紧密周期柱体域中Stokes流的均匀Inf-Sup范数等价性与鲁棒算子预处理
Uniform Inf-Sup Norm Equivalence and Robust Operator Preconditioning for Stokes Flow in tight domains with Periodic Pillars
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
- Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳大数据研究院工业与应用数学国际中心）
- SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学力学重点实验室）
- School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
- Shenzhen Loop Area Institute（深圳河套学院）
- Applied Mathematics and Computational Sciences, CEMSE Division, King Abdullah University of Science and Technology（阿卜杜拉国王科技大学CEMSE学院应用数学与计算科学系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
针对密集周期柱体域中的Stokes流，证明inf-sup范数与Brinkman型范数均匀等价，并基于此构建代数多重网格块预处理器，使迭代次数独立于网格和柱密度。
中文摘要 AI 辅助
许多微流控和多孔介质计算归结为同一核心任务：在由密集周期柱阵列穿孔的域上求解Stokes鞍点系统，例如确定性横向位移（DLD）粒子分选器。将装置缩放至单位尺寸后，几何由单个无量纲参数$m$控制——即横跨装置的柱数，等于周期的倒数。在实际装置中，$m$可达数百或数千，随着其增长，Stokes inf-sup常数按$m^{-1}$衰减，压力Schur补严重病态，标准块求解器速度随柱密度成比例下降。我们通过识别散度算子在此类几何上诱导的压力范数来消除此瓶颈。对于比例孔区域的周期柱阵列，我们证明该inf-sup范数在孔尺度$\sigma_\epsilon\asymp\epsilon$下与$L^2+\sigma_\epsilon H^1$ $K$-泛函范数均匀等价，常数与周期、柱数$m$及网格尺寸$h$无关。该等价将穿孔Stokes问题与均质渗透率下的Brinkman问题联系起来，其Riesz映射简化为压力质量逆加缩放刚度逆。结合算子预处理，这产生一个由标准代数多重网格求解构建的块预处理器，其迭代次数基本独立于网格尺寸和柱密度。二维Taylor-Hood实验证实了在网格细化、柱密度、几何尺度、紧密堆积和时间步方面的预测鲁棒性。
英文摘要
Many microfluidic and porous-media computations reduce to the same core task: solving a Stokes saddle-point system on a domain perforated by a dense periodic array of pillars, as in deterministic lateral displacement (DLD) particle sorters. After rescaling the device to unit size, the geometry is controlled by a single dimensionless parameter $m$---the number of pillars across the device, equal to the inverse period. In realistic devices $m$ reaches the hundreds or thousands, and as it grows the Stokes inf-sup constant decays like $m^{-1}$, the pressure Schur complement becomes severely ill-conditioned, and standard block solvers slow down in proportion to the pillar density. We remove this bottleneck by identifying the pressure norm that the divergence operator induces on such geometries. For periodic pillar arrays in the proportional-hole regime, we prove that this inf-sup norm is uniformly equivalent to the $L^2+σ_εH^1$ $K$-functional norm at the pore scale $σ_ε\asympε$, with constants independent of the period, the pillar count $m$, and the mesh size $h$. The equivalence identifies the perforated Stokes problem with a Brinkman problem at a homogenized permeability, and its Riesz map reduces to a pressure-mass inverse plus a scaled stiffness inverse. Combined with operator preconditioning, this yields a block preconditioner built from standard algebraic-multigrid solves whose iteration count is essentially independent of both mesh size and pillar density. Two-dimensional Taylor--Hood experiments confirm the predicted robustness in mesh refinement, pillar density, geometric scale, close packing, and time step.
