Direct methods in the calculus of variations
Direct methods in the calculus of variations
Weakly differentiable functions
Weakly differentiable functions
A uniformly accurate finite element method for the Reissner-Mindlin plate
SIAM Journal on Numerical Analysis
A periodic relaxation method for computing microstructures
Applied Numerical Mathematics
Error Estimates for the Finite Element Approximation of Problems in Unbounded Domains
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Multi-atomic Young measure and artificial boundary in approximation of micromagnetics
Applied Numerical Mathematics
Numerical Analysis for a Macroscopic Model in Micromagnetics
SIAM Journal on Numerical Analysis
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In this paper, a non-conforming finite element method coupled with an artificial boundary technique is developed in a multi-atomic Young measure approximation to solve the two-dimensional variational problem for the magnetization field in micromagnetics, which has an anisotropic potential energy and a nonconvex constraint and thus can develop microstructures. Compared with the conforming finite element approach, which turns out to be unstable in the sense that spurious numerical oscillations can occur in the discrete macroscopic magnetization field, the stability and convergence of the non-conforming finite element method can be established. It is also proved that, for the uniaxial energy density, two-atomic young measure is sufficient to approximate the macroscopic magnetization field. The efficiency of the method is illustrated by some numerical examples.