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Analytic regularity and Solution Approximation for a semilinear elliptic partial differential equation in a polygon

by Y. He and Ch. Schwab

(Report number 2022-43)

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

Keywords: Semilinear elliptic equations, analytic regularity, conical singularities, weighted Sobolev spaces

@Techreport{HS22_1031,
  author = {Y. He and Ch. Schwab},
  title = {Analytic regularity and Solution Approximation for a semilinear elliptic partial differential equation in a polygon},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-43.pdf },
  year = {2022}
}

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