General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperaturedependent 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.
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.
Truncation errors are considered for approximate differential operators with a class of particle methods. Introducing sufficient conditions for the weight function and a regularity of the family of discrete parameters leads to truncation error estimates of approximate gradient and Laplace operators with a particle method based on the Voronoi decomposition. Moreover, some numerical results agree well with theoretical ones.
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