3 research outputs found
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
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
Global existence of weak solutions to viscoelastic phase separation: Part II Degenerate Case
The aim of this paper is to prove global in time existence of weak solutions
for a viscoelastic phase separation. We consider the case with singular
potentials and degenerate mobilities. Our model couples the diffusive interface
model with the Peterlin-Navier-Stokes equations for viscoelastic fluids. To
obtain the global in time existence of weak solutions we consider appropriate
approximations by solutions of the viscoelastic phase separation with a regular
potential and build on the corresponding energy and entropy estimates.Comment: 29 pages, 28 figure