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Adaptive mesh refinement

by Hermann Forster

To solve the partial differential equations we use the finite element method. The magnetic wire is divided into tetrahedral finite elements. The space discretization leads to a system of coupled magnetic moments sitting at the nodes of the finite element mesh. The effective field at the nodes follows from the derivative of the Gibbs' free energy with respect to the magnetic moment. The Gilbert equation of motion has to be solved for each magnetic moment. The equations are coupled by the exchange and magnetostatic interactions between the magnetic moments.

In order to resolve a magnetic domain wall the element size has to be smaller than the characteristic length, lc, given by the minimum of the exchange length lex and the Bloch parameter. If the element size is too big a so-called domain wall collapse will occur: the magnetization becomes aligned anti-parallel at neighboring nodes and the torque on the magnetization vanishes. A large number of finite elements is required for the study of wall motion in magnetic wires using a uniform fine grid with an element size h < lc. In order to keep the number of finite elements small and avoid the domain wall collapse, an adaptive refinement scheme can be applied.

The outline of adaptive algorithms is as follows. Starting from an initial finite element mesh, we produce a sequence of refined grids, until the estimated error is below a given tolerance. Generally we have to distinguish three refinement strategies. First there is the possibility of moving the nodes from positions with nearly uniform magnetization to the wall or vortices regions (r-refinement). Then there is the possibility of adding new nodes and elements into the elements having a big error indicator (h-refinement). The third way is to interpolate the direction cosines by polynomials of higher order instead of linear functions (p-refinement). The aim of an adaptive refinement algorithm is to get the ``optimal'' mesh, where the number of nodes is as small as possible while keeping the error below a given tolerance. All  strategies are based on the idea of an equidistribution of the local error indicator to all mesh elements. Babuska and Rheinboldt state that a mesh is almost optimal when the local errors are approximately equal for all elements. Adaptive mesh algorithms have to identify the regions where a higher spatial resultion is required. Therefore so-called error indicators are computed from the current finite element solution. In micromagnetics a reliable error indicator is based on the constraint condition for the norm of the magnetization vector. Strong deviations of the norm show regions with non-uniform magnetization.

In the following we outline the h-refinement scheme used in the micromagnetic simulations of domain wall motion. The finite element grid is adjusted to the current wall position during the solution of the Gilbert equation of motion. The mesh is refined in regions with non-uniform magnetization, whereas elements are taken out where the magnetization is uniform. Thus the fine grid moves together with the wall, as the mesh can be coarsened as soon as the wall has passed by:
This algorithm guarantees that the simulation proceeds in time only if the space discretization error is below a certain threshold. Numerical studies showed that the moving mesh reduces the total CPU time by more than a factor of 4.

[ References ]

H. Forster, T. Schrefl, D. Suess, W. Scholz, V. Tsiantos, R. Dittrich, and J. Fidler, ``Domain wall motion in nano-wires using moving grids,'' J. Appl. Phys., vol. 91, pp. 6914-6919, 2002. 
H. Forster, ``Reversal modes in mesoscopic structures,'' PhD-Thesis, TU Vienna, 2003. 

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