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01.06.2021

A weighted and balanced FEM for singularly perturbed reaction-diffusion problems

verfasst von: Niall Madden, Martin Stynes

Erschienen in: Calcolo | Ausgabe 2/2021

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Abstract

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems \(-\varepsilon ^2\varDelta u +bu=f\) posed on bounded domains \(\varOmega \subset \mathbb {R}^k\) for \(k\ge 1\), with the Dirichlet boundary condition \(u=0\) on \(\partial \varOmega\), where \(0 <\varepsilon \ll 1\). The method is shown to be quasioptimal (on arbitrary meshes and for arbitrary conforming finite element spaces) with respect to a weighted norm that is known to be balanced when one has a typical decomposition of the unknown solution into smooth and layer components. A robust (i.e., independent of \(\varepsilon\)) almost first-order error bound for a particular FEM comprising piecewise bilinears on a Shishkin mesh is proved in detail for the case where \(\varOmega\) is the unit square in \(\mathbb {R}^2\). Numerical results illustrate the performance of the method.

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Nächster Artikel Second-order finite difference approximation for a nonlinear size-structured population model with an indefinite growth rate coupled with the environment

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Titel: A weighted and balanced FEM for singularly perturbed reaction-diffusion problems
verfasst von: Niall Madden
Martin Stynes
Publikationsdatum: 01.06.2021
Verlag: Springer International Publishing
Erschienen in: Calcolo / Ausgabe 2/2021
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-021-00421-w

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