Volume 4, Issue 4
Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control

Hans-Görg Roos & Christian Reibiger

Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 562-575.

Published online: 2011-04

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  • Abstract

We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection-diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.

  • Keywords

Convection-diffusion, linear finite elements, a priori analysis, layer-adapted meshes, singular perturbed, optimal control.

  • AMS Subject Headings

34E05, 34E20, 65L10, 65L11, 65L60, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-4-562, author = {}, title = {Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2011}, volume = {4}, number = {4}, pages = {562--575}, abstract = {

We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection-diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m1101}, url = {http://global-sci.org/intro/article_detail/nmtma/5983.html} }
TY - JOUR T1 - Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 562 EP - 575 PY - 2011 DA - 2011/04 SN - 4 DO - http://doi.org/10.4208/nmtma.2011.m1101 UR - https://global-sci.org/intro/article_detail/nmtma/5983.html KW - Convection-diffusion, linear finite elements, a priori analysis, layer-adapted meshes, singular perturbed, optimal control. AB -

We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection-diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.

Hans-Görg Roos & Christian Reibiger. (2020). Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control. Numerical Mathematics: Theory, Methods and Applications. 4 (4). 562-575. doi:10.4208/nmtma.2011.m1101
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