Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem (Q704194)

The author studies the numerical solution of the following reaction-diffusion problem: \[ -\mu^2(u_{xx}+ u_{yy}+ u_t= -f(P,t,u), \] where \[ P= (x,y),\quad (P,t)\in Q= \Omega\times (0,T],\quad \Omega= \{0< x< 1,0< y< 1\}. \] The initial-boundary conditions are \[ u(P,t)= g(P,t),\quad (P,t)\in\partial\Omega\times (0,T];\quad u(p,0)= u^0(P),\quad P\in\overline\Omega \] and \(\mu\) is a small parameter. The main result of the paper consists in constructing a monotone domain decomposition algorithm based on a multidimensional modification of the discrete Schwarz alternating method. Here the computational domain in the space variables is partitioned in many nonoverlapping subdomains and small interfacial subdomains are introduced near the interface and approximate boundary values computed on this interface. These are used then on the nonoverlapping subdomains. The rate of convergence of the monotone domain decomposition is investigated and results of the numerical experiments presented.