Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions

and Applied Analysis 3 e k 1 (β, t) = e k−1 2 (β, t) , t ∈ R + , ∂ t e k 2 − ∂ xx e k 2 = 0, (x, t) ∈ Ω 2 ×R + , e k 2 (x, 0) = 0, x ∈ Ω 2 , e k 2 (α, t) = e k−1 1 (α, t) , t ∈ R + , ∂ x e k 2 (1, t) − d 2 (?̃? k 2 , t) e k 2 (1, t) = 0, t ∈ R + , (6) where ?̃?k 1,2 are functions of x and twhich lie between u 1,2 and u k 1,2 and d 1,2 (u, t) are defined by d 1,2 (u, t) = ∂ u g 1,2 (u, t) . (7) For any function f ∈ L∞(Ω̃;R+) (where Ω̃ ⊂ R), we define 󵄩󵄩󵄩󵄩f (x, ⋅) 󵄩󵄩󵄩󵄩∞ = sup t>0 󵄨󵄨󵄨󵄨f (x, t) 󵄨󵄨󵄨󵄨 , 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩∞,∞ = sup x∈Ω̃,t>0 󵄨󵄨󵄨󵄨f (x, t) 󵄨󵄨󵄨󵄨 . (8) Theorem 2. Assume that 0 < α < β < 1 and that the nonlinear boundary functions g 1,2 satisfy ∂ u g 1,2 (u, t) ≤ μ 1,2 < 0, (i.e., d 1,2 ≤ μ 1,2 < 0) , (9) where μ 1,2 are constants. Then the error functions ek 1,2 of the SWR iteration (3) decay linearly to zero. Specifically, it holds that 󵄩󵄩󵄩󵄩 e 2k+1 1 󵄩󵄩󵄩󵄩∞,∞ ≤ ρ 󵄩󵄩󵄩󵄩 e 0 2 (α, ⋅) 󵄩󵄩󵄩󵄩∞ , 󵄩󵄩󵄩󵄩 e 2k+1 2 󵄩󵄩󵄩󵄩∞,∞ ≤ ρ 󵄩󵄩󵄩󵄩 e 0 1 (β, ⋅) 󵄩󵄩󵄩󵄩∞ , (10) where ρ is the called the convergence factor and is defined by ρ = (1 − μ 1 α) [μ 2 (β − 1) + 1] (1 − μ 1 β) [μ 2 (α − 1) + 1] . (11) Proof. Consider the following differential equations: ∂ t e k 1 − ∂ xx e k 1 = 0, (x, t) ∈ Ω 1 ×R + , e k 1 (x, 0) = 1 − μ 1 x 1 − μ 1 β 󵄩󵄩󵄩󵄩 e k−1 2 (β, ⋅) 󵄩󵄩󵄩󵄩∞ , x ∈ Ω 1 , ∂ x e k 1 (0, t) − μ 1 e k 1 (0, t) = −2μ 1 1 − μ 1 β 󵄩󵄩󵄩󵄩 e k−1 2 (β, ⋅) 󵄩󵄩󵄩󵄩∞ , t ∈ R + , e k 1 (β, t) = 󵄩󵄩󵄩󵄩 e k−1 2 (β, ⋅) 󵄩󵄩󵄩󵄩∞ , t ∈ R + , (12) ∂ t e k 2 − ∂ xx e k 2 = 0, (x, t) ∈ Ω 2 ×R + , e k 2 (x, 0) = μ 2 (x − 1) + 1 1 + μ 2 (α − 1) 󵄩󵄩󵄩󵄩 e k−1 1 (α, ⋅) 󵄩󵄩󵄩󵄩∞ , x ∈ Ω 2 , e k 2 (α, t) = 󵄩󵄩󵄩󵄩 e k−1 1 (α, ⋅) 󵄩󵄩󵄩󵄩∞ , t ∈ R + , ∂ x e k 2 (1, t) − μ 2 e k 2 (1, t) = 0, t ∈ R + . (13) It is easy to find that the solutions to (13) are the steady state solutions e k 1 (x, t) = 1 − μ 1 x 1 − μ 1 β 󵄩󵄩󵄩󵄩 e k−1 2 (β, ⋅) 󵄩󵄩󵄩󵄩∞ , x ∈ [0, β] , e k 2 (x, t) = μ 2 (x − 1) + 1 μ 2 (α − 1) + 1 󵄩󵄩󵄩󵄩 e k−1 1 (α, ⋅) 󵄩󵄩󵄩󵄩∞ , x ∈ [α, 1] . (14) Since μ 1,2 < 0, it is easy to know that ek 1 (x, t) ≥ 0 for all (x, t) ∈ [0, β]×[0, +∞) and ek 2 (x, t) ≥ 0 for all (x, t) ∈ [α, 1]×[0, +∞). Now, let Ek j = e k j − e k j and then by subtracting (6) from (13) we get ∂ t E k 1 − ∂ xx E k 1 = 0, (x, t) ∈ Ω 1 ×R + , E k 1 (x, 0) ≥ 0, x ∈ Ω 1 , ∂ x E k 1 (0, t) − d 1 E k 1 (0, t) = (μ 1 − d 1 ) e 1 (0, t)


Introduction
Schwarz waveform relaxation (SWR) algorithm can be regarded as a combination of the Schwarz domain decomposition (DD) method [1,2] and the waveform relaxation method [3,4].It is characterized by first partitioning the spatial domain into several subdomains and then solving subproblems simultaneously inside each subdomain through iterations (see [5][6][7] for the original idea of this approach).In the classical DD framework, one first discretizes the interested problem in time direction by using an implicit time integrator and then solves the state steady problems by using the DD iterative process.Therefore, compared to the classical DD approach, the SWR algorithm possesses the advantages that to be able to treat different subdomains numerically differently with an adapted procedure for each subdomain.
The SWR algorithm has been studied widely and deeply by many authors during the past decade.For example, Gander and his colleagues have investigated the convergence of the algorithm for linear heat equations [5,8], for the linear (advection) reaction diffusion equation [9][10][11].The nonlinear problems have also been investigated; see [6] for the reaction diffusion equation with a nonlinear reaction term and [12] for the nonlinear conservation laws.Nowadays, the SWR algorithm has been becoming more and more popular in scientific and engineering computing and has been adopted to solve complex problems arising from physics and engineering.For example, Martin [13] applied the algorithm to the shallowwater problem; Dolean et al. [14] and El Bouajaji et al. [15] investigated the application of the algorithm for the Maxwell's equations and Zhang and Jiang [16] applied the algorithm to time-periodic parabolic problems.
However, all of the aforementioned problems are initial value problems or initial boundary value problem with Dirichlet boundary conditions.In this paper, we consider the heat equations with nonlinear dynamical boundary conditions and according to our best knowledge there are no results about the SWR algorithm for this type of problems.This type of boundary conditions arises for many known equations of mathematical physics.They are motivated by problems in diffusion phenomena [17], reaction diffusion systems in phase-transition phenomena [18], models in climatology [19], and many others.For physical interpretations of dynamical boundary conditions for problem (1), the interested reader can refer to [20].
Here, we focus on the convergence properties of the algorithm on sufficient long time intervals and the dependence of the convergence rate on the overlap size and the nonlinearity of the boundary nonlinear functions.The remainder of this paper is organized as follows.In Section 2, we introduce
The comparison principle is central for the convergence analysis of the overlapping Schwarz waveform relaxation algorithm.For the Dirichlet type boundary conditions, this principle is known for long and can be found in every textbook concerning PDEs (see, e.g., [22]) and has been used widely in the SWR community [5,6,8,12].The comparison principle for dynamical boundary conditions is however much less known, until recently by von Below and de Coster [23].Rault [24,25], von Below and Pincet Mailly [26], and Pincet Mailly [27] used this comparison principle to study the blow-up phenomenon for strong nonlinear reaction diffusion equations.
Remark 3. Let  =  −  > 0 be the overlap size.Then, by substituting  =  +  into (11) we have It is easy to get Hence, increasing  (i.e., increasing the overlap size) will improve the convergence rate of the SWR algorithm.Moreover, the other two derivatives given in (21) imply that the SWR algorithm (3) is especially suitable for problem (1) that the nonlinear boundary functions have very large negative partial derivatives.The limiting case which is the convergence factor of the two-subdomain SWR algorithm for the heat equation ( 1) with Dirichlet boundary condition (see [5]).

Numerical Results
In this section, we present some numerical results to validate our theoretical predictions obtained in Section 3.For (1), we impose initial condition (, 0) = 0.The source function (, ) is chosen as where ũ(, ) = sin 1999 (4(1−)(√/(1+))). The nonlinear boundary functions are chosen as With the above choice, it is easy to know that the exact solution to (1) is (, ) = ũ(, ).We plot the exact solution ũ(, ) in Figure 2(a), where we can see that the exact solution is almost zero except around " = 0.5" due to the power "1999." The reason why we choose this function as the reference solution is that it is easy to intuitively see the evolution of the iterates    on subdomain  ∈ Ω  during iterations, as shown in Figure 1.We equally decompose the space domain into two subdomains with overlap size  = Δ, where  ≥ 1 is an integer and Δ is the space mesh size.Then, for each subproblem we discretize the governing equation by using a central finite difference in space with mesh parameter Δ = 0.025 and a backward Euler method in time with time step Δ = 0.025.The space partial derivatives appearing in boundaries are discretized by the upwind finite difference method.All of our experiments are performed in the time interval  ∈ [0, 30].
In this first set of experiments, we investigate the influence of the overlap size  on the convergence rate of the SWR algorithm.We choose  = 30 in ( 24) and then we show in Figure 1 the solution generated by the SWR algorithm after 2, 4, and 6 iterations with two different overlap sizes  = 4Δ (top) and  = 8Δ (bottom).Here, we just show the solution profile on the space-time domain (, ) ∈ [0, 1] × [0, 3] for better viewing purpose.The converged solution is plotted in Figure 2 on the left.We can see clearly that the overlap size has a remarkable influence on the convergence rate and in particular with a larger overlap size the algorithm converges faster.This observation confirms our theoretical predication given in Section 3. The measured error (between the converged solution and the iterates, in the sense of  ∞ norm) together with the linear bound predicted by Theorem 2 is shown in Figure 2 on the right.We see that the theoretical linear bound predicts the actual convergence rate very well.
Next, we study how the parameter  in (24) affects the convergence rate of the SWR algorithm.Note that, with the nonlinear boundary functions (24), the arguments  1 and  2 appearing in the convergence factor  are  + 3 and  + 4, respectively.We choose three different  and plot the measured error in Figure 3.We see that decreasing  has a positive effect on the convergence rate, which confirms our theoretical analysis done in Remark 3.

Conclusion
We have briefly analyzed the convergence properties of the overlapping SWR algorithm for heat equation with nonlinear dynamical boundary conditions, based on the recently established comparison principle for PDEs with dynamical boundary conditions [23].The linear convergence on sufficient long time intervals is proved and it is shown that the convergence rate ameliorates as one increases the overlap size.The dependence of the convergence rate on the nonlinearity of the nonlinear boundary functions is also revealed.
Further work includes generalizing the current work to many subdomains for practical parallel computation purpose.Also, to speed up the convergence rate dynamical transmission conditions, such as the popular Robin transmission conditions, should be utilized.Note that the transmission conditions used in this paper are Dirichlet type.For time dependent PDEs (initial problems on unbounded space domain), it is well known that the dynamical transmission conditions always yield significant convergence rate for the SWR algorithm (see, e.g., [9,10]).Due to the nonlinearity arising from the boundaries and the boundedness of the space domain, it is impossible to straightforwardly generalize the techniques proposed for Dirichlet boundary conditions to dynamical boundary conditions and new notion is eagerly expected.

Figure 1 :Figure 2 :
Figure 1: Influence of the overlap size on the convergence rate of the SWR algorithm.Top: with  = 4Δ the solution generated by the SWR algorithm after 2, 4, and 6 iterations; bottom: the similar information for  = 8Δ.