∪ Ω 2 , 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 We associate with problem (3.1) the following system: find (u θ,k and u θ,k i = u θ,k /Ω i ; i = 1, 2

The discrete Schwartz sequences
As we have defined before, for i = 1, 2, let τ hi be a standard regular and quasiuniform finite element triangulation in Ω i ; h i , being the mesh size. The two meshes being mutually independent Ω 1 ∩ Ω 2 , a triangle belonging to one triangulation does not necessarily belong to the other and for every w ∈ C (Ω i ) , we set where π hi denote an interpolation operator on Γ 0i . Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (3.8) and (3.9) .
Indeed, let u 0h be the discrete analog of u 0 , defined in (3.7), we respectively, define by u θ,k,n+1 such that and u θ,k,n+1 such that

Maximum norm analysis of asymptotic behavior
We begin by introducing two discrete auxiliary sequences and prove a fundamental lemma.

Two auxiliary Schwarz sequences
For w 0 2h = u 0 2h , we define the sequences w θ,∞,n+1 1h and w θ,n+1 and w θ,n+1 respectively. It is then clear that w θ,∞,n+1 where C is a constant independent of both h and n.

Iterative discrete algorithm
We give our following discrete algorithm where u θ,k h is the solution of the problem (3.4) and the first iteration u 0 h is solution of (3.7).

Lemma 4.2. under assumption (3), there exists a constant C independent of both h and n such that
Proof. We know from standard error estimate on uniform norm for linear problem [24] that there exists a constant C independent of h such that Let us now prove (4.5) by induction. Indeed for n = 1, using the result of Proposition 1, we have in Ω 1 u θ,k,1 Similar for Ω 2 u θ,k,1 Now, let assume that u θ,k,n 1 − u θ,k,n 1h 1 ≤ Ch 2 | log h| and prove that, u θ,k,n+1 Which give that, u θ,k,n+1 Similar assume that u θ,k,n 2 − u θ,k,n 2h 2 ≤ Ch 2 | log h| and prove that, Putting an fundamental theorem in present paper, Proof. Let us give the proof for i = 1. The one for i = 2 is similar and so will be omitted. Indeed, Let δ = δ 1 δ 2 , then making use of Theorem 2 and Lemma 3, we get So, for n large enough, we have δ 2n ≤ h 2 (4.11) and thus u θ,k 1 − u θ,k,n+1 which is the desired result.

Asymptotic behavior
This section is devoted to the proof of main result of the present paper, where we prove the theorem of the asymptotic behavior in L ∞ -norm for parabolic variational inequalities, where we evaluate the variation in L ∞ between u h (T ) , the discrete solution calculated at the moment T = p∆t and u ∞ , the asymptotic continuous solution of (3.4). We begin by introducing new two discrete auxiliary sequences and prove a new fundamental lemma 4.3.1. New two auxiliary Schwarz sequences. For w 0 2h = u 0 2h , we define the sequences w θ,k,n+1 1h and w θ,k,n+1 2h such that u θ,n+1 and w θ,n+1 where ρ (A) is the spectral radius of the elliptic operator.
Proof. Using the proposition (1) and (3) , the assumption (3) which give that:  where C is a constant independent of h and k.
Proof. We have u θ,p,2n+1 Using the lemma 4 and theorem 3, we have for θ ≥ In the same proofing that (4.22) and (4.23) .