Masahisa Tabata

Mária Lukáčová-Medvid'ová, Hana Mizerová, Hirofumi Notsu et al.

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-Pitkäranta'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.

Mária Lukáčová-Medvid'ová, Hana Mizerová, Hirofumi Notsu et al.

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-Pitkäranta'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.

Hirofumi Notsu, Masahisa Tabata

Optimal error estimates of stable and stabilized Lagrange-Galerkin (LG) schemes for natural convection problems are proved under a mild condition on time increment and mesh size. The schemes maintain the common advantages of the LG method, i.e., robustness for convection-dominated problems and symmetry of the coefficient matrix of the system of linear equations. We simply consider typical two sets of finite elements for the velocity, pressure and temperature, P2/P1/P2 and P1/P1/P1, which are employed by the stable and stabilized LG schemes, respectively. The stabilized LG scheme has an additional advantage, a small number of degrees of freedom especially for three-dimensional problems. The proof of the optimal error estimates is done by extending the arguments of the proofs of error estimates of stable and stabilized LG schemes for the Navier-Stokes equations in previous literature.

Masahisa Tabata, Shinya Uchiumi

We show numerical results of triangular cavity flow problems solved by a Lagrange-Galerkin scheme free from numerical quadrature. The scheme has recently developed by us, where a locally linearized velocity and the backward Euler approximation are used in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as it is, the theoretical stability and convergence results are assured, while the conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature errors. The scheme is employed to solve cavity flow problems in triangular domains, where we observe the bifurcation of stationary solutions and the patterns of streamlines.

Hirofumi Notsu, Masahisa Tabata

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.

Masahisa Tabata, Shinya Uchiumi

We present a Lagrange--Galerkin scheme free from numerical quadrature for the Navier--Stokes equations. Our idea is to use a locally linearized velocity and the backward Euler method in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as it is, the theoretical stability and convergence results are assured. While the conventional Lagrange--Galerkin schemes may encounter the instability caused by numerical quadrature errors, the present scheme is genuinely stable. For the $\pk 2/\pk 1$- and $\mini$-finite elements optimal error estimates are proved in $\ell^\infty(H^1)\times \ell^2(L^2)$ norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show the genuine stability of the scheme.

Masahisa Tabata, Shinya Uchiumi

We present a Lagrange-Galerkin scheme free from numerical quadrature for convection-diffusion problems. Since the scheme can be implemented exactly as it is, theoretical stability result is assured. While conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature error, the present scheme is genuinely stable. We prove the stability and convergence of the best possible order. Numerical results reflect these estimates.