Magnetostatic problems in fractal domains

TL;DR

This paper investigates magnetostatic problems in fractal domains, proving existence, uniqueness, and convergence of solutions, and providing numerical approximations with error estimates, motivated by quantum physics applications.

Contribution

It establishes the mathematical foundation for magnetostatic problems in fractal domains, including existence, uniqueness, convergence, and numerical methods.

Findings

Proved existence and uniqueness for fractal and pre-fractal problems.

Demonstrated convergence of pre-fractal solutions to the fractal limit.

Provided numerical simulations with a priori error estimates.

Abstract

We consider a magnetostatic problem in a 3D "cylindrical" domain of Koch type. We prove existence and uniqueness results for both the fractal and pre-fractal problems and we investigate the convergence of the pre-fractal solutions to the limit fractal one. We consider the numerical approximation of the pre-fractal problems via FEM and we prove a priori error estimates. Some numerical simulations are also shown. Our long term motivation includes studying problems that appear in quantum physics in fractal domains.