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hp FEM for Reaction-Diffusion Equations. I: Robust Exponential Convergence

by J. M. Melenk and Ch. Schwab

(Report number 1997-03)

Abstract
A singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right hand side is analytic. We show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width $O(p \varepsilon)$ is inserted near the domain boundary, that is, the rate of convergence is $O({ exp} (-b p))$ and independent of the perturbation parameter $\varepsilon$. We also show that the Spectral Element Method based on the use of a Gauss-Lobatto quadrature rule of order O(p) for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence.

Keywords: boundary layer, singularly perturbed problem, asymptotic expansions, error bounds

@Techreport{MS97_208,
  author = {J. M. Melenk and Ch. Schwab},
  title = {hp FEM for Reaction-Diffusion Equations. I: Robust Exponential Convergence},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1997-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1997/1997-03.pdf },
  year = {1997}
}

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