Christian Merdon

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex,

Computational survey on a posteriori error estimators for nonconforming finite element methods for the Poisson problem

H(div)

On velocity errors due to irrotational forces in the Navier-Stokes momentum balance

-conforming finite ...

Effective postprocessing for equilibration a posteriori error estimators

This paper compares different a posteriori error estimators for nonconforming first-order Crouzeix-Raviart finite element methods for simple second-order partial differential equations. All suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants. Novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples.

Optimal and Pressure-Independent L^2 Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions

This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier-Stokes equations. Four simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. In fact, the velocity error can be arbitrarily large in such situations. Only pressure-robust mixed finite element methods, whose velocity error is pressure-independent, can avoid this influence. Indeed, the presented examples show that the pressure-dependent component in velocity error estimates for classical mixed finite element methods is sharp. In consequence, classical mixed finite element methods are not able to simulate some classes of real-world flows, even in cases where dominant convection and turbulence do not play a role.

Divergence-free Reconstruction Operators for Pressure-Robust Stokes Discretizations with Continuous Pressure Finite Elements

Guaranteed error control via fully discrete a posteriori error estimators is possible with typical overestimation between 1.25 and 2 in simple computer benchmarks. The equilibration techniques due to Braess and that due to Luce–Wohlmuth are efficient tools with an accuracy limited by the hyper-circle threshold. This motivates postprocessing strategies and the analysis of successive improvements of guaranteed upper error bounds with a few pcg iterations result in reduced overestimation factors between 1 and 1.25. Numerical simulations for three classes of applications illustrate the efficiency for the Poisson model problem with and without jumping coefficients or a simple obstacle problem.

Computational Survey on A Posteriori Error Estimators for the Crouzeix–Raviart Nonconforming Finite Element Method for the Stokes Problem

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a-priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete \(H^1\) velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure-independent \(L^2\) velocity error. Numerical examples confirm the analytical results.

On Really Locking-Free Mixed Finite Element Methods for the Transient Incompressible Stokes Equations

Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right-hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right-hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local

Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations

H(\mathrm{div})

An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data

-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations conf...