Construct an InterpolatingFunction:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns Indeterminate for queries outside the domain:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that extrapolates outside its domain and does not give a warning message:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns 0 outside its domain:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns Indeterminate outside its domain and does not give a warning message:

Query the InterpolatingFunction outside of its domain:

The default for NDSolve and the finite element method is to return InterpolatingFunction objects that do not extrapolate outside of the given domain:

Allow InterpolatingFunction objects generated by NDSolve to extrapolate when evaluated outside of the simulation domain and not warn about it:

While it is not generally possible to construct periodic interpolating functions for arbitrary meshes, one can mimic periodic interpolating functions based on rectangular regions by specifying an "ExtrapolatiopnHandler".

Generate data and a mesh:

Generate a temporary interpolating function from this mesh and data:

Use the extrapolation handler to map the coordinates outside of the meshed domain back into the domain and evaluate over the temporary interpolating function:

To verify that the interpolation is now periodic, visualize the function f both inside and outside the original domain: