Some topological and geometrical problems arising in projective-difference methods for the triangulation of a domain

In this paper the problem of the partition of a polygon $\Omega$ into quadrilaterals (quadrangles and triangles) is studied, for which four given boundary points $A_i(1\le i\le4)$ become the vertices of a quadrilateral, and the partition itself is topologically equivalent to a special partition of a rectangle $Q$ into rectangles with sides parallel to the sides of $Q$. This problem is closely connected with the problem of choosing a basis of piecewise linear functions in the projective-difference method, for which the projective-difference analog of the operator $-\Delta\equiv-(\partial^2/\partial x^2+\partial^2/\partial y^2)$ for a boundary-value problem in $\Omega$ turns out to be spectrally equivalent to its simplest difference analog in a rectangle (see [1–5]).