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Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²

by D. Schötzau and C. Marcati and Ch. Schwab

(Report number 2020-15)

Abstract
In a polygon \(\Omega\subset \mathbb{R}^2\), we consider mixed \(hp\)-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and \(hp\) spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in \(\Omega\), we prove exponential rates of convergence of the mixed \(hp\)-discontinuous Galerkin finite element method (\(hp\)-DGFEM), with respect to the number of degrees of freedom, for small data which is piecewise analytic.

Keywords: Mixed $hp$-FEM, discontinuous Galerkin methods, exponential convergence, Navier-Stokes equations

@Techreport{SMS20_888,
  author = {D. Sch\"otzau and C. Marcati and Ch. Schwab},
  title = {Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-15.pdf },
  year = {2020}
}

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