Fitted Numerical Methods for Singular Perturbation Problems

Miller, John J. H.; O’Riordan, Eugene; Shishkin, G. I.

doi:10.1142/2933

Cited by 591 publications

(165 citation statements)

References 0 publications

Supporting

Mentioning

165

Contrasting

Order By: Relevance

“…Therefore, we approximate the local Green function g i by a fitted mesh method of Shishkin type [4,5,7] and then use the resulting approximations in place of g i 's in Equation (3).…”

Section: Methodsmentioning

confidence: 99%

“…The first of these involves replacing the standard finite-difference operator by a difference operator which reflects the singularly perturbed nature of the differential operator. Such numerical methods are referred to, in general, as fitted operator finite-difference methods [4,5]. Typical derivation of such methods based on the discretization of the domain into a set of equidistant subintervals and the exact solution of a local BVP with an irregular data on a pair of adjacent subintervals.…”

Section: Introductionmentioning

confidence: 99%

“…The second major approach in the construction of the -uniform finite-difference method involves the use of a fitted mesh; a mesh that is adapted according to the singular perturbation [4,5]. Let us concentrate on the most efficient subclass of the fitted meshes methods: Shishkin mesh [6].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

Filiz

Neslitürk

Sendur

2012

International Journal of Computer Mathematics

View full text Add to dashboard Cite

We propose a fully discrete -uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a -uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any . Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.

show abstract

“…Therefore, we approximate the local Green function g i by a fitted mesh method of Shishkin type [4,5,7] and then use the resulting approximations in place of g i 's in Equation (3).…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

Filiz

Neslitürk

Sendur

2012

International Journal of Computer Mathematics

View full text Add to dashboard Cite

show abstract

“…[1][2][3] and the reference therein). Problems of this type arise in many areas of engineering, such as uid mechanics, heat transfer as well as problems in structural mechanics posed over thin domains.…”

Section: Introductionmentioning

confidence: 94%

A note onp/hp finite element methods for reaction-diffusion problems in polygonal domains

Xenophontos

2000

Commun. Numer. Meth. Engng.

View full text Add to dashboard Cite

We present guidelines on how to choose the mesh and polynomial degree distribution for the approximation of reaction-di usion problems in polygonal domains, in the context of the hp ÿnite element method. These guidelines take into account the presence of both boundary layers and corner singularities in the solution. We present the results of numerical computations showing that robust, (near) exponential rates of convergence can be achieved using this technique.

show abstract

“…From this, one easily establishes that u ε Ω C. The bounds on v ε and w ε and their derivatives are then established using the arguments given in [4] for the problem…”

Section: Singularly Perturbed Problemmentioning

confidence: 99%

A parameter-uniform Schwarz method for a singularly perturbed reaction–diffusion problem with an interior layer

Miller

O’Riordan

Шишкин

et al. 2000

Applied Numerical Mathematics

View full text Add to dashboard Cite

In this paper we consider numerical methods for a singularly perturbed reaction-diffusion problem with a discontinuous source term. We show that such a problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-uniform mesh. The mesh is fitted to the boundary and interior layers that occur in the solution of the problem. We show by extensive computations that, for this problem, this method is parameteruniform in the maximum norm, in the sense that the numerical solutions converge in the maximum norm uniformly with respect to the singular perturbation parameter. The semiconductor device equationsThe stationary behaviour of semiconductor devices can be described by the nonlinear system of second-order elliptic equations of van Roosbroeck [6]. In one dimension, for example, a simple p-n diode Ω = (0, 1) with two Ohmic contacts, an anode at x = 0 and a cathode at x = 1, is governed by the equationson Ω with appropriate boundary conditions at x = 0 and x = 1. Here ψ is the electrostatic potential, η and ρ are the electron and hole concentrations in the Slotboom variables, D denotes the doping function * Corresponding author.0168-9274/00/$ -see front matter © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved PII: S 0168-9274(99 )00 1 40 -3

show abstract

Fitted Numerical Methods for Singular Perturbation Problems

Cited by 591 publications

References 0 publications

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

A note onp/hp finite element methods for reaction-diffusion problems in polygonal domains

A parameter-uniform Schwarz method for a singularly perturbed reaction–diffusion problem with an interior layer

Contact Info

Product

Resources

About