Singularly perturbed reaction-diffusion problems with discontinuities in the initial and/or the boundary data

Numerical approximations to the solutions of three different problem classes of singularly perturbed parabolic reaction-diffusion problems, each with a discontinuity in the bound\-ary-initial data, are generated. For each problem class, an analytical function associated with the discontinuity in the data, is identified. Parameter-uniform numerical approximations to the difference between the analytical function and the solution of the singularly perturbed problem are generated using piecewise-uniform Shishkin meshes. Numerical results are given to illustrate all the theoretical error bounds established in the paper.Comment: 8 figure

Parameter-uniform numerical methods for singularly perturbed parabolic problems with incompatible boundary-initial data

Numerical approximations to the solution of a linear singularly perturbed parabolic reaction-diffusion problem with incompatible bound\-ary-initial data are generated, The method involves combining the computational solution of a classical finite difference operator on a tensor product of two piecewise-uniform Shishkin meshes with an analytical function that captures the local nature of the incompatibility. A proof is given to show almost first order parameter-uniform convergence of these numerical/analytical approximations. Numerical results are given to illustrate the theoretical error bounds.Comment: 28 pages with 4 figure