722 research outputs found
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
This is the second part of our error analysis of the stabilized
Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model.
Our scheme is a combination of the method of characteristics and
Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements,
which leads to an efficient computation with a small number of degrees of
freedom especially in three space dimensions. In this paper, Part II, we apply
a semi-implicit time discretization which yields the linear scheme. We
concentrate on the diffusive viscoelastic model, i.e. in the constitutive
equation for time evolution of the conformation tensor a diffusive effect is
included. Under mild stability conditions we obtain error estimates with the
optimal convergence order for the velocity, pressure and conformation tensor in
two and three space dimensions. The theoretical convergence orders are
confirmed by numerical experiments.Comment: See arXiv:1603.01339 for Part I: a nonlinear schem
An advection-robust Hybrid High-Order method for the Oseen problem
In this work, we study advection-robust Hybrid High-Order discretizations of
the Oseen equations. For a given integer , the discrete velocity
unknowns are vector-valued polynomials of total degree on mesh elements
and faces, while the pressure unknowns are discontinuous polynomials of total
degree on the mesh. From the discrete unknowns, three relevant
quantities are reconstructed inside each element: a velocity of total degree
, a discrete advective derivative, and a discrete divergence. These
reconstructions are used to formulate the discretizations of the viscous,
advective, and velocity-pressure coupling terms, respectively. Well-posedness
is ensured through appropriate high-order stabilization terms. We prove energy
error estimates that are advection-robust for the velocity, and show that each
mesh element of diameter contributes to the discretization error with
an -term in the diffusion-dominated regime, an
-term in the advection-dominated regime, and
scales with intermediate powers of in between. Numerical results complete
the exposition
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part I: A nonlinear scheme
We present a nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type
Peterlin viscoelastic model. Our scheme is a combination of the method of
characteristics and Brezzi-Pitk\"aranta's stabilization method for the
conforming linear elements, which yields an efficient computation with a small
number of degrees of freedom. We prove error estimates with the optimal
convergence order without any relation between the time increment and the mesh
size. The result is valid for both the diffusive and non-diffusive models for
the conformation tensor in two space dimensions. We introduce an additional
term that yields a suitable structural property and allows us to obtain
required energy estimate. The theoretical convergence orders are confirmed by
numerical experiments. In a forthcoming paper, Part II, a linear scheme is
proposed and the corresponding error estimates are proved in two and three
space dimensions for the diffusive model.Comment: See arXiv:1603.01074 for Part II: a linear schem
Collision in a cross-shaped domain --- A steady 2D Navier--Stokes example demonstrating the importance of mass conservation in CFD
In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed
Collision in a cross-shaped domain
In the numerical simulation of the incompressible Navier-Stokes
equations different numerical instabilities can occur. While instability in
the discrete velocity due to dominant convection and instability in the
discrete pressure due to a vanishing discrete LBB constant are well-known,
instability in the discrete velocity due to a poor mass conservation at high
Reynolds numbers sometimes seems to be underestimated. At least, when using
conforming Galerkin mixed finite element methods like the Taylor-Hood
element, the classical grad-div stabilization for enhancing discrete mass
conservation is often neglected in practical computations. Though simple
academic flow problems showing the importance of mass conservation are
well-known, these examples differ from practically relevant ones, since
specially designed force vectors are prescribed. Therefore we present a
simple steady Navier-Stokes problem in two space dimensions at Reynolds
number 1024, a colliding flow in a cross-shaped domain, where the instability
of poor mass conservation is studied in detail and where no force vector is
prescribed
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