Large-scale magnetostatic field calculation in finite element micromagnetics with H2-matrices

Magnetostatic field calculations in micromagnetic simulations can be numerically expensive, particularly in the case of large-scale finite element simulations. The established finite element / boundary element method (FEM/BEM) by Fredkin & Koehler involves a densely populated matrix with unacceptable numerical costs for problems involving a large number of degrees of freedom \(N\). By using hierarchical matrices of \(\mathcal{H}^2\) type, we show that the memory requirements for the FEM/BEM method can be reduced dramatically, effectively converting the quadratic complexity \(\mathcal{O}(N^2)\) of the problem to a linear one \(\mathcal{O}(N)\). We obtain matrix size reductions of nearly \(99\%\) in test cases with more than \(10^6\) degrees of freedom, and we test the computed magnetostatic energy values by means of comparison with analytic values. The efficiency of the \(\mathcal{H}^2\)-matrix compression opens the way to large-scale magnetostatic field calculations in micromagnetic modeling, all while preserving the accuracy of the established FEM/BEM formalism.