Relja Vulanović

Relja Vulanović

A linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically after its solution is decomposed as $u_0+w$, where $u_0$, the corresponding reduced solution, is treated as a function known exactly or approximately. The component $w$ is then calculated using the exponentially fitted Allen-Southwell-Il'in (ASI) scheme on the Shishkin mesh and its asymptotic version. We prove that this numerical method is highly accurate, with errors that diminish when the discretization parameter increases, and, in some cases, even when the perturbation parameter decreases. This is a theoretical confirmation of earlier numerical results showing that the ASI scheme outperforms the general class of Samarskii-type schemes to which it belongs. Even higher accuracy is proved when $u_0$ is linear, in which case, the decomposition is not needed. New numerical experiments are provided to illustrate all this.

Relja Vulanović

A class of quasilinear singularly perturbed boundary value problems with a turning point of attractive type is considered. The problems are solved numerically by a finite-difference scheme on a special discretization mesh which is dense near the turning point. The scheme is a combination of the standard central and midpoint schemes and is practically second-order accurate. Pointwise accuracy is uniform in the perturbation parameter and, moreover, L1 errors decrease when the perturbation parameter tends to 0. This is achieved by the use of meshes which generalize the piecewise equidistant Shishkin mesh. Two particular types of meshes are considered and compared.

Thái Anh Nhan, Relja Vulanović

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on a Bakhvalov-type mesh is analyzed. The preconditioning technique is used to obtain the pointwise convergence uniform in the perturbation parameter.