# A Quasi-Optimal Crouzeix-Raviart Discretization of the Stokes Equations

## TL;DR

This paper introduces a modified Crouzeix-Raviart discretization for the Stokes equations that achieves quasi-optimal error bounds, independent of pressure errors, with computational costs comparable to standard methods.

## Contribution

It proposes a quasi-optimal discretization of the Stokes equations that maintains error bounds independent of pressure and is computationally efficient.

## Key findings

- Velocity error is proportional to the best approximation error.
- Velocity and pressure can be computed separately in 2D domains.
- Numerical experiments confirm theoretical error bounds.

## Abstract

We present a modification of the Crouzeix-Raviart discretization of the Stokes equations in arbitrary dimension which is quasi-optimal, in the sense that the error of the discrete velocity field in a broken $H^1$-norm is proportional to the error of the best approximation to the analytical velocity field. In particular, the velocity error is independent of the pressure error and the discrete velocity field is element-wise solenoidal. Moreover, the sum of the velocity error times the viscosity plus the pressure $L^2$-error is proportional to the sum of the respective best errors. All proportionality constants are bounded in terms of shape regularity and do not depend on the viscosity. For simply connected two-dimensional domains, the velocity and pressure can be computed separately. The modification only affects the right-hand side aka load vector. The cost for building the modified load vector is proportional to the cost for building the standard load vector. Some numerical experiments illustrate our theoretical results.