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

Boglaev, Igor

J. Comput. Appl. Math. 172, No. 2, 313-335 (2004).

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.

Reviewer: Erwin Schechter (Moers)

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

Keywords:

Parabolic reaction-diffusion problem; Boundary layers; Monotone method; Monotone Schwarz alternating algorithm; singular perturbation; monotone iteration; monotone domain decomposition algorithm; convergence; numerical experiments

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