1,341 research outputs found
Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell's Equations
This work is on the numerical approximation of incoming solutions to
Maxwell's equations with dissipative boundary conditions whose energy decays
exponentially with time. Such solutions are called asymptotically disappearing
(ADS) and they play an importarnt role in inverse back-scatering problems. The
existence of ADS is a difficult mathematical problem. For the exterior of a
sphere, such solutions have been constructed analytically by Colombini, Petkov
and Rauch [7] by specifying appropriate initial conditions. However, for
general domains of practical interest (such as Lipschitz polyhedra), the
existence of such solutions is not evident.
This paper considers a finite-element approximation of Maxwell's equations in
the exterior of a polyhedron, whose boundary approximates the sphere. Standard
Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to
approximate the electric and magnetic fields. Discrete initial conditions
interpolating the ones chosen in [7] are modified so that they are (weakly)
divergence-free. We prove that with such initial conditions, the approximation
to the electric field is weakly divergence-free for all time. Finally, we show
numerically that the finite-element approximations of the ADS also decay
exponentially with time when the mesh size and the time step become small.Comment: 15 pages, 3 figure
On multiple frequency power density measurements II. The full Maxwell's equations
We shall give conditions on the illuminations such that the
solutions to Maxwell's equations satisfy certain non-zero qualitative properties inside
the domain , provided that a finite number of frequencies are
chosen in a fixed range. The illuminations are explicitly constructed. This
theory finds applications in several hybrid imaging problems, where unknown
parameters have to be imaged from internal measurements. Some of these examples
are discussed. This paper naturally extends a previous work of the author
[Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied.Comment: 24 page
On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems
In this paper, we consider the well-known Fattorini's criterion for
approximate controllability of infinite dimensional linear systems of type
. We precise the result proved by H. O. Fattorini in
\cite{Fattorini1966} for bounded input , in the case where can be
unbounded or in the case of finite-dimensional controls. More precisely, we
prove that if Fattorini's criterion is satisfied and if the set of geometric
multiplicities of is bounded then approximate controllability can be
achieved with finite dimensional controls. An important consequence of this
result consists in using the Fattorini's criterion to obtain the feedback
stabilizability of linear and nonlinear parabolic systems with feedback
controls in a finite dimensional space. In particular, for systems described by
partial differential equations, such a criterion reduces to a unique
continuation theorem for a stationary system. We illustrate such a method by
tackling some coupled Navier-Stokes type equations (MHD system and micropolar
fluid system) and we sketch a systematic procedure relying on Fattorini's
criterion for checking stabilizability of such nonlinear systems. In that case,
the unique continuation theorems rely on local Carleman inequalities for
stationary Stokes type systems
Some Key Developments in Computational Electromagnetics and their Attribution
Key developments in computational electromagnetics are proposed. Historical highlights are summarized concentrating on the two main approaches of differential and integral methods. This is seen as timely as a retrospective analysis is needed to minimize duplication and to help settle questions of attribution
Implicit a posteriori error estimates for the Maxwell equations
An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases. \u
International Conference on Nonlinear Differential Equations and Applications
Dear Participants, Colleagues and Friends
It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA).
This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics.
The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area
On the proof of Taylor's conjecture in multiply connected domains
In this Letter we extend the proof, by Faraco and Lindberg (2020), of Taylor's conjecture in multiply connected domains to cover arbitrary vector potentials and remove the need to impose restrictions on the magnetic field to ensure gauge invariance of the helicity integral. This extension allows us to treat general magnetic fields in closed domains that are important in laboratory plasmas and brings closure to a conjecture whose resolution has been open for almost 50 years. (C) 2021 Elsevier Ltd. All rights reserved.Peer reviewe
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
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