FEM–BEM coupling for the large-body limit in micromagnetics

Abstract

We present and analyze a coupled finite element–boundary element method for a model in stationary micromagnetics. The finite element part is based on mixed conforming elements. For two- and three-dimensional settings, we show well-posedness of the discrete problem and present an a priori error analysis for the case of lowest order elements.

Simultaneous quasi‐optimal convergence rates in FEM‐BEM coupling

“…Define Then, the condition V ϕ + (1/2 − K) u = 0 implies the following assertions (i)–(ii). γ ext u ext = u and . u ext satisfies the exterior Calderón system: Proof Taking the exterior trace in , we obtain A calculation (which is non‐trivial for the 2D case) shows that u ext satisfies the following, second representation formula (see, e.g., the proof of , Lemma 2.3] for details): …”

A simple formula of the magnetic potential and of the stray field energy induced by a given magnetization

“…There are several methods for the effective calculation of this energy. Some of these methods are based on solving the elliptic partial differential Equation ( 6) using finite differences method (see, e.g., previous research [8][9][10]) or finite elements method (see, e.g., earlier studies [11][12][13][14]). Other methods are based on the calculation of U from the integral formula (see, e.g., previous research [15][16][17][18][19][20]):…”

“…There are several methods for the effective calculation of this energy. Some of these methods are based on solving the elliptic partial differential Equation () using finite differences method (see, e.g., previous research [8–10]) or finite elements method (see, e.g., earlier studies [11–14]). Other methods are based on the calculation of $$$ U $$$ from the integral formula (see, e.g., previous research [15–20]): $$$$ U(x)=\frac{1}{4\pi }{\int}_{\Omega}\frac{\left(y-x\right)\cdotp M(y)}{{\left|y-x\right|}^3} dy.$$…”

Second-order semi-implicit projection methods for micromagnetics simulations

“…There has been a continuous progress of developing numerical algorithms in the past few decades; see for example [6,7] and references therein. The spatial derivative is typically approximated by the finite element method (FEM) [8,9,10,11,12,13,14,15,16,17,18] and the finite difference method [19,20,21,22,23]. As for the temporal discretization, explicit schemes [15,24], fully implicit schemes [25,26,20], and semi-implicit schemes [19,27,28,29,30,31,32] have been extensively studied.…”