On generalized solution of two-dimensional elliptic problem with piecewise constant coefficients based on splitting a~differential operator and using specific basis functions

V. V. Smelov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 6 (2003) no. 1, pp. 59-72

Résumé

An alternative method with respect to difference and variational-difference algorithms is offered. It is intended for solving a boundary value problem with the second order elliptic operator in a two-dimensional domain combined of rectangles. Coefficients of a differential operator are assumed to be piecewise constant, i.e., are constant inside each rectangle. An approximate solution of the problem is realized in a generalized version. The proposed method is based on the splitting of the differential operator, using a specific system of the basic functions which ensures approximation of the solution by means of their small number. The final objective is to reduce the problem to a solution of one-dimensional problems with the algorithm oriented to a sufficiently small dimension of algebraic systems of equations and, respectively, to the fast convergence rate of the iterative process as well as to the essentially decreased computer memory.