Daisuke Tagami

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

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.

A Numerical Method for 3-D Eddy Current Problems

An iterative procedure using finite element method without the Lagrange multiplier is proposed for three-dimensional eddy current problems, which is based on an iterative procedure derived from a perturbation problem of the magnetostatic problem. To consider the continuity of an excitation current density, a correction method is also proposed. Numerical results show that the BiConjugate Gradient (BiCG) method is applicable to the complex symmetric linear systems arising in the iterative procedure, and that approximate physical quantities are suitable.

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

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.

Large Scale Finite Element Analysis with a Balancing Domain Decomposition Method

This paper describes the parallel finite element analysis of large scale problems based on the Domain Decomposition Method with preconditioner using Balancing Domain Decomposition using a parallel computer. In order to solve the issue of memory shortage and computational time, the developed system employs a dynamic load balancing and hierarchical distributed data management technique. The present system is successfully applied to static elastic stress analyses with effective performances.

Finite element computation of magnetic field problems with the displacement current

The effectiveness of the A-φ method in eddy current problems is widely known. On the other hand, the demand of high frequency computations increases. In this paper, A-φ method is applied to finite element approximations for three-dimensional high frequency problems with the displacement current. Numerical results show that A-φ method is more appliable in wide range of frequencies.

3‐D eddy current computation for a transformer tank

A large scale computation of three‐dimensional eddy current problems is considered; their numbers of degrees of freedom are near one million. A parallel computing using the Hierarchical Domain Decomposition Method (HDDM) is introduced to compute large scale problems. A transformer model is considered as a numerical example, and HDDM is applicable to the model.

A Stabilization Technique for Steady Flow Problems

Finite element methods with stabilization techniques for the steady Navier–Stokes equations are studied. To solve the steady Navier–Stokes equations, the Newton method is used. To compute the problem at each step of the nonlinear iteration, a stabilization technique is introduced. The mixed interpolation, which satisfies the inf-sup condition, with stabilized terms is also considered to investigate its computational efficiency. Numerical results show that stabilized terms improve convergences of the Newton method especially in the case of high Reynolds numbers as well as those of the linear solver at each step of the nonlinear iteration.

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

A finite element analysis is performed for a stationary linearized problem of the Navier--Stokes equations with surface tension. Since the surface tension brings about a second-order derivative of the velocity in the boundary condition, the velocity space is equipped with a stronger topology than in the conventional case. Under the strong topology, conditions of the uniform solvability and the approximation are verified on some pairs of finite element spaces for the velocity and the pressure. Thus an optimal error estimate is derived. Some numerical results are shown, which agree well with theoretical ones.

Polygonal Hele?Shaw problem with surface tension

We study a polygonal analogue of the Hele–Shaw moving boundary problem with surface tension based on a framework of polygonal motion proposed by Beneš et al. [5]. A key idea is to introduce a polygonal Dirichlet-to-Neumann map. We study variational properties of the polygonal Dirichletto-Neumann map and show that our polygonal Hele–Shaw problem is a polygonal analogue of the original problem. Local solvability of a polygonal Hele–Shaw problem is also proved by means of the variational structure.

A Finite Element Method for 3-D Eddy Current Problems Using an Iterative Approach

Abstract An iterative method is proposed for a finite element approximation of three-dimensional eddy current problems. The method is based on an iterative method derived from a perturbation problem of magnetostatic problems. The TEAM model and a transformer model are considered as numerical examples. In both examples, BiConjugate Gradient (BiCG) method is applicable for the complex symmetric linear system arising in each step of the iterative procedure for a rather wide range of the perturbation parameter, and the present results seem to be suitable.