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