8,240 research outputs found
Discretization methods for homogeneous fragmentations
Homogeneous fragmentations describe the evolution of a unit mass that breaks
down randomly into pieces as time passes. They can be thought of as continuous
time analogs of a certain type of branching random walks, which suggests the
use of time-discretization to shift known results from the theory of branching
random walks to the fragmentation setting. In particular, this yields
interesting information about the asymptotic behaviour of fragmentations.
On the other hand, homogeneous fragmentations can also be investigated using
a powerful technique of discretization of space due to Kingman, namely, the
theory of exchangeable partitions of . Spatial discretization is especially
well-suited to develop directly for continuous times the conceptual method of
probability tilting of Lyons, Pemantle and Peres.Comment: 21 page
Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
We consider time discretization methods for abstract parabolic problems with
inhomogeneous linear constraints. Prototype examples that fit into the general
framework are the heat equation with inhomogeneous (time dependent) Dirichlet
boundary conditions and the time dependent Stokes equation with an
inhomogeneous divergence constraint. Two common ways of treating such linear
constraints, namely explicit or implicit (via Lagrange multipliers) are
studied. These different treatments lead to different variational formulations
of the parabolic problem. For these formulations we introduce a modification of
the standard discontinuous Galerkin (DG) time discretization method in which an
appropriate projection is used in the discretization of the constraint. For
these discretizations (optimal) error bounds, including superconvergence
results, are derived. Discretization error bounds for the Lagrange multiplier
are presented. Results of experiments confirm the theoretically predicted
optimal convergence rates and show that without the modification the (standard)
DG method has sub-optimal convergence behavior.Comment: 35 page
Discretization methods for extremely anisotropic diffusion
In fusion plasmas there is extreme anisotropy due to the high temperature and large magnetic field strength.
This causes diffusive processes, heat diffusion and energy/momentum loss due to viscous friction,
to effectively be aligned with the magnetic field lines. This alignment leads to different values
for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction,
to the extent that heat diffusion coefficients can be up to times larger in the parallel direction than
in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used
to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion
to be polluted by the numerical error in approximating the parallel diffusion.
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Currently the common approach is to apply magnetic field aligned grids, an approach that automatically takes care
of the directionality of the diffusive coefficients. This approach runs into problems in the case of crossing field lines,
e.g., x-points and points where there is magnetic reconnection.
This makes local non-alignment unavoidable. It is therefore useful to consider numerical schemes that are more tolerant
to the misalignment of the grid with the magnetic field lines,
both to improve existing methods and to help open the possibility of applying regular non-aligned grids.
To investigate this several discretization schemes are applied to the anisotropic heat diffusion equation on a cartesian grid
On the Convergence of Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws
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