G. Berikelashvili

Construction and Analysis of Difference Schemes for Some Elliptic Problems, and Consistent Estimates of the Rate of Convergence

abstract:
In the present work we present new results connected with the construction and analysis of difference schemes for:
(a) the second order elliptic equation (the Dirichlet problem, mixed boundary value problem, nonlocal problems);
(b) general systems of elliptic second order equations (the Dirichlet problem);
(c) systems of equations of the statical theory of elasticity (the first mixed, the third boundary-value, the nonlocal problems with integral
restriction);
(d) the fourth order elliptic equation (the first boundary-vale problem);
(e) the problem of bending of an orthotropic plate freely supported over the contour;

For the construction of difference schemes the Steklov averaging operators are used. The correctness is investigated by the energy method. The
estimate of the rate of convergence is based on the corresponding a priori estimates and on the generalized Bramble-Hilbert lemma. Investigation of the solvability of nonlocal problems for the second order elliptic equation is based on the Lax-Milgram lemma.

Mathematics Subject Classification: 65N06, 65N12, 35J25

Key words and phrases: Difference schemes, elliptic equation, nonlocal boundary-value problem, weighted spaces, high accuracy, elasticity theory, a priori estimates