Useful upper and lower bounds to the magnetostatic self-energy

The magnetostatic energy of a spherical particle is shown to be bounded between two specific expressions which contain local variables only. Evaluating such expressions in numerical computations of magnetization configurations should be much simpler than the evaluation of the actual magnetostatic energy. An approximation in which the error can be controlled, and is known at each step of the minimization procedure is suggested. It is based on general inequalities which place the rigorous value of the magnetostatic energy between upper and lower bounds. Using these upper and lower bounds is therefore expected to reduce the total computation time by orders of magnitude