An Overlapping Schwarz Method for Singularly Perturbed Fourth-Order Convection-Diﬀusion Type

. In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diﬀusion type. The method splits the original domain into two overlapping subdomains. A hybrid diﬀerence scheme is proposed in which on the boundary layer region we use the central ﬁnite diﬀerence scheme on a uniform mesh while on the non-layer region we use the mid-point diﬀerence scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are suﬃcient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.


Introduction
problems depends adversely on the singular perturbation parameter. Most of these works have concentrated on second-order single differential equations ( [4] and the references therein), but for fourth-order equations only few results are reported in the literature [2,15,16,17].
Numerical methods for singularly perturbed problems comprising domain decomposition and Schwarz iterative techniques have been examined by various authors, for example, in [1,6,7,8,9,10,18,20]. In [10], the authors examined a continuous overlapping Schwarz method for a singularly perturbed convectiondiffusion equation with arbitrary fixed interface positions and found it to be uniformly convergent with respect to the perturbation parameter. In [20], an analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems with distinct small positive parameters is presented. The authors of [20] found a flaw in the analysis of domain decomposition methods explored in [6,13,18]. The authors observation is that the constant C is not independent of the iteration number k and it is growing at each induction step in their proof of [6,13,18]. But in [20] the authors have presented an alternate analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems with two parameters and problems in [18].
The authours of [8,9] have concluded that the numerical solution of classical finite difference scheme used in Schwarz method does not converge to the exact solution of their test problem which is a single equation. But our proposed scheme used in Schwarz method [3] has overcome the fundamental difficulty mentioned by the authours of [8,9]. In [8,9], the authors used the same scheme in both the layer and non-layer regions, whereas in our case we used different schemes in each region.
As far as the authors knowledge goes fourth-order SPPs have not yet been examined for higher-order of convergence. Therefore, we are interested in constructing a numerical method for fourth-order SPPs. Of primary interest we have been proved that when appropriate subdomains are used the method produce convergence of almost second-order.
Motivated by the works of [2,10,15,16,17] we have examined experimentally the performance of Schwarz method to the fourth-order Singularly Perturbed Boundary Value Problems (SPBVPs) described as below.
−εy iv (x) + a(x)y ′′′ (x) + b(x)y ′′ (x) − c(x)y(x) = −f (x), x ∈ Ω, (1.1) where a(x), b(x), c(x) are sufficiently smooth functions satisfying the following conditions: with 0 < ε ≪ 1, Ω = (0, 1),Ω = [0, 1] and y ∈ C (4) (Ω) ∩ C (2) (Ω), which have important applications in fluid dynamics, have been studied in [5], and the references therein. The SPBVPs (1.1)-(1.2) can be transformed into an equivalent weakly coupled system of two ODEs subject to suitable boundary conditions of the form: where y = (y 1 , y 2 ) T and a(x), b(x), c(x) are sufficiently smooth functions satisfying (1.3)-(1.5). The above weakly coupled system can be written in the matrix-vector form as In this paper, of primary interest we have proved that discrete Schwarz method converge to the solution of the continuous problem. The method is shown to be of almost second-order convergence. Furthermore, we show that, just two iterations are required to achieve the expected accuracy. Remark 1. The solution of the problem (1.1)-(1.2) exhibits a boundary layer at x = 1 which is less severe because the boundary conditions are prescribed for the derivative of the solution [14]. The condition (1.3) says that (1.1)-(1.2) is a non-turning point problem. The condition (1.4) is known as the quasimonotonicity condition [14]. The maximum principle theorem for the above system (1.1)-(1.2) and for the corresponding discrete problem are established using the conditions (1.3)-(1.4) and using this principle, we can establish a stability result.
The outline of rest of the paper is as follows. In Section 2, the continuous Schwarz method is described. The derivative estimates are obtained in Section 3. In Section 4, the discrete Schwarz method is described. The maximum pointwise error bounds are obtained in Section 5. Numerical experiments are presented in Section 6 and finally, conclusions are included in Section 7. Notations: Throughout the paper we use C, with or without subscript to denote a generic positive constant independent of ε, the iteration k and the discretization parameter N .
Let y : D → R, D ⊆ R. The appropriate norm for studying the convergence of the numerical solution to the exact solution of a SPP is y D = sup x∈D |y(x)|.
For a vector valued function z = (z 1 , z 2 ) T , define z Ω = max{ z 1 Ω , z 2 Ω }. Given any two vector valued functions, z and y, z ≥ y if z j ≥ y j for all j = 1, 2. For a vector of mesh functions Z(
Case 1: (y 1 + ξs 1 )(x 0 ) = 0, for x 0 ∈ Ω p . This implies that y 1 + ξs 1 attains its minimum at x = x 0 . Therefore, which is a contradiction. Case 2: (y 2 + ξs 2 )(x 0 ) = 0, for x 0 ∈ Ω p . This implies that y 2 + ξs 2 attains its minimum at x = x 0 . Therefore, which is a contradiction. Hence it can be conclude that y(x) ≥ 0, ∀x ∈Ω. ⊓ ⊔ An immediate consequence of this is the following stability result. Estimates of derivatives In Section 5 we establish the convergence of the discrete Schwarz method described in Section 4. To prove convergence of the numerical solution, we need the following stronger results on the estimates of the derivatives of the components of the solution of the BVPs (1.6)-(1.7). Now, decompose the solution y(x) of (1.6)-(1.7) into smooth and singular components v(x) and w(x) respectively as T is the solution of the reduced problem of the BVPs (1.6)-(1.7) given by and w(x) = (w 1 (x), w 2 (x)) T is a layer correction term given by The following lemma gives estimates of the derivatives of these components.
Proof. It is easy to check that

Discrete Schwarz method
The continuous overlapping Schwarz method described in Section 2 is discretized by introducing uniform meshes on each subdomain. The domain Ω = (0, 1) is divided into two overlapping subdomains as Ω c = (0, 1 − τ ) and Ω r = (1 − 2τ, 1). The subdomain transition parameter τ is chosen to be the Shishkin transition point τ = min 1 3 , 4ε α ln N as in ( [10], p.91). In each subdomain, In the proposed scheme we use the central finite difference scheme with a uniform mesh on the subdomain Ω r and the mid-point difference scheme with a uniform mesh on the subdomain Ω c . Then in each subdomain Ω N p , p = {c, r}, the corresponding discretization is, Then the algorithm for discrete Schwarz method is defined as follows.
Step1: We choose the initial mesh function Step2: We compute the mesh functions Y p [k] , p = {r, c} which are the solutions of the following discrete problems Step4: If the stopping criterion Y [k+1] − Y [k] ΩN ≤ tol is reached, then stop; otherwise go to Step 2. Here tol is the user prescribed accuracy. For each p = {c, r}, the matrix associated with L N is M-matrix, and hence it satisfies the following discrete maximum principle.
Proof. Please refer to [11,12] and [19]. ⊓ ⊔ An immediate consequence of this lemma is the following stability result.

Error estimates
In this Section, we estimate the error in discrete Schwarz iterates and prove that two iterations are required to attain almost second-order convergence. Following the method of analysis adapted in [18] and [20] we derive error estimates. The analysis proceeds as follows.
Lemma 5. Let y be the solution of (1.6)-(1.7) and let Y [k] be the k th iterate of the discrete Schwarz method described as in Section 4. Then, there are constants C such that Clearly, there are constants C such that Thus, the result holds for k = 0 and the proof is now completed by induction. Assume that, for an arbitrary integer k ≥ 0, there exists C such that Case (i): Error bound estimation onΩ N r . In the proposed scheme we use the central finite difference scheme onΩ N r . One can deduce the following truncation error estimate as in [12] In order to find a bound on L N (Y [k+1] r − y ) Ω N r we must decompose y as in (3.1). Consider 2) For the first term on the right-hand side of (5.2), we use the local truncation error estimate (5.1), h r ≤ CN −1 , ε ≤ CN −1 , and Lemma 2 to get For the second term on the right-hand side of (5.2), when τ = 4ε α ln N , using the local truncation error estimate (5.1), and h r ≤ CεN −1 ln N , we have Using the above estimates in (5.2), we have for some C. The end point of the subdomain Ω N r is 1 − 2τ , which is in general is not in Ω N = {x 1 < x 2 < x 3 < . . . < x N −1 }, so we use a piecewise linear interpolant of the previous iterate to determine Y [k+1] r (1 − 2τ ). Now, using our inductive argument, we have whereȳ is the piecewise linear interpolant of y using grid points ofΩ N c . For the second term on the right-hand side of (5.3), using solution decomposition y as in (3.1), we get Note that (1 − 2τ ) lies inΩ c . For any z ∈ C 2 (Ω c ), standard argument of piecewise linear interpolantz gives For the first term on the right-hand side of (5.4), we use the first bound of (5.5), h c ≤ CN −1 , and Lemma 2 to get For the second term on the right-hand side of (5.4), when τ = 4ε α ln N , note that the layer function w is monotonically increasing in the region (1/3, 1 − τ ) ⊂Ω c . Hence using the second bound of (5.5), we have Consider the mesh function where C is positive constants suitably chosen so that the following are satisfied. Note that, Ψ ± (1 − 2τ ) > 0, Ψ ± (1) > 0 and L N Ψ ± (x i ) > 0. Using the discrete maximum principle for the operator L N onΩ N r we get, Consequently, But since τ = 4ε α ln N , this gives Case (ii): Error bound estimation onΩ N c . We use solution decomposition as in Lemma 2 at each point c − y ) can be written in the form Suppose that (1 − τ ) lies inΩ r . For any z ∈ C 2 (Ω r ), standard argument of piecewise linear interpolantz gives In the proposed scheme we use the mid-point difference scheme onΩ N c . One can deduce the following truncation error estimate as in [12] .

Subcase (i):
For the first term on the right-hand side of (5.8), using the above local truncation error estimate, h c ≤ CN −1 , ε ≤ CN −1 and Lemma 2, we get Now, using our inductive argument, the bound of (5.9), h r ≤ CN −1 , ε ≤ CN −1 , and Lemma 2, we get where we have used the fact that (1 − τ ) is the mesh point ofΩ N r . Consider the mesh function where C is positive constants to be choosen suitably, so that the following expressions are satisfied.
We use the discrete maximum principle for the operator L N onΩ N c to get Subcase (ii): For the second term on the right-hand side of (5.8), when τ = 4ε α ln N , using the arguments discussed as in ( [11], Lemma 6) for x i ∈ Ω N c we get Now, using error bound for the smooth and layer parts we get Combining the error bounds (5.7) and (5.10), we have This completes the proof. ⊓ ⊔ Now we will show that the discrete Schwarz iterates converge at a higher rate than that suggested by Lemma 5.
Lemma 6. Let Y [k] (x i ) be the k th iterate of the discrete Schwarz method described in Section 4. Then there exists some C such that Therefore, we use Lemma 4 to obtain Y [1] r ΩN Therefore, we can apply Lemma 4 to get Y [1] c ΩN c ≤ C. Combining all these estimates we obtain Thus, the result holds for k = 0 and the proof is now completed by induction argument. Assume that for an arbitrary integer k ≥ 0 Here we used Therefore we get and consequently r ΩN r \Ωc ≤ Cν k+1 . (5.14) Finally note that Using our inductive hypothesis and (5.13), we have where we have used the fact that (1 − τ ) is the mesh point ofΩ N c . Therefore, we can apply Lemma 4 to get Combining the estimates (5.14) and (5.15) we obtain, For τ = 4ε α ln N using the arguments given in Lemma 4.1 of [10] we obtain,

⊓ ⊔
The following theorem is the main result of this paper, combining Lemmas 5 and 6 we prove that two iterations are sufficient to attain almost second-order convergence.
Theorem 2. Let y(x) be the solution to (1.6)-(1.7) and Y [k] (x i ) be the k th iterate of the discrete Schwarz method described in Section 4. If τ = 4ε α ln N and N > 2, then Proof. From Lemma 6 there exists Y such that Y := lt k→∞ Y [k] . We know from Lemma 5 that there exists C such that Also from Lemma 6 that there exists C such that Consequently, for N ≥ 2, there exists C such that Thus, using (5.16) and (5.17), we conclude that ⊓ ⊔

Numerical experiments
In this section, we consider one example to illustrate the theoretical results for the BVPs (1.1)-(1.2). The stopping criterion for the iterative procedure is taken to be We normally omit the superscript k on the final Schwarz iterate and write simply Y N j . Let Y N j be a Schwarz numerical approximation for the exact solution y j on the mesh Ω N and N is the number of mesh points. For a finite set of values of ε = {2 0 , . . . , 2 −30 }, we compute the maximum point-wise two mesh difference errors for j = 1, 2 is the numerical solution obtained on a mesh with the same transition points, but with 2N intervals in each subdomain. From these quantities the ε-uniform order of convergence is computed from The computed maximum pointwise errors D N j , (j = 1, 2) and the computed order of convergence p N j , (j = 1, 2) and k (the number of iterations computed) for various values of N and ε are tabulated in Table 1 and Table 2. The nodal errors are plotted as graphs in Figure 1. We can see that the errors decrease as N increases. The computed rates of convergence are almost second-order, with the usual ln N factor associated with these techniques.
The numerical results are presented in Table 1 and Table 2.
Numerical experiment validate the theoretical result. The graphs plotted in the figure is convergent curves in the maximum norm at nodal points for the different values of ε and N for the example considered. This graph clearly indicate that the optimal error bound is of order O(N −k + N −2 ln 3 N ) as predicted.