[Paper Review] Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well-posedness, Singular Limit, and Error Estimates

Jiajun Tong|arXiv (Cornell University)|Apr 21, 2019

Lattice Boltzmann Simulation Studies39 references4 citations

TL;DR

This paper introduces a regularized version of the 2D Stokes immersed boundary problem using a mollified delta function to smooth singular forcing and flow coupling. It establishes global well-posedness, proves convergence of the regularized string dynamics to the unregularized problem as the regularization parameter vanishes, and derives error estimates showing improved accuracy when the mollifier is suitably chosen.

ABSTRACT

Inspired by the numerical immersed boundary method, we introduce regularized Stokes immersed boundary problems in two dimensions to describe regularized motion of a 1-D closed elastic string in a 2-D Stokes flow, in which a regularized $δ$-function is used to mollify the flow field and singular forcing. We establish global well-posedness of the regularized problems, and prove that as the regularization parameter diminishes, string dynamics in the regularized problems converge to that in the Stokes immersed boundary problem with no regularization. Viewing the un-regularized problem as a benchmark, we derive error estimates under various norms for the string dynamics. Our rigorous analysis shows that the regularized problems achieve improved accuracy if the regularized $δ$-function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation.

Motivation & Objective

• To rigorously analyze a regularized continuous formulation of the 2D Stokes immersed boundary problem with mollified delta functions.
• To establish global well-posedness of the regularized system for 1D elastic strings in 2D Stokes flow.
• To justify the convergence of regularized string dynamics to the unregularized problem as the regularization parameter tends to zero.
• To derive quantitative error estimates between regularized and unregularized solutions, depending on the regularization parameter and mollifier choice.
• To suggest potential improvements for the numerical immersed boundary method based on the analytical error bounds.

Proposed method

• Formulates a regularized 2D Stokes IB problem using a smooth, compactly supported mollifier in place of the Dirac delta function to regularize the singular forcing and velocity field.
• Derives a contour dynamic equation for the string motion by eliminating the fluid variables, expressing the string velocity as a singular integral involving the fundamental solution of the 2D Stokes equation.
• Employs Hölder and Sobolev-type estimates in periodic Sobolev spaces to control differences in string configurations and their derivatives.
• Uses the method of characteristics and integral estimates to bound the difference between regularized and unregularized string dynamics over time.
• Applies sharp pointwise and $L^2$-type estimates on the difference of the velocity kernels to control the error in the string evolution.
• Derives error bounds in various norms (e.g., $L^2$, $ ilde{H}^1$) that depend explicitly on the regularization parameter $ ho$, the mollifier's properties, and the solution's regularity.

Experimental results

Research questions

• RQ1 Does the regularized 2D Stokes immersed boundary problem admit global solutions for initial string configurations in $H^{5/2}$?
• RQ2 Do the solutions of the regularized problem converge to those of the unregularized problem as the regularization parameter $ ho \to 0$?
• RQ3 How do the error estimates between regularized and unregularized string dynamics depend on the choice of the mollifier?
• RQ4 Can the regularized formulation achieve better accuracy than the standard numerical immersed boundary method?
• RQ5 What is the optimal regularized delta function that minimizes the error in the string dynamics?

Key findings

• The regularized problem is globally well-posed for initial string configurations in $H^{5/2}(bT)$, ensuring existence and uniqueness of solutions for all time.
• As the regularization parameter $\rho \to 0$, the string dynamics in the regularized problem converge to those in the unregularized problem in the $L^2$ and $\tilde{H}^1$ norms.
• Error estimates are derived showing that the difference in string dynamics is bounded by $C\lambda^{-2}\|Y_{1}^{\prime\prime}\|_{\dot{H}^\theta}^2$ and higher-order terms, where $\lambda$ controls the mollifier width.
• The error bound depends critically on the choice of the mollifier: better-constructed mollifiers (e.g., with higher-order smoothness) yield improved convergence rates.
• The analysis shows that the regularized formulation can achieve improved accuracy over the standard numerical method, suggesting a path for methodological enhancement.
• A refined error estimate in $\dot{H}^1$-type norm is derived as $\|g_{Y_1} - g_{Y_2}\|_{\dot{H}^1} \leq C\lambda^{-3}(\|Y_1\|_{\dot{H}^{2+\theta}} + \|Y_2\|_{\dot{H}^{2+\theta}})^3\|\delta Y\|_{\dot{H}^{3/2}}$, which is sharper than weaker alternatives.