Abstract
The \(hp\)-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed on an interval or a two-dimensional domain with an analytic boundary. On suitably designed Spectral Boundary Layer meshes, robust exponential convergence in a “balanced” norm is shown. This “balanced” norm is an \(\varepsilon \)-weighted \(H^1\)-norm, where the weighting in terms of the singular perturbation parameter \(\varepsilon \) is such that, in contrast to the standard energy norm, boundary layer contributions do not vanish in the limit \(\varepsilon \rightarrow 0\). Robust exponential convergence in the maximum norm is also established. We illustrate the theoretical findings with two numerical experiments.
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Notes
Note the subtle point that \(S_0(\lambda ,p)\subset H_{0}^{1}(I)\); in contrast, the reduced problem doesn’t involve boundary conditions.
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Melenk, J.M., Xenophontos, C. Robust exponential convergence of \(hp\)-FEM in balanced norms for singularly perturbed reaction-diffusion equations. Calcolo 53, 105–132 (2016). https://doi.org/10.1007/s10092-015-0139-y
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DOI: https://doi.org/10.1007/s10092-015-0139-y