1,330 research outputs found
Breaking spaces and forms for the DPG method and applications including Maxwell equations
Discontinuous Petrov Galerkin (DPG) methods are made easily implementable
using `broken' test spaces, i.e., spaces of functions with no continuity
constraints across mesh element interfaces. Broken spaces derivable from a
standard exact sequence of first order (unbroken) Sobolev spaces are of
particular interest. A characterization of interface spaces that connect the
broken spaces to their unbroken counterparts is provided. Stability of certain
formulations using the broken spaces can be derived from the stability of
analogues that use unbroken spaces. This technique is used to provide a
complete error analysis of DPG methods for Maxwell equations with perfect
electric boundary conditions. The technique also permits considerable
simplifications of previous analyses of DPG methods for other equations.
Reliability and efficiency estimates for an error indicator also follow.
Finally, the equivalence of stability for various formulations of the same
Maxwell problem is proved, including the strong form, the ultraweak form, and a
spectrum of forms in between
Using entanglement against noise in quantum metrology
We analyze the role of entanglement among probes and with external ancillas
in quantum metrology. In the absence of noise, it is known that unentangled
sequential strategies can achieve the same Heisenberg scaling of entangled
strategies and that external ancillas are useless. This changes in the presence
of noise: here we prove that entangled strategies can have higher precision
than unentangled ones and that the addition of passive external ancillas can
also increase the precision. We analyze some specific noise models and use the
results to conjecture a general hierarchy for quantum metrology strategies in
the presence of noise.Comment: 7 pages, 4 figures, published versio
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
- …