A direct higher order discretization in singular perturbations via domain split -- a computational approach
type: Item
instance of: scholarly article
title: A direct higher order discretization in singular perturbations via domain split -- a computational approach
zbMATH Open document ID: 1223.65059
author: Christian Grossmann
author: R. K. Mohanty
author: Hans-G. Roos
published in: Applied Mathematics and Computation
publication date: 2011-06-27T00:00:00Z
review text: The authors consider a specified form of a singularly perturbed two-point boundary value problem. As a means of solving the equation considered, two subproblems are generated over suitably chosen subintervals and then solved numerically. This then allows for the use of standard methods on each of the subintervals. The novelty of the work is that the authors do not treat the coupling iteratively, but rather introduce a linear equation which approximates the coupling condition to be solved exactly. Two different types of couplings similar to domain decomposition are studied: one based upon the notion of non-overlapping the domain, and the other on an overlapping domain. In each of the subintervals that are obtained in both the non-overlapping and overlapping domain they find amendable conditions for applying standard discretization techniques in each subinterval separately. A fourth-order compact finite difference method generating a linear discrete problem is applied. In splitting the domain into two subdomains the discretization is applied on a uniform grid separately in each domain. Given that approximations of the first order derivatives are required, the coupling of the different domains is performed via discrete approximations of the derivatives in the case of the non-overlapping domain split. In the case of the overlapping domain split the value at the one boundary is required and hence the coupling is devised via coinciding function values. Theoretical results are provided with regards to these two cases. Numerical experiments are also conducted, establishing the effect of certain chosen parameter values, associated computational errors and which technique used over a chosen type of domain produces the best numerical results. In comparison to the implementation of direct discretization on Shishkin meshes over the full interval, the authors verify that the separate treatment of the boundary value problems generated on the related subintervals produces a signifi…
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L11
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 65L50
zbMATH DE Number: 5913742
zbMATH Keywords: higher order discretization
zbMATH Keywords: difference methods
zbMATH Keywords: domain decomposition
zbMATH Keywords: singular perturbation
zbMATH Keywords: two-point boundary value problem
zbMATH Keywords: numerical experiments
zbMATH Keywords: numerical results
zbMATH Keywords: Shishkin meshes
reviewed by: Charis Harley
MaRDI profile type: Publication
full work available at URL: j.amc.2011.04.013
OpenAlex ID: W1978179090
cites work: Q5610080
cites work: The solution of a singularly perturbed convection--diffusion problem by an iterative domain decomposition method
cites work: High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type
cites work: Alternating Group Explicit Method For The Numerical Solution Of Non-Linear Singular Two-Point Boundary Value Problems Using A Fourth Order Finite Difference Method
cites work: Q3321316
cites work: Q3867093
cites work: Q2724540
cites work: A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers
cites work: Q4379541
cites work: A note on the conditioning of upwind schemes on Shishkin meshes
cites work: Robust Numerical Methods for Singularly Perturbed Differential Equations
cites work: A domain decomposition method in the case of nonoverlapping subdomains for a singularly perturbed convection-diffusion equation
cites work: Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry
cites work: Q4825126
cites work: Higher order methods for a singularly perturbed problem
DOI: 10.1016/J.AMC.2011.04.013
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description: scientific article; zbMATH DE number 5913742