Computing electric fields of an arbitrary point in space, using the finite element method (for a partial ordinary equation) with Python/C++?

I need to solve a partial differential equation using the finite element method (FEM). But I am very new to FEM and have a basic question about using the FEM result.

Suppose the input of my FEM method is a partial equation on electric fields, and the output, namely, the FEM result is a mapping from tetrahedrons into E-field values. Then, once we obtain the FEM result, how can we systematically compute the E-field for an arbitrary point in the space?

I can imagine one can obtain the E-fields in the following algorithm. Given the point in space, we check which tetrahedron contains the underlying point and then use the E-field corresponding to that tetrahedron as our E-field.

The problem is the E-field obtained in this way would not be smooth. So, what would be the “normal”, textbook algorithm to compute values from the FEM result? Should we use interpolation techniques, and are there existing tools for this in Python/C++?