How to integrate scattered data with discontinuity?

I have the following problem: I have one domain rectangle domain divided in a circle, and the corresponding rectangle with the circular hole where belong the circular domain. The function variable describing the phenomenon in this domain is discontinuous along the interface between domains. So I have this problem, I have the corresponding value of the variable in the interface for each subdomain (two values in the same position), but when i'm going to interpolate with scatteredInterpolant it says: "Warning: Duplicate data points have been detected and removed - corresponding values have been averaged." Then, the interpolation is smooth along the interface.

Is it there a good way to address this problem of integration of scattered data with a curve line of discontinuity?

Thank you very much in advance.

[Walter Roberson]

Please post a couple of sample images.

[Hector Palacios]

Well, I can give you two plots. The first one is what I want to obtain in matlab, is the result of another program where I did the pre process, the geometry. The second one is the plot that matlab gives averaging the points in the interface and smoothing the interpolation in the discontinuity. So I need be able to interpolate correctly the discontinuity between subdomains for integrating correctly.