Volume 5, Issue 4
Construction of Boundary Layer Elements for Singularly Perturbed Convection-Diffusion Equations and $L^2$-Stability Analysis

Int. J. Numer. Anal. Mod., 5 (2008), pp. 729-748.

Published online: 2008-05

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• Abstract

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the $H^1$-approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the $L^2$-stability analysis of the scheme and derive the $L^2$-error approximations.

• Keywords

boundary layer, boundary layer element, finite elements, singularly perturbed problem, convection-diffusion, stability, enriched subspaces, exponentially fitted splines.

65N30, 34D15, 76N20

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@Article{IJNAM-5-729, author = {}, title = {Construction of Boundary Layer Elements for Singularly Perturbed Convection-Diffusion Equations and $L^2$-Stability Analysis}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {4}, pages = {729--748}, abstract = {

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the $H^1$-approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the $L^2$-stability analysis of the scheme and derive the $L^2$-error approximations.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/835.html} }
TY - JOUR T1 - Construction of Boundary Layer Elements for Singularly Perturbed Convection-Diffusion Equations and $L^2$-Stability Analysis JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 729 EP - 748 PY - 2008 DA - 2008/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/835.html KW - boundary layer, boundary layer element, finite elements, singularly perturbed problem, convection-diffusion, stability, enriched subspaces, exponentially fitted splines. AB -

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the $H^1$-approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the $L^2$-stability analysis of the scheme and derive the $L^2$-error approximations.

C.-Y. Jung & R. Temam. (2019). Construction of Boundary Layer Elements for Singularly Perturbed Convection-Diffusion Equations and $L^2$-Stability Analysis. International Journal of Numerical Analysis and Modeling. 5 (4). 729-748. doi:
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