Numerical solution of singularly perturbed 2-D convection-diffusion elliptic interface PDEs with Robin-type boundary conditions

We consider a singularly perturbed two-dimensional convection-diffusion elliptic interface problem with Robin boundary conditions, where the source term is a discontinuous function. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ε, is a positive parameter which can be arbitrarily small. Due to the discontinuity in the source term and the presence of the diffusion parameter, the solutions to such problems have, in general, boundary, corner and weak-interior layers. In this work, a numerical approach is carried out using a finite-difference technique defined on an appropriated layer-adapted piecewise uniform Shishkin mesh to provide a good estimate of the error. We show some numerical results which corroborate in practice that these results are sharp

A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis

Fitted mesh method for a weakly coupled system of singularly perturbed reaction–convection–diffusion problems with discontinuous source term

In this article, a parameter-uniform numerical method for a weakly coupled system of singularly perturbed reaction–convection–diffusion problems with discontinuous source term containing two small parameters multiplied to the highest and second highest derivative is presented. A fitted mesh method using an upwind finite difference scheme on piecewise-uniform Shishkin mesh is constructed. Error analysis is undertaken and numerical results are provided to support the theoretical error bounds. Keywords: Singular perturbation problem, Reaction–convection–diffusion system, Fitted mesh method, Shishkin mesh, Discontinuous source term, Parameter-unifor