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Published in:
2020 | OriginalPaper | Chapter
First-Order System Least Squares Finite-Elements for Singularly Perturbed Reaction-Diffusion Equations
Authors : James H. Adler, Scott MacLachlan, Niall Madden
Published in: Large-Scale Scientific Computing
Publisher: Springer International Publishing
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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.