Convergence analysis of the overlapping Schwarz waveform relaxation algorithm for reaction-diffusion equations with time delay (Q2882360)

The authors consider the following reaction-diffusion equation with a discrete delay NEWLINE\[NEWLINE\tau> 0: \quad{\partial u\over\partial t}- \nu{\partial^2u\over\partial x^2}+ au(x,t)+ du(x,t,-\tau)= f(t,x)\quad\text{in }\mathbb{R}\times\mathbb{R}^+,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,t)= u_0(x,t)\quad\text{in }\mathbb{R}\times [-\tau,0],\quad u(\pm\infty, t)= 0\quad\text{in }\mathbb{R}^+,NEWLINE\]NEWLINE where \(\nu\), \(a\), \(d\) are constants with \(\nu> 0\), \(a\neq 0\).NEWLINENEWLINE The spatial domain is decomposed into two overlapping subdomains \(\Omega_1= (-\infty,L]\), \(\Omega_2= [0,\infty)\) with \(L>0\). Then subproblems in \(\Omega_j\times \mathbb{R}^+\) \((j= 1,2)\) are solved simultaneously iteratively. It is proved that the above overlapping Schwarz waveform relaxation (OSWR) algorithm is well-posed in the anisotropic Sobolev space \(H^{2,1}(\mathbb{R}\times [-\tau, 0])\) if \(u_0\) belongs to this space and \(f\in L^2(\mathbb{R}\times[-\tau, 0])\).NEWLINENEWLINE Next the authors show linear convergence of the algorithm at a ratio depending on the size of the overlap, the time-delay and the coefficients of the equation. The case \(a= 0\), \(\nu=1\) is examined giving interesting results.NEWLINENEWLINE Then the spatial domain is assumed to be a bounded one. The reaction-diffusion equation is discretized in space using the centred second-order finite difference scheme and the OSWR algorithm is applied. It turns out that the linear convergence of the method remains valid and the convergence is robust with respect to mesh refinement.NEWLINENEWLINE The case of an arbitrary number of subdomains is also studied and numerical experiments are reported.