We give the review of recent results in relative perturbation theory for eigenvalue and singular value problems and highly accurate algorithms which compute eigenvalues and singular values to the highest possible relative accuracy.

In this paper, we investigate a singular perturbation problem including a fourth order O.D.E. with general linear boundary conditions. Firstly, we obtain the necessary conditions of solution of O.D.E. by making use of fundamental solution, then by compatibility of these conditions with boundary conditions, we determine that, for given perturbation problem, whether boundary layer is formed or not.

Implicit time stepping procedures for the time dependent Stokes problem lead to stationary singular perturbation problems at each time step. These singular perturbation problems are systems of saddle point type, which formally approach a mixed formulation of the Poisson equation as the time step tends to zero. Preconditioners for discrete analogous of these systems are discussed. The preconditi...

We consider a singular elliptic perturbation of a Hammerstein integral equation with singular nonlinear term at the origin. The compactness of the solutions to the perturbed problem and, hence, the existence of a positive solution for the integral equation are proved. Moreover, these results are applied to nonlinear singular homogeneous Dirichlet problems.

We consider the approximation of singularly perturbed systems of reaction–diffusion problems, with the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. In this article we...

In this article we consider the approximation of singularly perturbed boundary value problems using a local adaptive grid h-refinement for finite element method, the variation iteration method and the homotopy perturbation method. The solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulti...