Mixed elements and a two-phase free-boundary problem in magnetostatics

Magnetostatics consists in finding vector fields b and h such that div b = 0, rot h = j (j is given), and b = Г(h), where Г is the subgradient of a convex functional U (the magnetic coenergy). When U is the sum of a quadratic functional and of a support functional, a 2-phase free-boundary problem results, a “vector” Stefan-like problem so to speak, because the unknown is a vector-field, not a function like e.g., temperature. The special structure of the equations calls for special “mixed” finite elements. We show how they help keep at the discrete level some interesting “complementarily” properties of the problem which are present at the continuous level.