Large-scale magnetostatic field calculation in finite element micromagnetics with H2-matrices
This work addresses the memory bottleneck in large-scale micromagnetic simulations, enabling simulations with more degrees of freedom than previously feasible.
The authors reduce the computational cost of magnetostatic field calculations in finite element micromagnetics from quadratic to linear complexity using H2-matrices, achieving nearly 99% matrix size reduction for problems with over 10^6 degrees of freedom while preserving accuracy.
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.