First-Order System Least Squares Finite-Elements for Singularly Perturbed Reaction-Diffusion Equations
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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.
KeywordsFirst-order system least squares (FOSLS) finite elements Singularly perturbed differential equations Parameter-robust discretizations
- 13.Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems, revised edn. World Scientific Publishing Co., Pte. Ltd., Hackensack (2012). https://doi.org/10.1142/9789814390743