Sven Christophersen

Riccardo Hertel, Sven Christophersen, Steffen Börm

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.

Steffen Börm, Sven Christophersen

We consider a compression method for boundary element matrices arising in the context of the computation of electrostatic fields. Green cross approximation combines an analytic approximation of the kernel function based on Green's representation formula and quadrature with an algebraic cross approximation scheme in order to obtain both the robustness of analytic methods and the efficiency of algebraic ones. One particularly attractive property of the new method is that it is well-suited for acceleration via general-purpose graphics processors (GPUs).

Steffen Börm, Sven Christophersen

We present a fast algorithm that constructs a data-sparse approximation of matrices arising in the context of integral equation methods for elliptic partial differential equations. The new algorithm uses Green's representation formula in combination with quadrature to obtain a first approximation of the kernel function and then applies nested cross approximation to obtain a more efficient representation. The resulting $\mathcal{H}^2$-matrix representation requires $\mathcal{O}(n k)$ units of storage for an $n\times n$ matrix, where $k$ depends on the prescribed accuracy.