APPROXIMATION OF THE LAPLACE AND STOKES OPERATORS WITH DIRICHLET BOUNDARY CONDITIONS THROUGH VOLUME PENALIZATION: A SPECTRAL VIEWPOINT ∗†

Abstract. 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 condi-tions in the limit when a penalization parameter, η, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of η, both in the continuous case and af-ter applying Fourier or finite difference discretization schemes. For fixed η, we find that only the part of the spectrum corresponding to eigenvalues λ. η−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 η and λ. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision O(η), Navier slip boundary conditions with slip length equal to η. Moreover, for a given discretization, we show that there exists a value of η, 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