Fabian Rohner | Search arXiv

arXiv2026

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

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.

math.NA↗

arXiv2026

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

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 $\exp(-σ\, p_{\mathrm{FE}}^{1/\mathfrak{s}_{x}})$ 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.

math.NA↗