The Maxwell-Landau-Lifshitz-Gilbert System: Mathematical Theory and Numerical Approximation

This thesis deals with the mathematical theory and numerical approximation of the
Landau--Lifshitz--Gilbert equation coupled to the Maxwell equations without artificial
boundary conditions.
As a starting point, the physical equations are stated on the unbounded three dimensional
space and reformulated in a mathematically precise way to a coupled partial
differential -- boundary integral system.
We derive a weak form of the whole coupled system, state the relation to the strong
form and show uniqueness of the Maxwell part of the solution. A numerical algorithm is
proposed based on the tangent plane scheme for the LLG part and using a finite element
and boundary element coupling as spatial discretization and the backward Euler method
and Convolution Quadrature as time discretization for the interior Maxwell part and the
boundary, respectively. Under minimal assumptions on the regularity of solutions, we
present well-posedness and convergence of the numerical algorithm.
For the pure Maxwell equations without the coupling to the LLG equation, we are
able to show stronger results than in the coupled case. We derive a weak form for the
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Maxwell transmission problem and demonstrate existence and uniqueness of the weak
solutions as well as equivalence with a strong solution. The proposed algorithm of finite-element/
boundary-element coupling via Convolution Quadrature converges with only
minimal assumptions on the regularity of the input data.
Again for the full Maxwell--LLG system, we show a-priori error bounds in the situation
of a sufficiently regular solution. This is done by a combination of the known linearly implicit
backward difference formula time discretizations with higher order non-conforming
finite element space discretizations for the LLG equation and the leapfrog and Convolution
Quadrature time discretization with higher order discontinuous Galerkin elements
and continuous boundary elements for the boundary integral formulation of Maxwell's
equations. The precise method of coupling allows us to solve the system at the cost of the
individual parts, with the same convergence rates under the same regularity assumptions
and the same CFL conditions as for an uncoupled examination.
Numerical experiments illustrate and expand on the theoretical results and demonstrate
the applicability of the methods.
For the formulation of the boundary integral equations, the study of the Laplace transform
is inevitable. We collect and extend the properties of the Laplace transform from
literature. In the suitable functional analytic setting, we give extensive proofs in a self
contained way of all the required properties.