Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains

We establish local Gevrey regularity for the weak solution to parametric divergence-form diffusion elliptic PDEs, assuming the diffusion coefficient itself possesses local Gevrey parametric regularity over a non-compact domain.

Here "local Gevrey regularity" means that the regularity is determined in a neighborhood of each parametric point and depends on that point.

Explicit bounds on parametric derivatives of the weak solution are proved in the $H^1$-norm.

Building on this local Gevrey framework, we develop a novel theoretical treatment of the dimension truncation error for infinite-dimensional integration.

We prove convergence rates of quasi-Monte Carlo quadrature for the finite-dimensional Gaussian-weighted integration of integrands having local Gevrey regularity.

Together with the finite element discretization error, the errors arising from dimension truncation and quasi-Monte Carlo quadrature yield a complete error analysis and convergence rates for the fully discrete approximation of a bounded linear functional of the weak solution.

Numerical experiments confirming the theoretical convergence rates are presented.