VESTNIK
Bashkirskogo universiteta

RUSSIAN
ISSN 1998-4812

# Archive | Volume 20, 2015, No. 3.

## ON FINITE DIFFERENCE METHOD OF SOLVING IMPERFECT CONTACT DIRICHLET’S PROBLEM FOR NON-LINEAR ELLIPTIC EQUATIONS

Vestnik Bashkirskogo Universiteta. 2015. Vol. 20. No. 3. Pp. 795-806.
Manapova A. R.
Bashkir State University
32 Zaki Validi St., 450076 Ufa, Republic of Bashkortostan, Russia.
Email: aygulrm@mail.ru

#### Abstract

The present work is devoted to computational aspects of solving non-linear boundary value problems for elliptic equations in inhomogeneous anisotropic media with discontinuous coefficients and a solution, where imperfect-contact matching condition is given at the inner boundary between media. I.e., the problems having a jump of the coefficients and the solution on the inner surface of body contact; the jump of the solution is proportional to the normal component of the flux. We develop approximate method for solving non-linear elliptic equations with imperfect-contact matching condition. Iterative processes with iterations on the inner boundary of the domain, where the coefficients and the solution are discontinuous, reduce the initial problem to solving non-linear boundary value problems in each contacting sub-domain of an integral domain at each iteration. By applying iteration method with a parameter, we reduce the non-linear problems in each of the sub-areas to linear ones. We implement iterative processes based on the upper relaxation method. Results from computations for model examples with known analytical solutions are presented in order to demonstrate the effectiveness of the proposed method. Computer experiments are included, using IDE Embarcadero Delphi.

#### Keywords

• • contact problems
• • non-linear elliptic equations
• • discontinuous coefficients and solution
• • finite-difference method
• • iterative method
• • imperfect-contact matching condition

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