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H(curl)-based approximation of the Stokes problem with slip boundary conditions

by W.M. Boon and R. Hiptmair and W. Tonnon and E. Zampa

(Report number 2024-24)

Abstract
The equations governing incompressible Stokes flow are reformulated such that the velocity is sought in the space H(curl). This relaxed regularity assumption leads to conforming finite element methods using spaces common to discretizations of Maxwell's equations. A drawback of this approach, however, is that it is not immediately clear how to enforce Navier-slip boundary conditions. By recognizing the slip condition as a Robin boundary condition, we show that the continuous system is well-posed, propose finite element methods, and analyze the discrete system by deriving stability and a priori error estimates. Numerical experiments in 2D confirm the derived, optimal convergence rates.

Keywords: Stokes, FEM, H(curl), Boundary, Objectivity

@Techreport{BHTZ24_1106,
  author = {W.M. Boon and R. Hiptmair and W. Tonnon and E. Zampa},
  title = {H(curl)-based approximation of the Stokes problem with slip boundary conditions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-24.pdf },
  year = {2024}
}

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