This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problems amounts to determining the coefficients of some interface boundary conditions so that the domain decomposition algorithm converges rapidly. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is the first terms of the asymptotic expansion around the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of some coefficients in the interface boundary conditions, to be tested numerically. This final numerical step is presented in a companion article. This part focuses on the method used to compute these coefficients and provides detailed examples on a model problem. A MODIFIED VERSION OF THIS PREPRINT HAS BEEN PUBLISHED IN CALCOLO