Abstract
We consider the third boundary value problem of static elasticity theory (stiff contact problem) in a rectangle, for solution of which we use a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme is convergent at the rate $O(|h|^{m})$ in the discrete $L_{2}$ -norm, provided that the solution of the original problem belongs to the Sobolev space with exponent $m\in[2, 4]$ .
Highlights
1 Introduction The problem of accuracy is the main problem in the theory of difference schemes as well as in applications
One approach for obtaining solutions with improved accuracy is represented by the idea of refinement by differences of higher order, which was offered by Fox [ ]: the right-hand side of the difference scheme is corrected by the solution obtained on the first stage and the scheme is repeatedly solved on the same grid
The first one consists of finding the correcting addend, and the second is related to obtaining an a priori estimate of convergence. To overcome these difficulties we use the method for derivation of estimates for the convergence rate of difference schemes developed in the last years by Samarskii and other authors, in which the convergence rate is compatible with the smoothness of the solution of the original differential problem
Summary
The problem of accuracy is the main problem in the theory of difference schemes as well as in applications. One approach for obtaining solutions with improved accuracy is represented by the idea of refinement by differences of higher order, which was offered by Fox [ ]: the right-hand side of the difference scheme is corrected by the solution obtained on the first stage and the scheme is repeatedly solved on the same grid This empirical idea is simple, its theoretical foundation encounters essential difficulties. The first one consists of finding the correcting addend, and the second is related to obtaining an a priori estimate of convergence To overcome these difficulties we use the method for derivation of estimates for the convergence rate of difference schemes developed in the last years by Samarskii and other authors (e.g., see [ – ]), in which the convergence rate is compatible with the smoothness of the solution of the original differential problem. It is proved that the corrected scheme converges at the rate O(|h|m) in the discrete L (ω)-norm provided that the exact solution belongs to the Sobolev space W m( ), m ∈ [ , ]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
