Microfluidic “lab on a chip” devices are small devices that operate on small length scales on small volumes of fluid; these devices find uses in a variety of applications. Designs for microfluidic chips are generally composed of standardized and often repeated components connected by long, thin, straight fluid channels. We propose a novel meshing algorithm for use in simulating the linear incompressible stationary Stokes equations on geometry with these features, which produces sparse symmetric positive indefinite systems with many repeated matrix blocks. We use a discretization that is formally third order accurate for velocity and second order accurate for pressure in the $L^{\infty }$ norm. We also propose a novel linear system solver based on cyclic reduction, reordered sparse Gaussian elimination, and operation caching that is designed to efficiently solve systems with repeated matrix blocks. We demonstrate that the resulting fluid solver is significantly faster than existing methods up to resolutions of a few million degrees of freedom for microfluidic problems.

微流体“芯片实验室”设备是小型设备，可在小规模规模上以小体积流体工作；这些设备可用于各种应用中。微流控芯片的设计通常由标准化且经常重复的组件组成，这些组件通过长，细，直的流体通道连接。我们提出了一种新颖的网格划分算法，用于在具有这些特征的几何体上模拟线性不可压缩平稳斯托克斯方程，从而生成具有许多重复矩阵块的稀疏对称正不定系统。我们使用的离散化形式对于速度在速度上是三阶精确的，而在压力上是二阶精确的${\mathrm{大号}}^{\infty }$规范。我们还提出了一种新颖的线性系统求解器，该算法基于循环归约，重新排序的稀疏高斯消除和运算缓存，旨在有效地求解具有重复矩阵块的系统。我们证明，所产生的流体求解器比现有方法快得多，可解决微流体问题的几百万个自由度。