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 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\, pFE^1/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.