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We report the results of a detailed study of the spectral properties ofLaplace and Stokes operators, modified with a volume penalization term designedto approximate Dirichlet conditions in the limit when a penalization parameter,$\eta$, tends to zero. The eigenvalues and eigenfunctions are determined eitheranalytically or numerically as functions of $\eta$, both in the continuous caseand after applying Fourier or finite difference discretization schemes. Forfixed $\eta$, we find that only the part of the spectrum corresponding toeigenvalues $\lambda \lesssim \eta^{-1}$ approaches Dirichlet boundaryconditions, while the remainder of the spectrum is made of uncontrolled,spurious wall modes. The penalization error for the controlled eigenfunctionsis estimated as a function of $\eta$ and $\lambda$. Surprisingly, in the Stokescase, 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 existsa value of $\eta$, corresponding to a balance between penalization anddiscretization errors, below which no further gain in precision is achieved.These results shed light on the behavior of volume penalization schemes whensolving the Navier-Stokes equations, outline the limitations of the method, andgive indications on how to choose the penalization parameter in practicalcases.

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