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hp-FEM for reaction-diffusion equations. II: Robust exponential convergence for multiple length scales in corner domains

by L. Banjai and J.M. Melenk and Ch. Schwab

(Report number 2020-28)

Abstract
In bounded, polygonal domains Ω ⊂ R2 with Lipschitz boundary ∂Ω consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze hp-FEM discretizations of a linear, second order, singularly perturbed reaction diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these hp-FEM afford exponential convergence, in the natural "energy" norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the exponential convergence.

Keywords: anisotropic hp–refinement, geometric corner refinement, exponential convergence

@Techreport{BMS20_901,
  author = {L. Banjai and J.M. Melenk and Ch. Schwab},
  title = {hp-FEM for reaction-diffusion equations.   II: Robust exponential convergence for multiple length scales in corner domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-28.pdf },
  year = {2020}
}

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