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

TL;DR

This paper analyzes the spectral properties of Laplace and Stokes operators with volume penalization to approximate Dirichlet boundary conditions, revealing the spectrum's structure, error estimates, and boundary condition approximations.

Contribution

It provides a detailed spectral analysis of penalized operators, showing how eigenvalues and eigenfunctions behave as the penalization parameter varies, and clarifies the boundary conditions approximated.

Findings

Only eigenvalues up to ~η^{-1} approximate Dirichlet conditions.

Eigenfunctions satisfy Navier slip conditions with slip length √η.

There is an optimal η balancing penalization and discretization errors.

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 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.