In this article, we deal with some problems involving a class of singularly perturbed elliptic operators. We prove the asymptotic preserving general Galerkin method associated to semilinear problem. use particular approximation estimate convergence rate on whole domain, for linear Finally, study behaviour semigroup generated.

The paper is devoted to the investigation of the relationship between slow integral manifolds of singularly perturbed differential equations and critical phenomena in chemical kinetics and laser models. We tried to consider different problems using the techniques of canards and black swans. The mathematical language of singular perturbations seems to apply to all critical phenomena even in the ...

We consider the approximation of curvature dependent geometric front evolutions by singularly perturbed parabolic double obstacle problems with small parameter ". We give a simpliied proof of optimal interface error estimates of order O(" 2), valid in the smooth regime, which is based on constructing precise barriers, perturbing the forcing term and exploiting the maximum principle.

For singularly perturbed convection-diffusion problems with the perturbation parameter εmultiplying the highest derivatives, we construct a scheme based on the defect correction method and its parallel variant that converge ε-uniformly with second-order accuracy in the time variable. We also give the conditions under which the parallel computation accelerates the solution process with preservin...

We describe the behaviour of minimum problems involving non-convex surface integrals in 2D singularly perturbed by a curvature term. We show that their limit is described by functionals which take into account energies concentrated on vertices of polygons. Non-locality and non-compactness e ects are highlighted.

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

We analyze existing discontinuous Galerkin methods on quasi-uniform meshes for singularly perturbed problems. We prove weighted L2 error estimates. We use the weighted estimates to prove L2 error estimates in regions where the solution is smooth. We also prove pointwise estimates in these regions.

We analyze existing discontinuous Galerkin methods on quasiuniform meshes for singularly perturbed problems. We prove weighted L2 error estimates. We use the weighted estimates to prove L2 error estimates in regions where the solution is smooth. We also prove pointwise estimates in these regions.

The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.

we consider the numerical approximation of a singularly perturbed reaction-diffusion problem over a square. Two different. approaches are compared namely: adaptive isotropic mesh refinement and anisotropic mesh refinement. Thus, we compare the h-refinement and the Shishkin mesh approaches numerically with PLTMG software [l]. It is shown how isotropic elements lead to over-refinement. and how an...