Related papers: A fitted second--order difference scheme on a modi…
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow…
We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of…
The singularly perturbed reaction-diffusion problem $\varepsilon^2\Delta^2 u - \mathrm{div}\left(c\nabla u\right) = f$ is considered on the unit square $\Omega$ in $\mathbb{R}^2$ with homogenous Dirichlet boundary conditions. Its solution…
We consider a singularly perturbed semilinear boundary value problem of a general form that allows various types of turning points. A solution decomposition is derived that separates the potential exponential boundary layer terms. The…
A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A…
A finite difference method is constructed for a singularly perturbed convection diffusion problem posed on an annulus. The method involves combining polar coordinates, an upwind finite difference operator and a piecewise-uniform Shishkin…
In this article, we address singularly perturbed two-parameter parabolic problem of the reaction-convection-diffusion type in two dimensions. These problems exhibit discontinuities in the source term and convection coefficient at particular…
This work develops an epsilon-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By moving one node arbitrarily in between its adjacent nodes, the new…
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More…
A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used…
We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and $H_{div}$-conforming elements for the second component we provide a…
We consider a non-polynomial cubic spline to develop the classes of methods for the numerical solution of singularly perturbed two-point boundary value problems. The proposed methods are second and fourth order accurate and applicable to…
A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for…
We consider a second order, two-point, singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and we obtain regularity results for its solution. First we establish classical…
We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called…
A two-parameter singularly perturbed problem with discontinuous source and convection coefficient is considered in one dimension. Both convection coefficient and source term are discontinuous at a point in the domain. The presence of…
Error estimates of finite element methods for reaction-diffusion Problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^1$ seminorm leads…
A singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise-uniform Shishkin mesh. Scaled discrete derivatives (with scaling…
We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted graded meshes…
We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f $ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial…