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...

A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a continuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these...

In this paper, we have proposed an ε-uniform initial value technique for singularly perturbed convection-diffusion problems in which an asymptotic expansion approximation of the solution of boundary value problem is constructed using the basic idea of WKB method. In this computational technique, the original problem reduces to combination of an initial value problem and a terminal value problem...

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional L-shaped domain Ω subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space C2/3(Ω̄) and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3π/2. We...

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...

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 not necessarily equal. The components of the solution exhibit overlapping layers. A Shishkin piecewise– uniform mesh is constructed, which is used, in conjunction with a cl...

A linear singularly perturbed time dependent convection–diffusion problem is examined. The initial condition is designed to have steep gradients in the vicinity of the inflow point, which are transported in time, thus creating a moving interior shock layer. The location of this interior layer is tracked by the characteristics of the reduced first order problem. A numerical method is designed an...

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 ...

A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a continuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these...

In this work, superconvergent approximation of singularly perturbed two-point boundary value problems of reaction-diiusion type and convection-diiusion type are studied. By applying the standard nite element method on the Shishkin mesh, superconvergent error bounds of (N ?1 ln(N +1)) p+1 in a discrete energy norm are established. The error bounds are uniformly valid with respect to the singular...