A Quasi-Optimal Crouzeix-Raviart Discretization of the Stokes Equations
This work provides a theoretical improvement for finite element discretizations of Stokes equations, ensuring optimal convergence rates without coupling between velocity and pressure errors.
The paper presents a modification of the Crouzeix-Raviart discretization for Stokes equations that achieves quasi-optimal error bounds, where the velocity error is proportional to the best approximation error and independent of pressure error, with constants independent of viscosity. Numerical experiments confirm the theory.
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.