@PhdThesis{duepublico_mods_00071918,
  author = 	{Marquardt, Pascal},
  title = 	{Adaptive spectral least-squares techniques for reaction-diffusion equations},
  year = 	{2020},
  month = 	{Jun},
  day = 	{18},
  keywords = 	{Numerische Mathematik; Spektrale Methoden; Adaptive Gitterverfeinerung; Reaktion-Diffusionsgleichung; Turing-Mechanismen},
  abstract = 	{In this thesis we are proposing a new numerical method for solving systems of reactiondiffusion
equations. In particular we are focussing on Turing mechanisms, which are
a special form of reaction-diffusion equations introduced by Alan Turing in 1952. The
speciality of these equations is that diffusion is considered destabilising. Thus starting
with a fully stable steady state so-called Turing patterns are established in the solution of
the system of partial differential equations.

Our approach is based on the triangular spectral element methods. To enhance the
performance of the triangular spectral element methods, we are extending them with
algorithms for spatial adaptation of the considered domain. We have seen in past studies
that this provides excellent improvements to spectral methods. For a good comparison
of different refinement criteria we are taking account of one heuristic criterion and two
spectral criteria closely related to our method.

In difference to recent research we are now considering time-dependent problems
with triangular spectral element methods. For the temporal discretisation of the Turing
mechanisms we are applying a generic theta-integration scheme with different values for theta. To
deal with problems arising from a static time step size, we are additionally introducing an
algorithm for an adaptive time step control.

Finally we are going to validate our method in two ways. First we are considering three
different time-independent model problems for which we already know the exact solution.
This will be done to generally verify the correctness of the method and the implementation.
Second we are applying our approach to the Turing mechanisms. For these equations we do
not know the solution, therefore we compare our results to theoretical assumptions. When
validating the method we are also comparing different configurations of our approach. We
will see in chapter 7 that our method combined with a spectral refinement criterion and
using adaptive time step size with the Crank-Nicolson method for temporal discretisation
provides good results for any Turing mechanism.},
  doi = 	{10.17185/duepublico/71918},
  url = 	{https://duepublico2.uni-due.de/receive/duepublico_mods_00071918},
  url = 	{https://doi.org/10.17185/duepublico/71918},
  file = 	{:https://duepublico2.uni-due.de/servlets/MCRFileNodeServlet/duepublico_derivate_00071750/Diss_Marquardt.pdf:PDF},
  language = 	{en}
}