The divergence constraint of the incompressible Navier--Stokes equations is revisited in mixed finite element framework. While many stable and convergent elements have been developed throughout past four decades, most classical methods relax only enforce condition discretely. As a result, these introduce pressure-dependent consistency error which can potentially pollute computed velocity. These are not robust sense that contribution from right-hand side, influences pressure continuous...

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier–Stokes equations using continuous velocity fields. With a particular mesh construction, Scott–Vogelius pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise conservation. We present herein first numerical tests this time dependent equations. prove that limit grad-div stabilized Taylor–Hood solutions...

Standard mixed finite element methods for the incompressible Navier–Stokes equations that relax divergence constraint are not robust against large irrotational forces in momentum balance and velocity error depends on continuous pressure. This robustness issue can be completely cured by using divergence-free elements which deliver pressure-independent estimates. However, construction of H1-conforming, is rather difficult. Instead, we present a novel approach arbitrary order errors. The does...

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

An improved understanding of the divergence-free constraint for incompressible Navier–Stokes equations leads to observation that a semi-norm and corresponding equivalence classes forces are fundamental their nonlinear dynamics. The recent concept pressure-robustness allows distinguish between space discretisations discretise these appropriately or not. This contribution compares accuracy pressure-robust non-pressure-robust transient high Reynolds number flows, starting from in generalised...

During the development of a convergence theory for Nicolaides’ extension classical MAC scheme incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that new kind finite volume discretizations biharmonic problem would be very useful tool in analysis generalized scheme. Therefore, we present and analyze schemes approximation with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove piecewise constant approximate solution...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 8 July 2020Accepted: 14 June 2021Published online: 28 October 2021Keywordsincompressible Navier--Stokes equations, divergence-free mixed finite element methods, pressure-robustness, convection stabilization, Galerkin least squares, vorticity equationAMS Subject Headings65N30, 65N12, 76D07Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society...

Abstract In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax divergence constraint and are discretely inf-sup stable. Though relaxation of was claimed to be harmless since beginning 1970s, Poisson locking is just replaced by another more subtle kind phenomenon, which sometimes called poor mass conservation led in computational practice exclusion with low-order pressure approximations like Bernardi–Raugel or Crouzeix–Raviart finite...

Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads a pressure-dependent contribution in velocity error, which is proportional inverse of viscosity, thus giving rise (different) locking phenomenon. recently proposed modification right-hand side alone discretization really locking-free; i.e., its error converges with optimal order and independent...