ROM for Viscous, Incompressible Flow in Polygons -- exponential $n$-width bounds and convergence rate

TL;DR

This paper proves exponential convergence rates for reduced order models of stationary incompressible Navier-Stokes equations in polygonal domains, supported by theoretical bounds and numerical experiments.

Contribution

It establishes exponential bounds for Kolmogorov n-widths and convergence rates of POD Galerkin methods for mixed boundary value problems in polygonal domains.

Findings

Exponential convergence of ROM approximations demonstrated.

Kolmogorov n-widths decay exponentially for analytic data.

Numerical experiments confirm theoretical exponential rates.

Abstract

We demonstrate exponential convergence of Reduced Order Model (ROM) approximations for mixed boundary value problems of the stationary, incompressible Navier-Stokes equations in plane, polygonal domains $\Omega$. Admissible boundary conditions comprise mixed BCs, no-slip, slip and open boundary conditions, subject to corner-weighted analytic boundary data and volume forcing. The small data hypothesis is assumed to ensure existence of a unique weak solution in the sense of Leray-Hopf. Recent results on corner-weighted, analytic regularity of velocity and pressure fields in $\Omega$, imply exponential convergence rates of so-called mixed $hp$-Finite Element Methods in $H^1(\Omega)^2\times L^2(\Omega)$ on sequences of geometric partitions of $\Omega$, with corner-refinement. Based on these exponential convergence rate bounds, we infer exponential bounds for the Kolmogorov $n$-widths of solution sets for analytic forcing and boundary data. This implies corresponding exponential convergence rates of POD Galerkin methods that are based on truth solutions which are obtained offline from low-order, divergence stable mixed Finite Element discretizations. Numerical experiments confirm the exponential rates and the theoretical results.