Higher-Order QMC hp-FEM UQ for Singular Perturbation: Reaction-Diffusion with Gevrey-Parametric Reaction Coefficient
Abstract
We analyze the combined discretization of singularly perturbed reaction-diffusion problems with perturbation parameter ε∈ (0,1] and with parametric, Gevrey-regular reaction term, by hp-Galerkin FE methods and by higher-order polynomial lattice rules. We establish ε-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 sy-Gevrey regularity of the reaction coefficient expansion. On the interval (−1,1), we establish an ε-robust, root-exponential convergence rate exp(−σ pFE1/sx) for hp-Galerkin FEM for reaction-diffusion problems under sx-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.
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BibTeX
@misc{rohner2026higherorder,
title = {{Higher-Order QMC $hp$-FEM UQ for Singular Perturbation: Reaction-Diffusion with Gevrey-Parametric Reaction Coefficient}},
author = {Fabian Rohner and Christoph Schwab},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39738},
eprint = {2610.39738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39738},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-Order QMC hp-FEM UQ for Singular Perturbation: Reaction-Diffusion with Gevrey-Parametric Reaction Coefficient
AU - Fabian Rohner
AU - Christoph Schwab
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39738
UR - https://arxiv.org/abs/2610.39738
AB - We analyze the combined discretization of singularly perturbed reaction-diffusion problems with perturbation parameter ε∈ (0,1] and with parametric, Gevrey-regular reaction term, by hp-Galerkin FE methods and by higher-order polynomial lattice rules. We establish ε-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 s_y-Gevrey regularity of the reaction coefficient expansion. On the interval (−1,1), we establish an ε-robust, root-exponential convergence rate exp(−σ p_FE^1/s_x) for hp-Galerkin FEM for reaction-diffusion problems under 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.
ER -