The interior low-frequency electromagnetic dipole excitation of a dielectric sphere is utilized as a simplified but realistic model in various biomedical applications. Motivated by these considerations, in this paper, we investigate analytically a near-field inverse scattering problem for the electromagnetic interior dipole excitation of a dielectric sphere. First, we obtain, under the low-frequency assumption, a closed-form approximation of the series of the secondary electric field at the dipole's location. Then, we utilize this derived approximation in the development of a simple inverse medium scattering algorithm determining the sphere's dielectric permittivity. Finally, we present numerical results for a human head model, which demonstrate the accurate determination of the complex permittivity by the developed algorithm.

The paper is concerned with the inverse problem for reconstructing a 3D penetrable object in a shallow water waveguide from the far-field data of the scattered fields with many acoustic point source incidences. An indicator sampling method is analyzed and presented for fast imaging the size, shape and location of such a penetrable object. The method has the advantages that a priori knowledge is avoided for the geometrical and material properties of the penetrable obstacle and the many complicated iterative techniques are avoided during the inversion. Numerical examples are given of successful shape reconstructions for several 3D penetrable obstacles having a variety of shapes. In particular, numerical results show that the proposed method is able to produce a good reconstruction of the size, shape and location of the penetrable target even for the case where the incident and observation points are restricted to some limited apertures.

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two-grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size $H$, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size $h = O(H^2)$ or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size $h = O(|log h|^{1/2}H^3)$. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size $h$. Some numerical results are presented to demonstrate the efficiency of the proposed methods.

We analyze finite volume schemes of arbitrary order $r$ for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as $(N^{-1}ln(N+1))^r$, where $2N$ is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order $(N^{-1}ln(N+1))^{2r}$, while at the Gauss points, the derivative error super-converges with order $(N^{-1}ln(N+1))^{r+1}$. All the above convergence and superconvergence properties are independent of the perturbation parameter $ε$. Numerical results are presented to support our theoretical findings.

In this article we propose an overlapping Schwarz domain decomposition method for solving a singularly perturbed semilinear reaction-diffusion problem. The solution to this problem exhibits boundary layers of width $O(\sqrt{\varepsilon}ln(1/\sqrt{\varepsilon}))$ at both ends of the domain due to the presence of singular perturbation parameter $\varepsilon$. The method splits the domain into three overlapping subdomains, and uses the Numerov or Hermite scheme with a uniform mesh on two boundary layer subdomains and a hybrid scheme with a uniform mesh on the interior subdomain. The numerical approximations obtained from this method are proved to be almost fourth order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Furthermore, it is proved that, for small $\varepsilon$, one iteration is sufficient to achieve almost fourth order uniform convergence. Numerical experiments are given to illustrate the theoretical order of convergence established for the method.

In this paper, we estimate the constants in the inverse inequalities for the finite element functions. Furthermore, we obtain the least upper bounds of the constants in inverse inequalities for the low-order finite element functions. Such explicit estimates of the constants can be used as computable error bounds for the finite element method.

In this paper, we propose a local multilevel preconditioner for the mortar finite element approximations of the elliptic problems. With some mesh assumptions on the interface, we prove that the condition number of the preconditioned systems is independent of the large jump of the coefficients but depends on the mesh levels around the cross points. Some numerical experiments are presented to confirm our theoretical results.