87 research outputs found
On a reduced sparsity stabilization of grad-div type for incompressible flow problems
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar
Pressure-induced locking in mixed methods for time-dependent (Navier-)Stokes equations
We consider inf-sup stable mixed methods for the time-dependent
incompressible Stokes and Navier--Stokes equations, extending earlier work on
the steady (Navier-)Stokes Problem. A locking phenomenon is identified for
classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart
elements by a novel, elegant and simple numerical analysis and corresponding
numerical experiments, whenever the momentum balance is dominated by forces of
a gradient type. More precisely, a reduction of the convergence order for
high order methods, and even a complete stall of the convergence order
for lowest-order methods on preasymptotic meshes is predicted by the analysis
and practically observed. On the other hand, it is also shown that
(structure-preserving) pressure-robust mixed methods do not suffer from this
locking phenomenon, even if they are of lowest-order. A connection to
well-balanced schemes for (vectorial) hyperbolic conservation laws like the
shallow water or the compressible Euler equations is made.Comment: 5 page
Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form
We consider {\it local} balances of momentum and angular momentum for the
incompressible Navier-Stokes equations. First, we formulate new weak forms of
the physical balances (conservation laws) of these quantities, and prove they
are equivalent to the usual conservation law formulations. We then show that
continuous Galerkin discretizations of the Navier-Stokes equations using the
EMAC form of the nonlinearity preserve discrete analogues of the weak form
conservation laws, both in the Eulerian formulation and the Lagrangian
formulation (which are not equivalent after discretizations). Numerical tests
illustrate the new theory
Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation
In this paper, we consider the recently introduced EMAC formulation for the
incompressible Navier-Stokes (NS) equations, which is the only known NS
formulation that conserves energy, momentum and angular momentum when the
divergence constraint is only weakly enforced. Since its introduction, the EMAC
formulation has been successfully used for a wide variety of fluid dynamics
problems. We prove that discretizations using the EMAC formulation are
potentially better than those built on the commonly used skew-symmetric
formulation, by deriving a better longer time error estimate for EMAC: while
the classical results for schemes using the skew-symmetric formulation have
Gronwall constants dependent on with the Reynolds
number, it turns out that the EMAC error estimate is free from this explicit
exponential dependence on the Reynolds number. Additionally, it is demonstrated
how EMAC admits smaller lower bounds on its velocity error, since {incorrect
treatment of linear momentum, angular momentum and energy induces} lower bounds
for velocity error, and EMAC treats these quantities more accurately.
Results of numerical tests for channel flow past a cylinder and 2D
Kelvin-Helmholtz instability are also given, both of which show that the
advantages of EMAC over the skew-symmetric formulation increase as the Reynolds
number gets larger and for longer simulation times.Comment: 21 pages, 5 figure
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