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Erscheinungsjahr
- 2023 (2) (entfernen)
Sprache
- Englisch (2) (entfernen)
Schlagworte
- Partielle Differentialgleichung (2) (entfernen)
THE NONLOCAL NEUMANN PROBLEM
(2023)
Instead of presuming only local interaction, we assume nonlocal interactions. By doing so, mass
at a point in space does not only interact with an arbitrarily small neighborhood surrounding it,
but it can also interact with mass somewhere far, far away. Thus, mass jumping from one point to
another is also a possibility we can consider in our models. So, if we consider a region in space, this
region interacts in a local model at most with its closure. While in a nonlocal model this region may
interact with the whole space. Therefore, in the formulation of nonlocal boundary value problems
the enforcement of boundary conditions on the topological boundary may not suffice. Furthermore,
choosing the complement as nonlocal boundary may work for Dirichlet boundary conditions, but
in the case of Neumann boundary conditions this may lead to an overfitted model.
In this thesis, we introduce a nonlocal boundary and study the well-posedness of a nonlocal Neu-
mann problem. We present sufficient assumptions which guarantee the existence of a weak solution.
As in a local model our weak formulation is derived from an integration by parts formula. However,
we also study a different weak formulation where the nonlocal boundary conditions are incorporated
into the nonlocal diffusion-convection operator.
After studying the well-posedness of our nonlocal Neumann problem, we consider some applications
of this problem. For example, we take a look at a system of coupled Neumann problems and analyze
the difference between a local coupled Neumann problems and a nonlocal one. Furthermore, we let
our Neumann problem be the state equation of an optimal control problem which we then study. We
also add a time component to our Neumann problem and analyze this nonlocal parabolic evolution
equation.
As mentioned before, in a local model mass at a point in space only interacts with an arbitrarily
small neighborhood surrounding it. We analyze what happens if we consider a family of nonlocal
models where the interaction shrinks so that, in limit, mass at a point in space only interacts with
an arbitrarily small neighborhood surrounding it.
Coastal erosion describes the displacement of land caused by destructive sea waves,
currents or tides. Due to the global climate change and associated phenomena such as
melting polar ice caps and changing current patterns of the oceans, which result in rising
sea levels or increased current velocities, the need for countermeasures is continuously
increasing. Today, major efforts have been made to mitigate these effects using groins,
breakwaters and various other structures.
This thesis will find a novel approach to address this problem by applying shape optimization
on the obstacles. Due to this reason, results of this thesis always contain the
following three distinct aspects:
The selected wave propagation model, i.e. the modeling of wave propagation towards
the coastline, using various wave formulations, ranging from steady to unsteady descriptions,
described from the Lagrangian or Eulerian viewpoint with all its specialties. More
precisely, in the Eulerian setting is first a steady Helmholtz equation in the form of a
scattering problem investigated and followed subsequently by shallow water equations,
in classical form, equipped with porosity, sediment portability and further subtleties.
Secondly, in a Lagrangian framework the Lagrangian shallow water equations form the
center of interest.
The chosen discretization, i.e. dependent on the nature and peculiarity of the constraining
partial differential equation, we choose between finite elements in conjunction
with a continuous Galerkin and discontinuous Galerkin method for investigations in the
Eulerian description. In addition, the Lagrangian viewpoint offers itself for mesh-free,
particle-based discretizations, where smoothed particle hydrodynamics are used.
The method for shape optimization w.r.t. the obstacle’s shape over an appropriate
cost function, constrained by the solution of the selected wave-propagation model. In
this sense, we rely on a differentiate-then-discretize approach for free-form shape optimization
in the Eulerian set-up, and reverse the order in Lagrangian computations.