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

Title: 拉普拉斯算子和斯托克斯算子在狄利克雷边界条件下的体积惩罚近似：一种谱观点

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. Show Chinese abstract

Abstract: 我们报告了对拉普拉斯算子和斯托克斯算子谱性质的详细研究结果，这些算子通过一个体积惩罚项进行了修改，该惩罚项旨在当惩罚参数$\eta$趋近于零时逼近狄利克雷条件。 特征值和特征函数根据$\eta$的函数关系确定，无论是连续情况还是应用傅里叶或有限差分离散化方案之后的情况。 对于固定的$\eta$，我们发现只有对应于特征值$\lambda \lesssim \eta^{-1}$的谱部分趋近于狄利克雷边界条件，而其余谱由不受控制的虚假壁面模式组成。 受控特征函数的惩罚误差作为$\eta$和$\lambda$的函数进行估计。 令人惊讶的是，在斯托克斯情况下，我们证明特征函数近似满足，精度为$O(\eta)$，无滑移边界条件，滑移长度等于$\sqrt{\eta}$。 此外，对于给定的离散化，我们证明存在一个$\eta$的值，对应于惩罚和离散化误差之间的平衡，低于该值时不再获得精度的提升。 这些结果揭示了在求解纳维-斯托克斯方程时体积惩罚方案的行为，概述了该方法的局限性，并给出了如何在实际情况下选择惩罚参数的指示。