# Magneto-Fluid Dynamics Seminar

#### The mimetic finite difference method and the mass-lumped finite element method for the Landau-Lifshitz equation

**Speaker:**
Eugenia Kim, University of California, Berkeley

**Location:**
Warren Weaver Hall 905

**Date:**
Tuesday, October 17, 2017, 11 a.m.

**Synopsis:**

Micromagnetics studies magnetic behavior of ferromagnetic materials at sub-micrometer

length scales. The dynamics of the magnetic distribution in a ferromagnetic material are

governed by the Landau-Lifshitz equation. This equation is highly nonlinear, has a non-

convex constraint, has several equivalent forms, and involves solving an auxiliary problem

in the infinite domain, all of which pose interesting challenges in developing numerical

methods. We introduce numerical methods that preserve the properties of the underlying

PDE. First, we discuss the explicit and implicit mimetic finite difference method for the

Landau-Lifshitz equation, which is a new spatial discretization method which works on

arbitrary polygonal and polyhedral meshes. These schemes provide enormous flexibility

in modeling magnetic devices of various shapes. We present convergence tests for the

schemes on general meshes such as distorted and randomized meshes. We also provide

numerical experiments for the NIST standard problem #4 and formation of domain wall

structures in a thin film. We further present a high-order mimetic finite difference method

for the Landau-Lifshitz equation and compare the efficiencies of the low and high order

mimetic finite difference method using skyrmion simulation. Lastly, we present a new

class of convergent mass-lumped finite element method for the Landau-Lifshitz equation

that deals with weak solutions. This is a joint work with Konstantin Lipnikov and Jon

Wilkening.