Isogeometric Residual Minimization (iGRM) for Non-Stationary Stokes and Navier-Stokes Problems

TL;DR

This paper introduces an efficient FEM-based solver for non-stationary Stokes and Navier-Stokes equations that achieves linear computational cost using a residual minimization technique and smooth B-spline functions.

Contribution

The paper develops a novel linear-cost FEM solver employing residual minimization and B-splines for stable, efficient simulation of non-stationary fluid flow problems.

Findings

Achieves linear computational cost for complex flow problems.

Effectively handles high Reynolds number instabilities.

Validated on manufactured solutions and cavity flow problems.

Abstract

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier-Stokes equations. Our method employs a technique developed by Guermond and Minev, which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step, we solve a PDE having weak-derivatives in one direction only (which allows for the linear computational cost), at the expense of handling strong second-order derivatives of the previous time step solution, on the right-hand side of these PDEs. This motivates the use of smooth functions such as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To deal robustly with these instabilities, we propose to use a residual minimization technique. We test our method on problems having manufactured solutions, as well as on the cavity flow problem.