Finite Difference Schemes for Convection-diffusion Problems with a Concentrated Source and a Discontinuous Convection Field

2001 ◽  
Vol 2 (1) ◽  
pp. 41-49 ◽  
Author(s):  
Torsten Linß
Keyword(s):  
Numerical Experiments ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Convection Diffusion ◽  
Concentrated Source ◽  
General Meshes ◽  
Theoretical Results ◽  
Diffusion Problems

AbstractA singularly perturbed convection-diffusion problem with a concentrated source is considered. The problem is solved numerically using two upwind difference schemes on general meshes. We prove convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm on Shishkin and Bakhvalov meshes. Numerical experiments complement our theoretical results.

scholarly journals The Sdfem for a Convection-diffusion Problem with Two Small Parameters

2003 ◽  
Vol 3 (3) ◽  
pp. 443-458 ◽  
Author(s):  
Hans-Görg Roos ◽  
Zorica Uzelac
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Small Parameters ◽  
Perturbation Parameters ◽  
Theoretical Results

AbstractA singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

scholarly journals Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Thái Anh Nhan ◽  
Relja Vulanović
Keyword(s):  
Uniform Convergence ◽  
Condition Number ◽  
Discrete System ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Proof Techniques ◽  
Consistency Error ◽  
Diffusion Problems

A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.

scholarly journals Supercloseness Result of Higher Order FEM/LDG Coupled Method for Solving Singularly Perturbed Problem on S-Type Mesh

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shenglan Xie ◽  
Huonian Tu ◽  
Peng Zhu
Keyword(s):  
Numerical Experiments ◽  
Diffusion Problem ◽  
Higher Order ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Piecewise Polynomial ◽  
Coupled Method ◽  
Convection Diffusion ◽  
Polynomial Approximations ◽  
Theoretical Results

we present a first supercloseness analysis for higher order FEM/LDG coupled method for solving singularly perturbed convection-diffusion problem. Based on piecewise polynomial approximations of degreek  (k≥1), a supercloseness property ofk+1/2in DG norm is established on S-type mesh. Numerical experiments complement the theoretical results.

Spectral analysis of finite difference schemes for convection diffusion equation

2017 ◽  
Vol 150 ◽  
pp. 95-114 ◽  
Author(s):  
V.K. Suman ◽  
Tapan K. Sengupta ◽  
C. Jyothi Durga Prasad ◽  
K. Surya Mohan ◽  
Deepanshu Sanwalia
Keyword(s):  
Spectral Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

scholarly journals A STABILIZING SUBGRID FOR CONVECTION–DIFFUSION PROBLEM

2006 ◽  
Vol 16 (02) ◽  
pp. 211-231 ◽  
Author(s):  
ALI I. NESLITURK
Keyword(s):  
Numerical Experiments ◽  
A Priori ◽  
Diffusion Problem ◽  
Triangular Element ◽  
Discrete Solution ◽  
Convergence Properties ◽  
Small Diffusion ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Priori Error Estimates

A stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection–diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.

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