where ν is the outward unit normal vector on ∂Ω; ν exists a.e. for Lipschitz domains. The goal of this paper is to understand the asymptotic behavior of Λ(γ) as γ → ∞ when ∂Ω ∈ C1. Since Λ(γ) → ∞ when γ → ∞, (2) can be viewed as a singularly perturbed linear eigenvalue problem. The asymptotic behavior of Λ(γ) was first studied by Lacey, Ockendon and Sabina in [3], where they investigated some r...

As a simpliied model for contact problems, we study a mixed Neumann-Robin boundary value problem for the Laplace operator in a smooth domain in R 2. The Robin condition contains a small parameter " inducing boundary layers of corner type at the transition points as proved in 4]. We present an integral equation for the numerical solution of this problem together with estimates of the error. We i...

In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the nume...

A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion–reaction problem with a diffusion parameter tending to zero. To solve numerically the singularly perturbed diffusion– reaction problem, monotone approximations are employed. Numerical examples are presented f...

We study singularly perturbed Fredholm equations of the second kind. We give sufficient conditions for existence and uniqueness of solutions and describe the asymptotic behavior of the solutions. We examine the relationship between the solutions of the perturbed and unperturbed equations, exhibiting the degeneration of the boundary layer to delta functions. The results are applied to several ex...

We consider a spline difference scheme on a piecewise uniform Shishkin mesh for a singularly perturbed boundary value problem with two parameters. We show that the discrete minimum principle holds for a suitably chosen collocation points. Furthermore, bounds on the discrete counterparts of the layer functions are given. Numerical results indicate uniform convergence. AMS Mathematics Subject Cla...

Second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered. The behavior of the ordinary Galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed.

A fourth-order finite-difference method for a semilinear singularly perturbed boundary value problem is studied. This method is based on Hermitian approximation of the second derivative on special new discretization mesh of Bakhvalov type. Numerical examples which demonstrate the effectiveness of the method are presented. AMS Mathematics Subject Classification (2000): 65L10

A singularly perturbed time-dependent convection-diffusion problem is examined on non-rectangular domains. The nature of the boundary and interior layers that arise depends on the geometry of the domains. For problems with different types of layers, various numerical methods are constructed to resolve the layers in the solutions and the numerical solutions are shown to converge independently of...

In this paper, we have proposed a numerical method for singularly perturbed  fourth order ordinary differential equations of convection-diffusion type. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and  finite difference method. In order to get a numerical solution for the derivative of the solution, the given interval is divided  in...