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Exponential Convergence of hp-FE Algorithms for incompressible flow in polygons with mixed boundary conditions

by Y. He and P. Houston and C. Schwab

(Report number 2026-41)

Abstract
We prove exponential error vs. work bounds for numerical solvers of the stationary Navier-Stokes equations (NSE) in bounded, polygonal domains \(\Omega \subset \mathbb{R}^2\), subject to analytic (in \(\overline{\Omega}\)) source terms, and a combination of mixed Dirichlet (``no-slip''), Neumann (``open'') and slip boundary conditions. The solution algorithms are fully discrete, and are based on conforming (velocity-pressure) mixed \(hp\)-finite element (FE) Galerkin discretizations of the NSE. The corresponding velocity-pressure subspace pairs are assumed divergence-stable on sequences of regular triangulations with geometric refinement towards corners, and with inf-sup stability constants that may degenerate algebraically with respect to the polynomial degree \(k\). A finitely terminated fixed-point iteration for the approximation of the resulting nonlinear system of algebraic equations is introduced and error reduction per iteration is established independent of the \(hp\)-FE discretization parameters. The mathematical arguments are based on recent results \cite{he2024analytic} on corner-weighted, analytic regularity of velocity-pressure pairs solving the stationary NSE in \(\Omega\), and on exponential consistency of \(hp\)-FE spaces on such solution pairs. Numerical experiments confirm the theory.

Keywords: Stationary Navier–Stokes equations; mixed boundary conditions; hp-finite element methods; exponential convergence

@Techreport{HHS26_1196,
  author = {Y. He and P. Houston and C. Schwab},
  title = {Exponential Convergence of hp-FE Algorithms for incompressible flow in polygons with mixed boundary conditions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-41.pdf },
  year = {2026}
}

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