Optimal uniform-convergence results for convection–diffusion problems in one dimension using preconditioning

Optimal uniform-convergence results for convection–diffusion problems in one dimension using... A linear one-dimensional convection–diffusion problem with a small singular perturbation parameter ε is considered. The problem is discretized using finite-difference schemes on the Shishkin mesh. Generally speaking, such discretizations are not consistent uniformly in ε , so ε -uniform convergence cannot be proved by the classical approach based on ε -uniform stability and ε -uniform consistency. This is why previous proofs of convergence have introduced non-classical techniques (e.g., specially chosen barrier functions). In the present paper, we show for the first time that one can prove optimal convergence inside the classical framework: a suitable preconditioning of the discrete system is shown to yield a method that, uniformly in ε , is both consistent and stable. Using this technique, optimal error bounds are obtained for the upwind and hybrid finite-difference schemes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational and Applied Mathematics Elsevier

Optimal uniform-convergence results for convection–diffusion problems in one dimension using preconditioning

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier B.V.
ISSN
0377-0427
eISSN
1879-1778
D.O.I.
10.1016/j.cam.2018.02.012
Publisher site
See Article on Publisher Site

Abstract

A linear one-dimensional convection–diffusion problem with a small singular perturbation parameter ε is considered. The problem is discretized using finite-difference schemes on the Shishkin mesh. Generally speaking, such discretizations are not consistent uniformly in ε , so ε -uniform convergence cannot be proved by the classical approach based on ε -uniform stability and ε -uniform consistency. This is why previous proofs of convergence have introduced non-classical techniques (e.g., specially chosen barrier functions). In the present paper, we show for the first time that one can prove optimal convergence inside the classical framework: a suitable preconditioning of the discrete system is shown to yield a method that, uniformly in ε , is both consistent and stable. Using this technique, optimal error bounds are obtained for the upwind and hybrid finite-difference schemes.

Journal

Journal of Computational and Applied MathematicsElsevier

Published: Aug 15, 2018

References

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