2,673 research outputs found

### Numerical method for solving electromagnetic wave scattering by one and many small perfectly conducting bodies

In this paper, we investigate the problem of electromagnetic (EM) wave
scattering by one and many small perfectly conducting bodies and present a
numerical method for solving it. For the case of one body, the problem is
solved for a body of arbitrary shape, using the corresponding boundary integral
equation. For the case of many bodies, the problem is solved asymptotically
under the physical assumptions $a\ll d \ll \lambda$, where $a$ is the
characteristic size of the bodies, $d$ is the minimal distance between
neighboring bodies, $\lambda=2\pi/k$ is the wave length and $k$ is the wave
number. Numerical results for the cases of one and many small bodies are
presented. Error analysis for the numerical method are also provided

### Applications of potential theoretic mother bodies in Electrostatics

Any polyhedron accommodates a type of potential theoretic skeleton called a
mother body. The study of such mother bodies was originally from Mathematical
Physics, initiated by Zidarov and developed by Bj\"{o}rn Gustafson and Makoto
Sakai. In this paper, we attempt to apply the brilliant idea of mother body to
Electrostatics to compute the potentials of electric fields

### A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Particles

Scalar wave scattering by many small particles of arbitrary shapes with
impedance boundary condition is studied. The problem is solved asymptotically
and numerically under the assumptions a << d << lambda, where k = 2pi/lambda is
the wave number, lambda is the wave length, a is the characteristic size of the
particles, and d is the smallest distance between neighboring particles. A fast
algorithm for solving this wave scattering problem by billions of particles is
presented. The algorithm comprises the derivation of the (ORI) linear system
and makes use of Conjugate Orthogonal Conjugate Gradient method and Fast
Fourier Transform. Numerical solutions of the scalar wave scattering problem
with 1, 4, 7, and 10 billions of small impedance particles are achieved for the
first time. In these numerical examples, the problem of creating a material
with negative refraction coefficient is also described and a recipe for
creating materials with a desired refraction coefficient is tested

### Finite element analysis for identifying the reaction coefficient in PDE from boundary observations

This work is devoted to the nonlinear inverse problem of identifying the
reaction coefficient in an elliptic boundary value problem from single Cauchy
data on a part of the boundary. We then examine simultaneously two elliptic
boundary value problems generated from the available Cauchy data. The output
least squares method with the Tikhonov regularization is applied to find
approximations of the sought coefficient. We discretize the PDEs with piecewise
linear finite elements. The stability and convergence of this technique are
then established. A numerical experiment is presented to illustrate our
theoretical findings.Comment: Reaction coefficient identification, Finite element method, Boundary
observation, Tikhonov regularization, Neumann problem, Mixed problem,
Ill-posed proble

### Variational method for multiple parameter identification in elliptic PDEs

In the present paper we investigate the inverse problem of identifying
simultaneously the diffusion matrix, source term and boundary condition as well
as the state in the Neumann boundary value problem for an elliptic partial
differential equation (PDE) from a measurement data, which is weaker than
required of the exact state. A variational method based on energy functions
with Tikhonov regularization is here proposed to treat the identification
problem. We discretize the PDE with the finite element method and prove the
convergence as well as analyse error bounds of this approach.Comment: 22 pages, 13 figures; Journal of Mathematical Analysis and
Applications, 201

### Matrix Coefficient Identification in an Elliptic Equation with the Convex Energy Functional Method

In this paper we study the inverse problem of identifying the diffusion
matrix in an elliptic PDE from measurements. The convex energy functional
method with Tikhonov regularization is applied to tackle this problem. For the
discretization we use the variational discretization concept, where the PDE is
discretized with piecewise linear, continuous finite elements. We show the
convergence of approximations. Using a suitable source condition, we prove an
error bound for discrete solutions. For the numerical solution we propose a
gradient-projection algorithm and prove the strong convergence of its iterates
to a solution of the identification problem. Finally, we present a numerical
experiment which illustrates our theoretical results

### A reaction coefficient identification problem for fractional diffusion

We analyze a reaction coefficient identification problem for the spectral
fractional powers of a symmetric, coercive, linear, elliptic, second-order
operator in a bounded domain $\Omega$. We realize fractional diffusion as the
Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the
semi-infinite cylinder $\Omega \times (0,\infty)$. We thus consider an
equivalent coefficient identification problem, where the coefficient to be
identified appears explicitly. We derive existence of local solutions,
optimality conditions, regularity estimates, and a rapid decay of solutions on
the extended domain $(0,\infty)$. The latter property suggests a truncation
that is suitable for numerical approximation. We thus propose and analyze a
fully discrete scheme that discretizes the set of admissible coefficients with
piecewise constant functions. The discretization of the state equation relies
on the tensorization of a first-degree FEM in $\Omega$ with a suitable $hp$-FEM
in the extended dimension. We derive convergence results and obtain, under the
assumption that in neighborhood of a local solution the second derivative of
the reduced cost functional is coercive, a priori error estimates

### Finite element approximation of source term identification with TV-regularization

In this paper we investigate the problem of recovering the source term in an
elliptic system from a measurement of the state on a part of the boundary. For
the particular interest in reconstructing probably discontinuous sources, we
use the standard least squares method with the total variation regularization.
The finite element method is then applied to discretize the minimization
problem, we show the stability and the convergence of this technique.
Furthermore, we have proposed an algorithm to stably solve the minimization
problem. We prove the iterate sequence generated by the derived algorithm
converging to a minimizer of the regularization problem, and that convergence
measurement is also established. Finally, a numerical experiment is presented
to illustrate our theoretical findings.Comment: Inverse source problem, boundary observation, total variation
regularization, ill-posedness, finite element method, stability and
convergence, elliptic boundary value proble

### Equations defining recursive extensions as set theoretic complete intersections

Based on the fact that projective monomial curves in the plane are complete
intersections, we give an effective inductive method for creating infinitely
many monomial curves in the projective $n$-space that are set theoretic
complete intersections. We illustrate our main result by giving different
infinite families of examples. Our proof is constructive and provides one
binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces
cutting out the curve in question

### Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data

The aim of this paper is to present the global estimate for gradient of
renormalized solutions to the following quasilinear elliptic problem:
\begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \
\Omega, \\ u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*}
in Lorentz-Morrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$),
$\mu$ is a finite Radon measure, $A$ is a monotone Carath\'eodory vector valued
function defined on $W^{1,p}_0(\Omega)$ and the $p$-capacity uniform thickness
condition is imposed on the complement of our domain $\Omega$. It is remarkable
that the local gradient estimates has been proved firstly by G. Mingione in
\cite{Mi3} at least for the case $2 \le p \le n$, where the idea for extending
such result to global ones was also proposed in the same paper. Later, the
global Lorentz-Morrey and Morrey regularities were obtained by N.C.Phuc in
\cite{55Ph1} for regular case $p>2 - \frac{1}{n}$. Here in this study, we
particularly restrict ourselves to the singular case $\frac{3n-2}{2n-1}<p\le
2-\frac{1}{n}$. The results are central to generalize our technique of
good-$\lambda$ type bounds in previous work \cite{MP2018}, where the local
gradient estimates of solution to this type of equation was obtained in the
Lorentz spaces. Moreover, the proofs of most results in this paper are
formulated globally up to the boundary results.Comment: arXiv admin note: substantial text overlap with arXiv:1807.1051

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