37,376 research outputs found
Robust globally divergence-free weak Galerkin finite element methods for natural convection problems
This paper proposes and analyzes a class of weak Galerkin (WG) finite element
methods for stationary natural convection problems in two and three dimensions.
We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity,
pressure, and temperature approximations in the interior of elements,
respectively, and piecewise polynomials of degrees l, k, l(l = k-1,k) for the
numerical traces of velocity, pressure and temperature on the interfaces of
elements. The methods yield globally divergence-free velocity solutions.
Well-posedness of the discrete scheme is established, optimal a priori error
estimates are derived, and an unconditionally convergent iteration algorithm is
presented. Numerical experiments confirm the theoretical results and show the
robustness of the methods with respect to Rayleigh number.Comment: 32 pages, 13 figure
Fracton topological order via coupled layers
In this work, we develop a coupled layer construction of fracton topological
orders in spatial dimensions. These topological phases have sub-extensive
topological ground-state degeneracy and possess excitations whose movement is
restricted in interesting ways. Our coupled layer approach is used to construct
several different fracton topological phases, both from stacked layers of
simple topological phases and from stacks of fracton topological
phases. This perspective allows us to shed light on the physics of the X-cube
model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be
obtained as the strong-coupling limit of a coupled three-dimensional stack of
toric codes. We also construct two new models of fracton topological order: a
semionic generalization of the X-cube model, and a model obtained by coupling
together four interpenetrating X-cube models, which we dub the "Four Color Cube
model." The couplings considered lead to fracton topological orders via
mechanisms we dub "p-string condensation" and "p-membrane condensation," in
which strings or membranes built from particle excitations are driven to
condense. This allows the fusion properties, braiding statistics, and
ground-state degeneracy of the phases we construct to be easily studied in
terms of more familiar degrees of freedom. Our work raises the possibility of
studying fracton topological phases from within the framework of topological
quantum field theory, which may be useful for obtaining a more complete
understanding of such phases.Comment: 20 pages, 18 figures, published versio
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