An algorithm of linear combinations: thermal conductivity

This paper presents the computational algorithms, which make it possible to overcome some complexities with the numerical solution of the boundary-value problems of thermal conductivity when the domain of the solution has a complex form or boundary conditions differ from standard ones.
Boundary contours are assumed to be broken lines (the flat case) or triangles (a 3D case). Boundary conditions and calculation results are presented as discrete functions whose values or their averaged values are given at geometric centers of boundary elements. Boundary conditions can be defined on the heat flows through boundary elements as well as on temperature, a linear temperature combination and heat flow intensity both at the boundary of the solution domain and inside it.
The solution to the boundary value problem is presented in the form of a linear combination of the fundamental solutions of the Laplace equation and their partial derivatives and, also, any solutions of these equations that are regular in the solution domain, the values of functions for which can be calculated at the points of the boundary of the solution domain and at its internal points. If the solution, which participates in the linear combination, is singular, then its average value according to this boundary element is considered.