In this paper a singularly perturbed convection–diffusion equation with a discontinuous source term is examined. Boundary and weak interior layers appear in the solution. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter.

We considered finite difference methods of higher order for semilinear singularly perturbed boundary value problems, consisted of constructing difference schemes on nonuniform meshes. Construction of schemes is presented and convergence uniform in perturbation parameter for one method is shown on Bakhvalov’s type of mesh. Numerical experiments demonstrated influence of different meshes on devel...

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

In this paper a singularly perturbed reaction-diffusion equation with a discontinuous source term is examined. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented which validate the theoretical results.

It is difficult to develop an algorithm which is able to generate the appropriate mesh around the interfaces in bimaterials. In this study, a corresponding algorithm is proposed for this class of unified structures made from different materials with arbitrary shapes. The non-uniform mesh is generated adaptively based on advancing front technique available in Abaqus software. Implementing severa...

We focus ourselves on the analysis of the solution of unsteady linear 2D singularly perturbed convection–diffusion equation. This type of equation can be considered as simplified model problem to many important problems, especially to Navier– Stokes equations. The space discretization of such a problem is a difficult task and it stimulated development of many stabilization methods (e.g. streaml...

We consider the numerical solution, by a Petrov–Galerkin finite-element method, of a singularly perturbed reaction–diffusion differential equation posed on the unit square. In Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50, 2729–2743), it is argued that the natural energy norm, associated with a standard Galerk...

The derivatives of the solution of singularly perturbed diierential equations become unbounded as the singular perturbation parameter " tends to zero. Therefore to approximate such derivatives, it is required to scale the derivatives in such a way that they are of order one for all values of the perturbation parameter. In practice , derivatives are related to the ux or drag and, hence, it is de...