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

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