Discrete approximations for singularly perturbed boundary value problems with parabolic layers

REPORTRAPPORT Discrete approximations for singularly perturbed boundary value problems with parabolic layers Abstract Singularly perturbed boundary value problems for equations of elliptic and parabolic type are studied. For small values of the perturbation parameter, parabolic boundary layers appear in these problems. If classical discretisation methods are used, the solution of the nite diierence scheme and the approximation of the diiusive ux derived from it do not converge uniformly with respect to this parameter. In particular, the relative error of the diiusive ux becomes unbounded as the perturbation parameter tends to zero. Using the method of special condensing grids, we can construct diierence schemes that allow approximation of the solution and the normalised diiusive ux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diiusion equations. Also for these problems we construct special nite diierence schemes, the solution of which converges "-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. Also for parabolic equations "-uniformly convergent approximations for the normalised uxes are constructed. Results of numerical experiments are discussed. Summarising, we consider: 1. Problems for Singularly Perturbed (SP) parabolic equation with discontinuous boundary conditions. 2. Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type. 3. Problems for SP parabolic equations, for which the solution and the normalised diiusive uxes are required. Introduction Consider a substance (or admixture) in a solution with a ux satisfying Fick's law, and with distribution given by a diiusion equation. Let the initial concentration of the admixture in the material as well as the concentration of the admixture on the boundary of the body be known. It is required to nd the distribution of admixture in the material at any given time and also the quantity of admixture (that is the diiusive ux) emitted from the boundaries into the exterior environment. Such problems are of interest in environmental sciences in determining the pollution entering the environment from manufactured sources, such as houses, factories and vehicles, and from industrial and agricultural waste disposal sites, and also in chemical kinetics where the chemical reactions are described by reaction-diiusion equations. In considering such problems, it is important to note that the diiusion Fourier number, which is given by the diiusion coeecient of the admixture in materials, can be suuciently small that large variations of concentration occur along the depth …