In the context of adaptive mesh in FEM, after a new mesh is created, the data on the old mesh are to be transferred to the new mesh. For the data on the integration points (IPs), it seems the usual way is to (1) transfer the data on IPs of the old mesh to the nodes of the old mesh, (2) use the shape function to transfer the data from the nodes of the old mesh to the nodes of the new mesh, (3) then use the shape function to transfer the data on the nodes of the new mesh to the IPs of the new mesh.
There are different techniques for the first step (see (1) above), and the most popular one seems to be the patch method or superconvergent patch recovery. In this method the IPs of the old mesh are used as the sampling points.
My question would be why not just simply use the patch method to transfer the data on the IPs of the old mesh to the IPs of the new mesh directly, as the IPs of the old mesh are used as sampling points anyway.