A singularly perturbed convection–diffusion problem, posed on the unit square in $${\mathbb {R}}^2$$ , is studied; its solution has both exponential and characteristic boundary layers. The problem solved numerically using local discontinuous Galerkin (LDG) method Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $$k>0$$ each variable, error between LDG true proved to...

We consider the reaction-diffusion equation with discontinues coefficients and singular sources in one dimension. In this work, we construct ε-uniformly convergent High Order Compact (HOC) monotone finite difference schemes defined on a priori Shishkin meshes, which have order two, three and four except for a logarithmic factor. Numerical experiments are presented and discussed.

In the first part of the paper we discuss minimal smoothness assumptions for the components of the solution decomposition which allow to prove robust convergence results in the energy norm for linear or bilinear finite elements on Shishkin meshes applied to convectiondiffusion problems with exponential boundary layers. In the corresponding derivation the standard Lagrange interpolant is used, i...

We consider the numerical approximation of singularly perturbed reaction-diffusion problems over twodimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi-optimal rate. The results of numerical computations showing agreement with the a...

We analyse a streamline diiusion scheme on a special piecewise uniform mesh for a model time-dependent convection-diiusion problem. The method with piecewise linear elements is shown to be convergent, independently of the diiusion parameter, with a pointwise accuracy of almost order 5=4 outside the boundary layer and almost order 3=4 inside the boundary layer. Numerical results are also given.

In the present work, three-step Taylor Galerkin finite element method(3TGFEM) and least-squares finite element method(LSFEM) have been discussed for solving parabolic singularly perturbed problems. For singularly perturbed problems, a small parameter called singular perturbation parameter is multiplied with the highest order derivative term. As this singular perturbation parameter approaches to...

We consider the numerical solution, by a Petrov–Galerkin finite-element method, of a singularly perturbed reaction–diffusion differential equation posed on the unit square. In Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50, 2729–2743), it is argued that the natural energy norm, associated with a standard Galerk...

In Russ. Acad. Dokl. Math., 48, 1994, 346–352, Shishkin presented a numerical algorithm for a quasilinear time dependent singularly perturbed differential equation, with an internal layer in the solution. In this paper, we implement this method and present numerical results to illustrate the convergence properties of this numerical method.