@phdthesis{Forster2016,
  author    = {Forster, Johannes},
  title     = {Variational Approach to the Modeling and Analysis of Magnetoelastic Materials},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147226},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {This doctoral thesis is concerned with the mathematical modeling of magnetoelastic materials and the analysis of PDE systems describing these materials and obtained from a variational approach. The purpose is to capture the behavior of elastic particles that are not only magnetic but exhibit a magnetic domain structure which is well described by the micromagnetic energy and the Landau-Lifshitz-Gilbert equation of the magnetization. The equation of motion for the material's velocity is derived in a continuum mechanical setting from an energy ansatz. In the modeling process, the focus is on the interplay between Lagrangian and Eulerian coordinate systems to combine elasticity and magnetism in one model without the assumption of small deformations. The resulting general PDE system is simplified using special assumptions. Existence of weak solutions is proved for two variants of the PDE system, one including gradient flow dynamics on the magnetization, and the other featuring the Landau-Lifshitz-Gilbert equation. The proof is based on a Galerkin method and a fixed point argument. The analysis of the PDE system with the Landau-Lifshitz-Gilbert equation uses a more involved approach to obtain weak solutions based on G. Carbou and P. Fabrie 2001.},
  subject      = {Magnetoelastizit{\"a}t},
  language  = {en}
}
@phdthesis{GallegoValencia2017,
  author    = {Gallego Valencia, Juan Pablo},
  title     = {On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-148874},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.},
  subject      = {Eulersche Differentialgleichung},
  language  = {en}
}
@phdthesis{Schnuecke2016,
  author    = {Schn{\"u}cke, Gero},
  title     = {Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-139579},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds' transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise \$\mathcal{P}^{k}\$ polynomial approximation space is used on the reference cell, the sub-optimal \$\left(k+\frac{1}{2}\right)\$ convergence for monotone fuxes and the optimal \$(k+1)\$ convergence for an upwind flux are proven in the \$\mathrm{L}^{2}\$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal \$\left(k+1\right)\$ convergence and in the two dimensional case the sub-optimal \$\left(k+\frac{1}{2}\right)\$ convergence are proven in the \$\mathrm{L}^{2}\$-norm, if a piecewise \$\mathcal{P}^{k}\$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed.},
  subject      = {Galerkin-Methode},
  language  = {en}
}