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...

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximation...

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general sett...

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergent rate N ?2 ln 2 N + N ?3=2 is established on a discrete energy norm. This rate is uniformly valid with respect to the singular perturbation parameter. As a by-product, an-uniform convergence of the same order is obtained for the L 2-nor...

We discuss some static and dynamic problems involving a competition between \bulk" and \surface" energy. In each case the bulk energy prefers microstructure, and the surface energy is a singular perturbation. It is natural to treat the surface energy density as a small parameter. In our static examples { involving branching of twins and magnetic domains { this leads to scaling laws for the ener...

We consider conforming finite element approximation of fourth-order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C1 finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of ( N−1ln(N + 1))p in a discrete energy norm is established. The error bound is uniformly valid with respect to the sing...