On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients

Mathematical statements of the optimal control problems for quasilinear elliptic equations with the controls in the variable coefficients of the equation of state are considered. Both local and integral constraints on the controls are considered. The objective functionals correspond to the optimization with respect to a certain number of quality indexes. Finite difference approximations of optimization problems are constructed, and estimates of the approximation error with respect to the state and to the objective functional are established. The weak convergence in control is proved. The approximations are regularized after Tikhonov. Interesting examples of some applied optimization problems that naturally lead to the nonlinear optimal control problems examined in this paper are considered.     

A new wavelet preconditioner for finite difference operators

This paper is devoted to the construction of a new multilevel preconditioner for operators discretized using finite differences. It uses the basic ingredients of a multiscale construction of the inverse of a variable coefficient elliptic differential operator derived by Tchamitchian [19]. It can be implemented fast and can therefore be easily incorporated in finite difference solvers for elliptic PDEs. Theoretical results, as well as numerical tests and implementation technical details are presented. This work has been partially supported by TMR Research Network Contract FMRX-CT98-0184.AMS subject classification 00A69, 65T60, 65Y99, 15A12     

A MODIFIED UPWIND DIFFERENCE SCHEME FOR NONLINEAR PARABOLIC EQUATIONS IN A VARIABLE DOMAIN

The nonlinear parabolic equations in a variable domain are considered. A modified up-wind difference scheme is given in the variable domain. Stability in l norm and error estimate in norm are obtained.     

A Locally Conservative Eulerian-Lagrangian Finite Difference Method for a Parabolic Equation

The object of this paper is to define a finite difference analogue of a locally conservative Eulerian—Lagrangian method based on mixed finite elements and to prove its convergence. The method is appropriate for convection-dominated diffusive processes; here, it will be considered in the case of a semilinear parabolic equation in a single space variable.This revised version was published online in October 2005 with corrections to the Cover Date.     

空间四阶的时间亚扩散方程的有限差分方法

提出了两个求解空间四阶的时间亚扩散方程的数值方法,其误差阶分别为O（τ＋h2）和O（τ2＋h2）.通过Fourier方法,发现两个差分格式均为无条件稳定的.最后,通过数值例子,验证了两个算法的有效性.     

On the stability of explicit finite difference methods for advection–diffusion equations

In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.