This article reviews some of the salient features of the piecewise-uniform Shishkin mesh. The central analytical techniques involved in the associated numerical analysis are explained via a particular class of singularly perturbed differential equations. A detailed discussion of the Shishkin solution decomposition is included. The generality of the numerical approach introduced by Shishkin is highlighted. The impact of Shishkin's ideas on the field of singularly perturbed differential is assessed in this selective review of his research output over the past thirty years.

A Dirichlet problem for a singularly perturbed steady-state convection-diffusion equation with constant coefficients on the unit square is considered. In the equation under consideration the convection term is represented by only a single derivative with respect to one coordinate axis. This problem is discretized by the classical five-point upwind difference scheme on a rectangular piecewise uniform mesh that is refined in the neighborhood of the regular and the characteristic boundary layers. It is proved that, for sufficiently smooth right-hand side of the equation and the restrictions of the continuous boundary function to the sides of the square, without additional compatibility conditions at the corners, the error of the discrete solution is $O(N^{-1}\ln^2 N)$ uniformly with respect to the small parameter, in the discrete maximum norm, where $N$ is the number of mesh points in each coordinate direction.

In this work we consider a parabolic system of two linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms. To obtain an efficient approximation of the exact solution we propose a numerical method combining the Crank-Nicolson method used in conjunction with the central finite difference scheme defined on a piecewise uniform Shishkin mesh. The method gives uniform numerical approximations of second order in time and almost second order in space. Some numerical experiments are given to support the theoretical results.

In this paper a problem arising in the modelling of semiconductor devices motivates the study of singularly perturbed differential equations of reaction-diffusion type with discontinuous data. The solutions of such problems typically contain interior layers where the gradient of the solution changes rapidly. Parameter-uniform methods based on piecewise-uniform Shishkin meshes are constructed and analysed for such problems. Numerical results are presented to support the theoretical results and to illustrate the benefits of using a piecewise-uniform Shishkin mesh over the use of uniform meshes in the simulation of a simple semiconductor device.

Continuing an earlier work in space dimension one, the aim of this article is to present, in space dimension two, a novel method to approximate stiff problems using a combination of (relatively easy) analytical methods and finite volume discretization. The stiffness is caused by a small parameter in the equation which introduces ordinary and corner boundary layers along the boundaries of a two-dimensional rectangle domain. Incorporating in the finite volume space the boundary layer correctors, which are explicitly found by analysis, the boundary layer singularities are absorbed and thus uniform meshes can be preferably used. Using the central difference scheme at the volume interfaces, the proposed scheme finally appears to be an efficient second-order accurate one.

We consider a one-dimensional convection-diffusion boundary value problem, whose solution contains a boundary layer at the outflow boundary, and construct a finite element method for its approximation. The finite element space consists of piecewise polynomials on a uniform mesh but is enriched by a finite number of functions that represent the boundary layer behavior. We show that this method converges at the optimal rate, independently of the singular perturbation parameter, when the error is measured in the energy norm associated with the problem. Numerical results confirming the theory are also presented, which also suggest that in the case of variable coefficients, the number of enrichment functions need not be as high as the theory suggests.

A singularly perturbed two-point boundary-value problem of reaction-convection-diffusion type is considered. The problem involves two small parameters that give rise to two boundary layers of different widths. The problem is solved using a streamline-diffusion FEM (SDFEM).
A robust a posteriori error estimate in the maximum norm is derived. It provides computable and guaranteed upper bounds for the discretisation error.
Numerical examples are given that illustrate the theoretical findings and verify the efficiency of the error estimator on a priori adapted meshes and in an adaptive mesh movement algorithm.

We present an analysis of an Alternating Direction Implicit (ADI) scheme for a linear, singularly perturbed reaction-diffusion equation. By providing an expression for the error that separates the temporal and spatial components, we can use existing results for steady-state problems to give a succinct analysis for the time-dependent problem, and that generalizes for various layer-adapted meshes. We report the results of numerical experiments that support the theoretical findings. In addition, we provide a numerical comparison between the ADI and Euler techniques, as well details of the computational advantage gained by parallelizing the algorithm.

The two-level local projection stabilization is considered as a one-level approach in which the enrichments on each element are piecewise polynomial functions. The dimension of the enrichment space can be significantly reduced without losing the convergence order. For example, using continuous piecewise polynomials of degree $r \geq 1$, only one function per cell is needed as enrichment instead of $r$ in the two-level approach. Moreover, in the constant coefficient case, we derive formulas for the user-chosen stabilization parameter which guarantee that the linear part of the solution becomes nodally exact.

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise-uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretization, to form a new numerical method for solving this problem. It is proved that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters. Numerical results are presented in support of the theory.

A singularly perturbed degenerate parabolic problem in one space dimension is considered. Bounds on derivatives of the solution are proved; these bounds depend on the two data parameters that determine how singularly perturbed and how degenerate the problem is. A tensor product mesh is constructed that is equidistant in time and of Shishkin type in space. A finite difference method on this mesh is proved to converge; the rate of convergence obtained depends on the degeneracy parameter but is independent of the singular perturbation parameter. Numerical results are presented.

A singularly perturbed quasilinear boundary-value problem is considered in the case when its solution has a boundary shock. The problem is discretized by an upwind finite-difference scheme on a mesh of Shishkin type. It is proved that this numerical method has pointwise accuracy of almost first order, which is uniform in the perturbation parameter.

This paper is concerned with the solution of the nonlinear system of equations arising from the A.M. Il'in's scheme approximation of a model semilinear singularly perturbed boundary value problem. We employ Newton and Picard methods and propose a new version of the two-grid method originated by O. Axelsson [2] and J. Xu [19]. In the first step, the nonlinear differential equation is solved on a "coarse" grid of size $H$. In the second step, the problem is linearized around an appropriate interpolation of the solution computed in the first step and the linear problem is then solved on a fine grid of size $h<<H$. It is shown that the algorithms achieve optimal accuracy as long as the mesh sizes satisfy $h = O(H^{2^m})$, $m=1,2,...$, where $m$ is the number of the Newton (Picard) iterations for the difference problem. We count the number of the arithmetical operations to illustrate the computational cost of the algorithms. Numerical experiments are discussed.

A parabolic initial-boundary value problem with solutions displaying exponential layers is solved using layer-adapted meshes. The paper combines finite elements in space, i.e., a pure Galerkin technique on a Shishkin mesh, with some standard discretizations in time. We prove error estimates as well for the $\theta$-scheme as for discontinuous Galerkin in time.