Hans-Goerg Roos

Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline-functions

Abstract For a self-adjoint second-order model singular perturbation problem global uniformly convergent schemes are constructed. The basic idea consists in replacing the coefficients by piecewise polynomials and solving the resulting problems exactly.

Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers

We consider an interior turning point problem with two exponential boundary layers. Its discretisation on a piecewise-uniform Shishkin mesh yields a scheme which is uniformly convergent (measured in the discrete maximum norm) of almost order one. Richardson extrapolation improves the accuracy to O ( N - 2 ln 2 N ) . Both can be proved under the assumption e ? C N - 1 .

A modified Bakhvalov mesh

Abstract We consider a few numerical methods for solving a one-dimensional convection–diffusion singularly perturbed problem. More precisely, we introduce a modified Bakvalov mesh generated using some implicitly defined functions. Properties of this mesh and convergence results for several methods on it are given. Numerical results are presented in support of the theoretical considerations.

Remarks on Balanced Norm Error Estimates for Systems of Reaction-Diffusion Equations

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 H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.

A direct higher order discretization in singular perturbations via domain split: A computational approach

In this paper the domain split concept as known from domain decomposition is combined with higher order difference discretizations to construct direct methods for the numerical treatment of singularly perturbed two-point boundary value problems.