We consider the following singularly perturbed semilinear elliptic problem: 2 u ? u + u p = 0 in ; u > 0 in and @u @ = 0 on @; where is a bounded smooth domain in R N , > 0 is a small constant and p is a sub-critical exponent. Let J u] := R (2 2 jruj 2 + 1 2 u 2 ? 1 p+1 u p+1)dx be its energy functional, where u 2 H 1 ((). Ni and Takagi ((15], 16]) proved that for a single boundary spike soluti...

Diiusion problems under singular perturbations of the domain or the boundary conditions are analyzed. The rst problem that we consider is the diiusion of a material from a domain that is nearly impermeable, having only several small patches on the boundary where the material can slowly leak out. The second problem that is studied is the diiusion of a material that originates from some localized...

We design non-standard finite difference schemes for self-adjoint singularly perturbed two-point boundary value problems. Essential physical properties (e.g., dissipativity) of the solutions of such problems are captured in the schemes by an appropriate renormalization of the denominator of the discrete derivative. The schemes are analyzed for -uniform convergence. Several numerical examples ar...

We consider the numerical approximation of singularly perturbed problems, and in particular reaction-di usion problems, by the h version of the nite element method. We present guidelines on how to design non-uniform meshes both in one and two dimensions, that are asymptotically optimal as the meshwidth tends to zero. We also present the results of numerical computations showing that robust, opt...

In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. It is assumed that both equations have a turning point at the same point. An appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

A finite element method for a singularly perturbed convection-diffusion problem with exponential boundary layers is analysed. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the ...

In this paper we consider singularly perturbed ordinary differential equations with convection terms. An ε-uniformly convergent finite difference scheme is constructed for such boundary value problems by using an upwind finite difference operator and a piecewise uniform mesh. Our primary interest is the design of a numerical method, for which a parallel technique will later be applicable. In th...

A heterogeneous domain-decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems. The method, which is parallelizable at various levels, uses several ideas of asymptotic analysis. The subdomains match the domains of validity of the local [ “inner” and “outer”) asymptotic expansions, and cut-off functions are used to match solutions in ...

We analyze finite volume schemes of arbitrary order r for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (Nln(N + 1)), where 2N is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order (Nln(N + 1)...

Linear singularly perturbed ordinary differential equations of convection diffusion type are considered. The convective coefficient varies in scale across the domain which results in interior layers appearing in areas where the convective coefficient decreases from a scale of order one to the scale of the diffusion coefficient. Appropriate parameter-uniform numerical methods are constructed. Nu...