M. Bubak et al. (Eds.): ICCS 2008, Part I, LNCS 5101, pp. 1022–1031, 2008. © Springer-Verlag Berlin Heidelberg 2008 Application of R-Functions Method and Parallel Computations to the Solution of 2D Elliptic Boundary Value Problems Marcin Detka and Czeslaw Cichoń Chair of Applied Computer Science, Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland {Marcin.Detka,Czeslaw.Cichon}@tu.kielce.pl Abstract. In the paper, the R-function theory developed by Rvachew is applied to solve 2D elliptic boundary value problems. Unlike the well-established FEM or BEM method, this method requires dividing the solution into two parts. In the first part, the boundary conditions are satisfied exactly and in the second part, the differential equation is satisfied in an approximate way. In such a way, it is possible to formulate in one algorithm the so-called general structural solution of a boundary-value problem and use it for an arbitrary domain and arbitrary boundary conditions. The usefulness of the proposed computational method is verified using the example of the solution of the Laplace equation with mixed boundary conditions. Keywords: structural solution, R-functions, parallel computations. 1 Introduction Mathematical models of engineering problems are often defined as boundary-value problems involving partial-differential equations. For the description of such problems it is required to have analytical information connected with the equation itself (or a set of equations) and geometrical information necessary to define boundary conditions. This information concerns the solution domain, shapes of particular parts of the boundary, distribution and forms of the imposed constraints and the like. It is accounted for in a different way in various solution methods. In the paper, such problems are solved in a uniform way using the R-function theory, developed by Rvachew et al. [3]. In this theory, the so-called structural solutions are constructed with the use of elaborated tools of the analytical geometry. As a result, the structural solution exactly satisfying the boundary conditions contains some unknown parameters that have to be computed. The paper is limited to elliptic problems in two dimensions. Such problems are still dealt with because of their well-known relation to many physical models. Furthermore, theoretical and numerical results obtained in this area are very useful in practice.