Magnetostatic problems in fractal domains
For mathematicians and physicists studying PDEs on fractal domains, this provides rigorous theoretical foundations and numerical methods for a specific class of problems.
The paper proves existence and uniqueness for magnetostatic problems in fractal Koch-type domains, establishes convergence of pre-fractal solutions to the fractal limit, and provides FEM error estimates with numerical simulations.
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.