Suppose I have a complicatedly shaped conductor with inhomogeneous conductivity. The conductor is modeled using the Finite Element Method. The conductor has electrical contacts on both ends. Those contacts are represented in the FEM mesh by a couple of nodes on either end of the conductor.
Now, I'd like to simulate current and voltage on both contacts. Basically, I could just solve the whole FEM system of the conductor, for example by inverting the whole system matrix. But in fact, I'm only interested in the values on the nodes representing the contacts.
So when inverting the system, I do not need to find a matrix that contains the relationship from any node to any node, but just a matrix, that contains the relationship between all contact nodes.
I still could compute the inverse of the whole system matrix, and then cut-out only the relevant rows and columns. But that would be a waste of resources.
Is there a way to directly compute the inverse of a matrix representing an equation system only for a subset of it's variables? Same for iterative solution of only a subset of variables.