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Higher-Order QMC Galerkin FEM for Singular Perturbation: Reaction--Diffusion with Affine-Parametric Reaction

by F. Rohner and C. Schwab

(Report number 2026-25)

Abstract
We analyze higher-order quasi--Monte Carlo (QMC) Galerkin discretizations for a singularly perturbed reaction--diffusion problem with an affine-parametric reaction coefficient. We derive error estimates that are robust with respect to the perturbation parameter \(\varepsilon\). To this end, we establish \(\varepsilon\)-uniform stability and mixed-parametric regularity estimates in an \(\varepsilon\)-weighted energy norm, with constants independent of the parameter dimension and of \(\varepsilon\). These estimates imply robust dimension truncation estimates and dimension-independent higher-order QMC error bounds for linear quantities of interest. Assuming an abstract, \(\varepsilon\)-robust finite element approximation property, we combine the parametric estimates with the spatial discretization analysis to derive a general fully discrete QMC--FE error estimate with constants independent of both the parameter dimension and \(\varepsilon\). We verify this approximation property for a one-dimensional model problem using continuous piecewise affine finite elements on Shishkin meshes. Numerical experiments illustrate the predicted robustness and convergence rates.

Keywords: singular perturbation, Quasi--Monte Carlo integration, Shishkin meshes, uncertainty quantification

@Techreport{RS26_1180,
  author = {F. Rohner and C. Schwab},
  title = {Higher-Order QMC Galerkin FEM for Singular Perturbation:
Reaction--Diffusion with Affine-Parametric Reaction},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-25.pdf },
  year = {2026}
}

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