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 finite difference scheme and the approximation of the diffusive flux derived from it do not converge uniformly with respect to this parameter. In particular, the relative error of the diffusive flux becomes unbounded as the perturbation parameter tends to zero. Using the method of special condensing grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $epsilon$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. Also for parabolic equations $epsilon$-uniformly convergent approximations for the normalised fluxes 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 diffusive fluxes are required.

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Department of Numerical Mathematics [NM]
Boundary-value problems, multigrid and defect correction

Farrell, P.A, Hemker, P.W, & Shishkin, G.I. (1995). Discrete approximations for singularly perturbed boundary value problems with parabolic layers. Department of Numerical Mathematics [NM]. CWI.