Acceleration of the Numerical Solution of ParabolicSingularly Perturbed Problems by Parallel MethodsBased on a Defect - Correction Technique

On the rectangular space-time domain G = D (0; T ] with the boundary S = GnG, where D = (0; d), we consider the Dirichlet problem for the singularly perturbed parabolic equation: " 2 @ @x a(x; t) @ @x u(x; t) ! ? c(x; t)u(x; t) ? p(x; t) @ @t u(x; t) = f(x; t); (1a) The perturbation parameter " may take any small positive value, say " 2 (0; 1]. If " = 0, the parabolic equation (1a) degenerates into a rst-order equation, in which only the time derivative remains. For small values of the parameter ", as " ! 0, the solution of (1) exhibits parabolic boundary layers (i.e., layers described by a parabolic equation) in some neighbourhood of the lateral boundary. The presence of thin layers gives rise to diiculties when classical discretization methods are used for the numerical solution of problem (1). The error in the approximate solution depends on the value of " and may be of a size comparable with