3,817 research outputs found
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
Random sampling of plane partitions
This article presents uniform random generators of plane partitions according
to the size (the number of cubes in the 3D interpretation). Combining a
bijection of Pak with the method of Boltzmann sampling, we obtain random
samplers that are slightly superlinear: the complexity is in
approximate-size sampling and in exact-size sampling
(under a real-arithmetic computation model). To our knowledge, these are the
first polynomial-time samplers for plane partitions according to the size
(there exist polynomial-time samplers of another type, which draw plane
partitions that fit inside a fixed bounding box). The same principles yield
efficient samplers for -boxed plane partitions (plane partitions
with two dimensions bounded), and for skew plane partitions. The random
samplers allow us to perform simulations and observe limit shapes and frozen
boundaries, which have been analysed recently by Cerf and Kenyon for plane
partitions, and by Okounkov and Reshetikhin for skew plane partitions.Comment: 23 page
A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain
The 3D incompressible Euler equation is an important research topic in the
mathematical study of fluid dynamics. Not only is the global regularity for
smooth initial data an open issue, but the behaviour may also depend on the
presence or absence of boundaries.
For a good understanding, it is crucial to carry out, besides mathematical
studies, high-accuracy and well-resolved numerical exploration. Such studies
can be very demanding in computational resources, but recently it has been
shown that very substantial gains can be achieved first, by using Cauchy's
Lagrangian formulation of the Euler equations and second, by taking advantages
of analyticity results of the Lagrangian trajectories for flows whose initial
vorticity is H\"older-continuous. The latter has been known for about twenty
years (Serfati, 1995), but the combination of the two, which makes use of
recursion relations among time-Taylor coefficients to obtain constructively the
time-Taylor series of the Lagrangian map, has been achieved only recently
(Frisch and Zheligovsky, 2014; Podvigina {\em et al.}, 2016 and references
therein).
Here we extend this methodology to incompressible Euler flow in an
impermeable bounded domain whose boundary may be either analytic or have a
regularity between indefinite differentiability and analyticity.
Non-constructive regularity results for these cases have already been obtained
by Glass {\em et al.} (2012). Using the invariance of the boundary under the
Lagrangian flow, we establish novel recursion relations that include
contributions from the boundary. This leads to a constructive proof of
time-analyticity of the Lagrangian trajectories with analytic boundaries, which
can then be used subsequently for the design of a very high-order
Cauchy--Lagrangian method.Comment: 18 pages, no figure
Shape-Driven Nested Markov Tessellations
A new and rather broad class of stationary (i.e. stochastically translation
invariant) random tessellations of the -dimensional Euclidean space is
introduced, which are called shape-driven nested Markov tessellations. Locally,
these tessellations are constructed by means of a spatio-temporal random
recursive split dynamics governed by a family of Markovian split kernel,
generalizing thereby the -- by now classical -- construction of iteration
stable random tessellations. By providing an explicit global construction of
the tessellations, it is shown that under suitable assumptions on the split
kernels (shape-driven), there exists a unique time-consistent whole-space
tessellation-valued Markov process of stationary random tessellations
compatible with the given split kernels. Beside the existence and uniqueness
result, the typical cell and some aspects of the first-order geometry of these
tessellations are in the focus of our discussion
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