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Article Dans Une Revue Numerische Mathematik Année : 2014

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

Romain Nguyen van Yen
  • Fonction : Auteur
Dmitry Kolomenskiy

Résumé

We report the results of a detailed study of the spectral properties of Laplace and Stokes operators, modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, $\eta$, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of $\eta$, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed $\eta$, we find that only the part of the spectrum corresponding to eigenvalues $\lambda \lesssim \eta^{-1}$ approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of $\eta$ and $\lambda$. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision $O(\eta)$, Navier slip boundary conditions with slip length equal to $\sqrt{\eta}$. Moreover, for a given discretization, we show that there exists a value of $\eta$, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier-Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

Dates et versions

hal-00830110 , version 1 (04-06-2013)

Identifiants

Citer

Romain Nguyen van Yen, Dmitry Kolomenskiy, Kai Schneider. Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint. Numerische Mathematik, 2014, 128 (2), pp.301-338. ⟨10.1007/s00211-014-0610-8⟩. ⟨hal-00830110⟩
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