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Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem. (English) Zbl 1070.65091

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.

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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