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We consider solutions to the time-harmonic Maxwell's Equations
of a TE (transverse electric) nature. For such solutions we provide
a rigorous derivation of the leading order boundary perturbations
resulting from the presence of a finite number of interior inhomogeneities
of small diameter. We expect that these formulas will form the basis for
very effective computational identification algorithms, aimed at determining
information about the inhomogeneities from electromagnetic
The present work is a mathematical analysis of two algorithms, namely
the Roothaan and the level-shifting algorithms, commonly used in
practice to solve the Hartree-Fock equations. The level-shifting
algorithm is proved to be well-posed and to converge provided the shift
parameter is large enough. On the contrary, cases when the Roothaan
algorithm is not well defined or fails in converging are
exhibited. These mathematical results are confronted to numerical
experiments performed by chemists.
The phase relaxation model is a diffuse interface model with
small parameter ε which
consists of a parabolic PDE for temperature
θ and an ODE with double obstacles
for phase variable χ.
To decouple the system a semi-explicit Euler method with variable
step-size τ is used for time discretization, which requires
the stability constraint τ ≤ ε. Conforming piecewise
linear finite elements over highly graded simplicial meshes
with parameter h are further employed for space discretization.
A posteriori error
estimates are derived for both unknowns θ and χ, which
exhibit the correct asymptotic order in terms of ε, h and
τ. This result circumvents the use of duality, which does not
even apply in this context.
Several numerical experiments illustrate the reliability of the
estimators and document the excellent performance of the ensuing
In dimension one it is proved that the solution to a total variation-regularized
least-squares problem is always a function which is "constant almost everywhere" ,
provided that the data are in a certain sense outside the range of the operator
to be inverted. A similar, but weaker result is derived in dimension two.
We discuss the stability of "critical" or "equilibrium" shapes of
a shape-dependent energy functional. We analyze a problem arising when
looking at the positivity of the second derivative in order to prove
that a critical shape is an optimal shape. Indeed, often when
positivity -or coercivity- holds, it does for a weaker norm than the
norm for which the functional is twice differentiable and local
optimality cannot be a priori deduced. We solve this problem for a
particular but significant example. We prove "weak-coercivity" of
the second derivative uniformly in a "strong" neighborhood of the
We are interested in the theoretical study of a spectral problem
arising in a physical situation, namely interactions of fluid-solid
type structure. More precisely, we study the existence of solutions
for a quadratic eigenvalue problem, which describes the vibrations of a
system made up of two elastic bodies, where a slip is allowed on their
interface and which surround a cavity full of an inviscid
and slightly compressible fluid. The problem shall be treated like a
generalized eigenvalue problem. Thus by using some functional analysis
results, we deduce the existence of solutions and prove a spectral
asymptotic behavior property, which allows us to compare the spectrum
of this coupled model and the spectrum associated to the problem without
transmission between the fluid-solid media.
Hermite polynomial interpolation is investigated.
Some approximation results are obtained. As an example, the Burgers
equation on the whole line is considered. The stability and the
convergence of proposed Hermite pseudospectral scheme are proved
strictly. Numerical results are presented.
In this paper, we study the long wave approximation for quasilinear
symmetric hyperbolic systems. Using the technics developed by
Joly-Métivier-Rauch for nonlinear geometrical optics, we prove that
under suitable assumptions the long wave limit is described by
KdV-type systems. The error estimate if the system is coupled appears to
be better. We apply formally our technics to Euler equations with free
surface and Euler-Poisson systems. This leads to new systems of KdV-type.