Cached Gaussian elimination for simulating Stokes flow on domains with repetitive geometry - Ding, Ounan; Schroeder, Craig (Journal of computational physics, 2020)
•Specialized efficient solver for fluids for microfluidics applications.•Solves Stokes equations with one million degrees of freedom in one second.•Special solver for sparse symmetric indefinite systems of linear equations.•Formally third order accurate in velocity and second order accurate in pressure. 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∞ 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.