Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint
This work provides theoretical insights into the limitations and optimal parameter selection for volume penalization methods in fluid dynamics simulations.
The paper analyzes the spectral properties of Laplace and Stokes operators with volume penalization for Dirichlet boundary conditions, finding that only eigenvalues below a threshold (λ ≲ η⁻¹) approximate Dirichlet conditions, while higher modes are spurious. In the Stokes case, eigenfunctions satisfy Navier slip conditions with slip length √η, and there is an optimal penalization parameter balancing discretization and penalization errors.
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, $η$, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of $η$, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed $η$, we find that only the part of the spectrum corresponding to eigenvalues $λ\lesssim η^{-1}$ 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 $\sqrtη$. 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.