Project Name :
Consistent estimates of convergence in the method of corrections with higher order differences
Donor Organization :
Shota Rustaveli National Science Foundation
Budget:
--
Grant number :
FR/406/5-106/12
Research Direction :
5 Mathematics, Mechanics
Sub Direction :
5-106 Computational Mathematics. Numerical methods.
Keyword(s) :
difference scheme; convergence rate;refinement
Description :
The method of finite differences is one of the most widespread and universal methods of numerical solution of boundary value
problems for differential equations, in particular, those of elliptic type.
For approximation methods the question of accuracy is significant. The concept of the present project constitutes a coalescence
of the following two directions of construction and investigation of finite difference schemes.
I Direction. Since for real problems the initial data is not always sufficiently smooth, one has to consider them in the Sobolev
spaces. Accordingly, during the last 30 years, in works of A.A. Samarskii, R.D.Lazarov, V.L.Makarov etc., a method for obtaining
convergence estimates (in terms of Sobolev spaces) whose convergence rate consistent with the smoothness of the desired
solution, is developed.
II Direction. In order to be economical in the amount of calculations it is desirable for the difference scheme to be
sufficiently good on rough meshes, i.e. to have high order accuracy. Therefore, construction of approximate solutions of high
order accuracy represents an urgent problem. In some cases, on a compact stencil, it is possible to construct a high order
accuracy difference schemes, but one cannot always construct schemes of such type. Therefore for construction of high order
accuracy solution we will use a method of corrections by differences of higher order which was offered empirically by
L. Fox. Foundation of this method in the case of Dirichlet problem for Poisson/Laplace equations is given in works of E.A. Volkov.
The goal of the Project is to combine the two directions of computational mathematics mentioned above in solving some
problems posed for elliptic equations and systems of equations.
The essence of the proposed method is the following. Using the solution of a difference scheme it is calculated the
corrected right hand side of the difference scheme. By appropriate selection of the corrector the solution of this difference
scheme will be of higher order accuracy, while for convergence it will be obtained the estimate consistent to the smoothness of
sought solution
Duration :
15/04/2013 - up to date
Project Leader :
Givi Berikelashvili
Project manager :
Project participant(s) :
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