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Estimates for the combined effect of boundary
approximation and numerical integration on the approximation of
(simple) eigenvalues and eigenvectors of 4th order
eigenvalue problems with variable/constant coefficients
in convex domains with curved boundary by an isoparametric mixed
finite element method, which,
in the particular case of bending problems of
aniso-/ortho-/isotropic plates with variable/constant
thickness, gives a simultaneous approximation to bending moment
tensor field $\Psi= (\psi_{ij})_{1 \le i,j \le 2}$ and
displacement field `u', have been developed.
Using systematically a tricky idea of N.V. Krylov, we obtain
general results on the rate of convergence of a certain class of
monotone approximation schemes for stationary
Hamilton-Jacobi-Bellman equations with variable coefficients.
This result applies in particular to control schemes based on the
dynamic programming principle and to finite difference schemes
despite, here, we are not able to treat the most general case.
General results have been obtained earlier by Krylov for
finite difference schemes in the stationary case with constant
coefficients and in the time-dependent case with variable
coefficients by using control theory and probabilistic methods.
In this paper we are able to handle variable coefficients by a
purely analytical method. In our opinion this way is far simpler
and, for the cases we can treat, it yields a better rate of
convergence than Krylov obtains in the variable coefficients case.
We study the asymptotic behavior of a semi-discrete numerical
approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary
conditions $\frac{\partial u}{\partial\eta} =
u^{p_{11}}v^{p_{12}}$, $\frac{\partial v}{\partial\eta} =
u^{p_{21}}v^{p_{22}}$ on ∂Ω x (0,T), where
Ω is a bounded smooth domain in ${\mathbb{R}}^d$. We focus in the
existence or not of non-simultaneous blow-up for a semi-discrete
approximation (U,V). We prove that if U blows up in finite time
then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1)
, which is the same condition as the one
for non-simultaneous blow-up in the continuous problem. Moreover,
we find that if the continuous
problem has non-simultaneous blow-up then the same is true for
the discrete one. We also prove some
results about the convergence of the scheme and the convergence
of the blow-up times.
This paper is concerned with the coupling of two models for the
propagation of particles in scattering media. The first model is a
linear transport equation of Boltzmann type posed in the phase space
(position and velocity). It accurately describes the physics but is
very expensive to solve. The second model is a diffusion equation
posed in the physical space. It is only valid in areas of high
scattering, weak absorption, and smooth physical coefficients, but
its numerical solution is much cheaper than that of transport. We
are interested in the case when the domain is diffusive everywhere
except in some small areas, for instance non-scattering or
oscillatory inclusions. We present a natural coupling of the two
models that accounts for both the diffusive and non-diffusive
regions. The interface separating the models is chosen so that the
diffusive regime holds in its vicinity to avoid the calculation of
boundary or interface layers. The coupled problem is analyzed
theoretically and numerically. To simplify the presentation, the
transport equation is written in the even parity form. Applications
include, for instance, the treatment of clear or spatially
inhomogeneous regions in near-infra-red spectroscopy, which is
increasingly being used in medical imaging for monitoring certain
properties of human tissues.
In this work, we investigate the Perfectly
Matched Layers (PML)
introduced by Bérenger [3] for designing
efficient numerical absorbing
layers in electromagnetism.
We make a mathematical analysis of this model, first via a modal
analysis with standard Fourier techniques, then via energy
techniques. We obtain uniform in time stability results (that make
precise some results known in the literature) and state some energy
decay results that illustrate the absorbing properties of the
model. This last technique allows us to prove the stability of the
Yee's scheme for discretizing PML's.
A modal synthesis method to solve the elastoacoustic vibration problem
is analyzed. A two-dimensional coupled fluid-solid system is considered;
the solid is described by displacement variables, whereas displacement
potential is used for the fluid. A particular modal synthesis leading to
a symmetric eigenvalue problem is introduced. Finite element discretizations
with Lagrangian elements are considered for solving the uncoupled problems.
Convergence for eigenvalues and eigenfunctions is proved,
error estimates are given, and numerical experiments exhibiting the good
performance of the method are reported.
This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996).
We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators
(Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).