Higher-Order QMC Galerkin FEM for Singular Perturbation: Reaction-Diffusion with Affine-Parametric Reaction
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 ε. To this end, we establish ε-uniform stability and mixed-parametric regularity estimates in an ε-weighted energy norm, with constants independent of the parameter dimension and of ε. These estimates imply robust dimension truncation estimates and dimension-independent higher-order QMC error bounds for linear quantities of interest. Assuming an abstract, ε-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 ε. 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.
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BibTeX
@misc{rohner2026higherorder,
title = {{Higher-Order QMC Galerkin FEM for Singular Perturbation: Reaction-Diffusion with Affine-Parametric Reaction}},
author = {Fabian Rohner and Christoph Schwab},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39734},
eprint = {2610.39734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39734},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-Order QMC Galerkin FEM for Singular Perturbation: Reaction-Diffusion with Affine-Parametric Reaction
AU - Fabian Rohner
AU - Christoph Schwab
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39734
UR - https://arxiv.org/abs/2610.39734
AB - 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 ε. To this end, we establish ε-uniform stability and mixed-parametric regularity estimates in an ε-weighted energy norm, with constants independent of the parameter dimension and of ε. These estimates imply robust dimension truncation estimates and dimension-independent higher-order QMC error bounds for linear quantities of interest. Assuming an abstract, ε-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 ε. 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.
ER -