Effective operators and their variational principles for discrete electrical network problems

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study general class problems (called Z-problems) that arise in effective media theory, especially within theory composites, for defining operator. A new unified approach is developed, based on block operator methods, obtaining solutions Z-problem, formulas terms Schur complement, associated variational principles (e.g., Dirichlet Thomson minimization principles) lead to upper lower bounds In case finite-dimensional spaces, this allows relaxation standard hypotheses positivity invertibility classes operators usually considered such replacing inverses with Moore–Penrose pseudoinverse. As develop show how it applies classical example from composites conductivity periodic problem continuum (2d 3d) under hypotheses. After that, consider following three important diverse examples (increasing complexity) discrete electrical network which our relaxed First, an operator-theoretic reformulation Dirichlet-to-Neumann (DtN) map finite linear graph given used relate DtN Z-problem. Second, essentially Finally, networks graphs analog equation continuum.