Exact pressure elimination for the Crouzeix-Raviart scheme applied to the Stokes and Navier-Stokes problems

TL;DR

This paper introduces an exact pressure elimination method for the Crouzeix-Raviart scheme applied to Stokes and Navier-Stokes problems, reducing the linear system size and improving computational efficiency.

Contribution

It presents a novel algebraic transformation that simplifies the coupled systems, leading to symmetric positive definite or invertible systems with the same stencil as velocity matrices.

Findings

Reduced system size with auxiliary variables

Symmetric positive definite or invertible systems

Enhanced solver performance

Abstract

We show that, using the Crouzeix-Raviart scheme, a cheap algebraic transformation, applied to the coupled velocity-pressure linear systems issued from the transient or steady Stokes or Navier-Stokes problems, leads to a linear system only involving as many auxiliary variables as the velocity components. This linear system, which is symmetric positive definite in the case of the transient Stokes problem and symmetric invertible in the case of the steady Stokes problem, with the same stencil as that of the velocity matrix, provides the exact solution of the initial coupled linear system. Numerical results show the increase of performance when applying direct or iterative solvers to the resolution of these linear systems.