Suppose I have the following integral:$$\int_{-1}^1 \int_{-1}^1 C_if(x,y)dxdy$$where\begin{equation}C_i=\begin{cases}C_1 \quad \text{in }\Omega_1\\C_2 \quad \text{in } \Omega_2\\\end{cases}\end{equation}So how can I compute it using Gauss quadrature without having to splitting the unit cell into two seperate parts or re-transforming them into another 2 unit cells. I mean, since the integrand is discontinuous over the unit cell, it seems to be not possible to use Guass quadrature directly, so what other information do I need or what treatment should I take to deal with it? Is it possible to accomplish this goal?
Using Gauss quadrature for a discontinuous integrand
numerical analysis;quadrature;integration
Your question may sound like one can not use Gauss quadrature for discontinuous integrand that is not necessary a case. The problem is that a straightforward application of it (without taking into account the discontinuity) can give unsatisfactory accuracy (e.g. adding more quadrature points decrease only slowly the quadrature error). To get an answer that will fit your goal, you should specify in your question  how your sets $\Omega_i$ are described. I can give a reference for one particular case. I am aware of such integration when using level set methods, where an interface can split a domain Omega into two parts, so the sets $\Omega_i$ are given implicitly by a sign of the level set function, because the interface itself is given by the zero level set.So if you can construct a smooth function $\phi$ (especially that $\nabla \phi$ is defined almost everywhere) such that $\Omega_1 = \{ x : \phi(x) < 0 \}$ and $\Omega_2 = \{x : \phi(x)>0\}$ then search in Google or in scientific databases for terms like quadrature methods implicitly defined volumes. I can not recommend a specific publication, because I have not practical experiences with this problem.Similarly the so called Volume of Fluid Method has to deal with such problem without using a level set function, but a characteristic functions for $\Omega_i$.Addition:As I wrote before, I did not apply numerical quadrature of this type before and I am only aware of the fact that such problem has been solved many times in level set methods, therefore I do not want to recommend any particular algorithm and I suggest to search for variants suitable to particular needs. Just a quick search gave me this recent PDF file being:High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in HyperrectanglesR. I. SayeSIAM Journal on Scientific Computing 2015 37:2, A993-A1019 Briefly checking it, it claims that the quadrature points (and corresponding positive weights) can be found such that they lie strictly in one subdomain using the information given by level set function. This is of course the direction one should follow in level set methods.