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First-Order System Least Squares Finite-Elements for Singularly Perturbed Reaction-Diffusion Equations

  • James H. Adler
  • Scott MacLachlanEmail author
  • Niall Madden
Conference paper
  • 88 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

Abstract

We propose a new first-order-system least squares (FOSLS) finite-element discretization for singularly perturbed reaction-diffusion equations. Solutions to such problems feature layer phenomena, and are ubiquitous in many areas of applied mathematics and modelling. There is a long history of the development of specialized numerical schemes for their accurate numerical approximation. We follow a well-established practice of employing a priori layer-adapted meshes, but with a novel finite-element method that yields a symmetric formulation while also inducing a so-called “balanced” norm. We prove continuity and coercivity of the FOSLS weak form, present a suitable piecewise uniform mesh, and report on the results of numerical experiments that demonstrate the accuracy and robustness of the method.

Keywords

First-order system least squares (FOSLS) finite elements Singularly perturbed differential equations Parameter-robust discretizations 

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsTufts UniversityMedfordUSA
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  3. 3.School of Mathematics, Statistics, and Applied MathematicsNational University of Ireland GalwayGalwayIreland

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