39 research outputs found
Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations
Error estimates with optimal convergence orders are proved for a stabilized
Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a
combination of Lagrange-Galerkin method and Brezzi-Pitkaranta's stabilization
method. It maintains the advantages of both methods; (i) It is robust for
convection-dominated problems and the system of linear equations to be solved
is symmetric. (ii) Since the P1 finite element is employed for both velocity
and pressure, the number of degrees of freedom is much smaller than that of
other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is
efficient especially for three-dimensional problems. The theoretical
convergence orders are recognized numerically by two- and three-dimensional
computations
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