Many Eulerian numerical methods dealing with interfaces resort to the definition of averaged physical properties in cells containing an interface by invoking different homogenization and/or averaging principles (Drew, 1983). For example the use of the arithmetic or the harmonic mean are two widely used functions typically used in heat transfer problems and the solution of the Poisson equation in incompressible solvers. In this work we discuss the errors introduced by any arbitrary regularization approach in a deterministic particular problem when solving an elliptic equation. By applying a multiscale decomposition analysis, we decompose the full regularization error problem into an outer and inner problem that can be solved at arbitrary order. Except in some particular cases, first order errors are generated at the interface and diffused into the bulk region. These errors appear as effective error jump conditions across the interface for the outer problem. In this work we provide first order expressions for the quantification of the error jump conditions as a function of the structure of the solution of the regularized problem. The performance of the proposed method is discussed using various test cases where an analytical solution is available. We show that in a general multidimensional case the optimal choice of the averaging rule depends on the problem at hand.