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Higher-Order QMC hp-FEM UQ for Singular Perturbation: Reaction--Diffusion with Gevrey-Parametric Reaction Coefficient

by F. Rohner and C. Schwab

(Report number 2026-33)

Abstract
We analyze the combined discretization of singularly perturbed reaction--diffusion problems with perturbation parameter \(\varepsilon\in (0,1]\) and with parametric, Gevrey-regular reaction term, by \(hp\)-Galerkin FE methods and by higher-order polynomial lattice rules. We establish \(\varepsilon\)-robust algebraic convergence rates for the combined QMC \(hp\)-FEM approximation that are not subject to the curse of dimensionality (CoD) in the parametric reaction coefficient, and only limited by the sparsity and the \(\mathfrak{s}_{y}\)-Gevrey regularity of the reaction coefficient expansion. On the interval \((-1,1)\), we establish an \(\varepsilon\)-robust, root-exponential convergence rate \(\exp(-\sigma \, p_{\mathrm{FE}}^{1/\mathfrak{s}_{x}})\) for \(hp\)-Galerkin FEM for reaction--diffusion problems under \(\mathfrak{s}_{x}\)-Gevrey regularity assumptions on the reaction coefficient and the source term, a result that can be of independent interest. Numerical experiments are consistent with the theoretical bounds.

Keywords: singular perturbation, Gevrey regularity, Quasi--Monte Carlo integration, reaction--diffusion equations, uncertainty quantification

@Techreport{RS26_1188,
  author = {F. Rohner and C. Schwab},
  title = {Higher-Order QMC hp-FEM UQ for Singular Perturbation: Reaction--Diffusion with Gevrey-Parametric Reaction Coefficient},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-33},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-33.pdf },
  year = {2026}
}

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