On the numerical solution of the distributed parameter identification problem

A new error estimate is derived for the numerical identification of a distributed parameter a(x) in a two point boundary value problem, for the case that the finite element method and the fit-to-data output-least-squares technique are used for the identifications. With a special weighted norm, we get a pointwise estimate. Prom the error estimate and also from the numerical tests, we find that if we decrease the mesh size, the maximum error between the identified parameter and the true parameter will increase. In order to improve the accuracy, higher order finite element spaces should be used in the approximations.

A parallel splitting up method and its application to Navier-Stokes equations

A parallel splitting-up method (or the so called alternating-direction method) is proposed in this paper. The method not only reduces the original linear and nonlinear problems into a series of one dimensional linear problems, but also enables us to compute all these one dimensional linear problems by parallel processors. Applications of the method to linear parabolic problem, steady state and nonsteady state Navier-Stokes problems are given. peerReviewed

Global extrapolation with a parallel splitting method

Extrapolation with a parallel splitting method is discussed. The parallel splitting method reduces a multidimensional problem into independent one-dimensional problems and can improve the convergence order of space variables to an order as high as the regularity of the solution permits. Therefore, in order to match the convergence order of the space variables, a high order method should also be used for the time integration. Second and third order extrapolation methods are used to improve the time convergence and it was found that the higher order extrapolation method can produce a more accurate solution than the lower order extrapolation method, but the convergence order of high order extr…

A linear approach for the nonlinear distributed parameter identification problem

In identifying the nonlinear distributed parameters we propose an approach, which enables us to identify the nonlinear distributed parameters by just solving linear problems. In this approach we just need to identify linear parameters and then recover the nonlinear parameters from the identified linear parameters. An error estimate for the finite element approximation is derived. Numerical tests are also presented.

Parallel finite element splitting-up method for parabolic problems

An efficient method for solving parabolic systems is presented. The proposed method is based on the splitting-up principle in which the problem is reduced to a series of independent 1D problems. This enables it to be used with parallel processors. We can solve multidimensional problems by applying only the 1D method and consequently avoid the difficulties in constructing a finite element space for multidimensional problems. The method is suitable for general domains as well as rectangular domains. Every 1D subproblem is solved by applying cubic B-splines. Several numerical examples are presented.

A parallel FE-splitting up method to parabolic problems