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.

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 Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case

2017 ◽  
Vol 9 (1) ◽  
pp. 68 ◽  
Author(s):  
Michel Fortin ◽  
Abdellatif Serghini Mounim
Keyword(s):  
Finite Volume ◽  
Numerical Approximation ◽  
Dimensional Case ◽  
Parabolic Problems ◽  
Finite Volume Schemes ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Hybrid Finite Element ◽  
The One ◽  
Diffusion Problems

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Renato M. Cotta ◽  
Carolina P. Naveira-Cotta ◽  
Diego C. Knupp
Keyword(s):  
Eigenvalue Problem ◽  
Eigenfunction Expansion ◽  
Space Variable ◽  
Eigenvalue Problems ◽  
Integral Transform ◽  
Diffusion Problem ◽  
Eigenfunction Expansions ◽  
Diffusion Operator ◽  
Convection Diffusion ◽  
Diffusion Problems

The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers' equation.

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