Finite element methods for Maxwell's equations.

hysteresis in magnetism (electromagnetism)

Our aim in this article is to discuss recent issues related with the Cahn-Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results.

The Cahn-Hilliard Equation with Logarithmic Potentials

The Landau-Lifshitz (LL) equation of micromagnetism governs rich variety of the evolution of magnetization patterns in ferromagnetic media. This is due to the complexity of physical quantities appearing in the LL equation. This complexity causes also interesting mathematical properties of the LL equation: nonlocal character for some quantities, nonconvex side-constraints, strongly nonlinear terms. These effects influence also numerical approximations. In this work, recent developments on the approximation of weak solutions, together with the overview of well-known methods for strong solutions, are addressed.

A survey on the Numerics and Computations for the Landau-Lifshitz Equation of Micromagnetism

Geometrical integration of Landau-Lifshitz-Gilbert equation based on the mid-point rule

Adaptive finite element methods for Cahn-Hilliard equations

We propose an implicit, fully discrete scheme for the numerical solution of the Maxwell-Landau-Lifshitz-Gilbert equation which is based on linear finite elements and satisfies a discrete sphere constraint as well as a discrete energy law. As numerical parameters tend to zero, solutions weakly accumulate at weak solutions of the Maxwell-Landau-Lifshitz-Gilbert equation. A practical linearization of the nonlinear scheme is proposed and shown to converge for certain scalings of numerical parameters. Computational studies are presented to indicate finite-time blow-up behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems.

A Convergent Implicit Finite Element Discretization of the Maxwell-Landau-Lifshitz-Gilbert Equation

We propose a convergent finite element based discretization of the stochastic Landau-Lifshitz-Gilbert equation. Solutions of the discretization satisfy the sphereconstraint at nodal points of the spatial triangulation, and have finite energies. Computational studies are included.

A convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation

In this paper we give an overview of the numerical methods for the solution of the Landau-Lifshitz-Gilbert equation. We discuss advantages of the presented methods and perform numerical experiments to demonstrate their performance. We also discuss the coupling with Maxwell's equations.

Numerical methods for the landau-lifshitz-gilbert equation

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by the nonlinear degenerate parabolic system $$\gamma\frac{\partial u}{\partial t}-\nabla.(b(u)\nabla[w+\alpha\phi])=0,\qquad w=-\gamma\Delta u+\gamma^{-1}\Psi'(u),\qquad\nabla.(c(u)\nabla\phi)=0$$ subject to an initial condition u 0(?)?[?1,1] on u and flux boundary conditions on all three equations. Here ???>0, ????0, ? is a non-smooth double well potential, and c(u):=1+u, b(u):=1?u 2 are degenerate coefficients. On extending existing results for the simplified two dimensional phase field model, we show stability bounds for our approximation and prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in three space dimensions. Furthermore, a new iterative scheme for solving the resulting nonlinear discrete system is introduced and some numerical experiments are presented.

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L'ubomír Baňas

Papers

Adaptive finite element methods for Cahn-Hilliard equations

A Convergent Implicit Finite Element Discretization of the Maxwell-Landau-Lifshitz-Gilbert Equation

A convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation

Numerical methods for the landau-lifshitz-gilbert equation

Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration