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Special basis functions adjusted for solving problems with elliptic operators that have discontinuous coefficients are proposed. A specific property of bases constructed here is the automatic fulfillment of jump conditions for those problems at the points of discontinuity of the coefficients of equations. Another essential property of such basis is high-accuracy approximation of approximate piecewise-smooth solutions to the above problems by a small number of basis functions. The justification of the results follows from two theorems presented here. This paper is a further development of the author's results [1, 2] concerning the expansions of functions into fast-convergent series composed of bases arranged from trigonometric functions. [ABSTRACT FROM AUTHOR]

In order to solve approximately integral equations with kernels of the form K( x − t) by the Galerkin method we propose a nonstandard trigonometric basis of a high approximating quality, which enables us to reduce the double integral in the Galerkin algorithm to rather simple single integration. [ABSTRACT FROM AUTHOR]

A specific variant of spherical functions having considerable advantages over generally used ones is proposed in the paper. This essentially simplifies identical operations in the problems where a solution depending on spherical coordinates is sought in the form of a decomposition over spherical functions. A complete list of identities for concomitant associated Legendre functions is presented. The application of the proposed variant of spherical functions is illustrated by a particular example. [ABSTRACT FROM AUTHOR]

A special basis based on trigonometric functions is proposed for solution of integral equations with kernels of the form K( x – t) by the Galerkin method. This basis possesses high approximation quality and allows one to reduce the double integral in the Galerkin algorithm to a very simple single integration. [ABSTRACT FROM AUTHOR]

New quadrature formulas are developed based on specific bases of trigonometric functions. Those quadratures are comparable with Gauss quadrature formulas, oriented to a class of functions common with those formulas, but have nothing to do with a quadrature for periodic functions based on trigonometric formulas as well. [ABSTRACT FROM AUTHOR]

A discrete maximum principle is presented for linear algebraic systems with coefficients of arbitrary signs and the number of unknowns exceeding (in the general case) the number of equations. The presentation uses the formal language of linear algebra (without any notions from the theory of difference schemes). This formal algebraic approach allows one to reveal easily the existence or absence of data for realization of an extended maximum principle in any algorithm represented in the form of a system of linear algebraic equations. The theorems formulated here directly imply the absolute stability of solutions to systems of equations. Variants of homogeneous algebraic systems and systems of equations with a right-hand side are considered. Theoretical results are illustrated by examples of actual problems. [ABSTRACT FROM AUTHOR]

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases. [ABSTRACT FROM AUTHOR]