Masahisa Tabata

A second order characteristic finite element scheme for convection-diffusion problems

Summary. A new characteristic finite element scheme is presented for It is of second order accuracy in time increment, symmetric, and unconditionally stable. Optimal error estimates are proved in the framework of n

A stabilized finite element method for the Rayleigh–Bénard equations with infinite Prandtl number in a spherical shell

L^2

Robustness of a Characteristic Finite Element Scheme of Second Order in Time Increment

-theory. Numerical results are presented for two examples, which show the advantage of the scheme.

Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients

Abstract A finite element scheme is developed and analyzed for a thermal convection problem of Boussinesq fluid with infinite Prandtl number in a spherical shell. This problem is a mathematical model of the Earths mantle movement and has been a topic of interest for geophysicists. It is described by the Rayleigh–Benard equations with infinite Prandtl number, that is, a system of the Stokes equations and the convection–diffusion equation coupled with the buoyancy and the convection terms. A stabilized finite element scheme with P1/P1/P1 element is presented, and an error estimate is established. The obtained theoretical convergence order is also recognized by a numerical result. Another numerical result is shown as an example of the Earths mantle movement simulation.

A Precise Computation of Drag Coefficients of a Sphere

In using the characteristic finite element method we have to pay much attention to the errors caused by numerical integration. Compared to the first order scheme, the second order scheme is robust with respect to the numerical integration error, which has been recognized by numerical results. We have also given a rough explanation for the robustness. Under a mild condition on the numerical integration we can show that the second order scheme is stable with the numerical integration. A detail discussion and a complete proof of the stability will be presented in the forthcoming paper.

Error Estimates for Finite Element Approximations of Drag and Lift in Nonstationary Navier-Stokes Flows

Summary.General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients. These variable coefficients turn the diffusion and the buoyancy terms to be nonlinear, which increases the nonlinearity of the problems. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Error estimates are also provided for schemes modified by approximate coefficients, which are used conveniently in practical computations.

Uniform solvability of finite element solutions in approximate domains

We present a computational method for drag coefficients of axisymmetric bodies. It is a kind of consistent flux method but the introduction of a proper test function enables us to establish an error estimate under some assumption. Applying the method, we obtain drag coefficients of a sphere for Reynolds numbers between 10 and 200, which are found between numerical upper and lower bounds.

Finite element approximation to infinite Prandtl number Boussinesq equations with temperature-dependent coefficients: thermal convection problems in a spherical shell

Error estimates are obtained for finite element approximations of the drag and the lift of a body immersed in nonstationary Navier-Stokes flows. By virtue of a consistent flux technique, the error estimates are reduced to those of the velocity as well as its first order derivatives and the pressure. Semi-implicit backward Euler method is used for the time integration and no stability condition is required. The error estimate in a square summation norm is optimal in the sense that it has the same order as the fundamental error estimate of the velocity. The error estimate in the supremum norm is not optimal in general but it is so for some finite elements.

Mathematical Modeling and Numerical Simulation of Earth’s Mantle Convection

Uniform solvability of finite element solutions in approximate domains is studied. An operator that extends functions in finite element spaces to the exact domain is constructed and some estimates in the boundary skin are presented. The extension operator is successfully used to prove uniform solvability in approximate domains for problems subject to slip boundary conditions and so on.

A Finite Element Analysis of a Linearized Problem of the Navier--Stokes Equations with Surface Tension

A stabilized finite element scheme for infinite Prandtl number Boussinesq equations with temperature-dependent coefficients is analyzed. The domain is a spherical shell and the P1-element is employed for every unknown function. The finite element solution is proved to converge to the exact one in the first order of the time increment and the mesh size. The scheme is applied to Earths mantle convection problems with viscosities strongly dependent on the temperature and some numerical results are shown.