Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

In an open, bounded Lipschitz polygon \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ , we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in \(\Omega \) $\mathrm{\Omega }$ . The boundary conditions on each edge of \(\partial \Omega \) $\partial \mathrm{\Omega }$ are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in \(H^1(\Omega )\) $H^1(\mathrm{\Omega })$ , quantized tensor formats and certain deep neural networks.