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 $\varepsilon\in (0,1]$ 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.