The concept of elastic moment tensor occurs in several interesting contexts, in particular in imaging small elastic inclusions and in asymptotic models of dilute elastic composites. In this paper, we compute the elastic moment tensors for ellipses and ellipsoids by using a systematic method based on layer potentials. Our computations reveal an underlying elegant relation between the elastic moment tensors and the single layer potential.

In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.

The main aim of this paper is to show that the quadrilateral mesh condition $RDP(N,\psi)$ is only sufficient but not necessary for the optimal order error estimate of the $Q_{1}$ isoparametric element in the $H^{1}$ norm.

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear Schrödinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.

We consider a Second Harmonic Generation (SHG) problem of an optical signal wave with an optical pump in a medium represented by a smooth bounded domain $\Omega\subset \mathbb{R}^d $, which is assumed to contain a heterogeneous material: a compactly imbedded subdomain $B^r\subset \subset \Omega$ in the shape of a small ball contains a nonlinear material, while $\Omega\setminus \overline {B^r}$ is filled with a linear material. We begin by proving existence and uniqueness of the solution to the TE approximation of SHG for arbitrary bounded susceptibilities, thus improving the result obtained by Bao and Dobson ( Eur. J. Appl. Math. 6 (1995), 573-590) under small enough susceptibilities assuption. We then establish an existence and uniqueness result of a solution to the TM approximation problem. In both parts we study the asymptotic behavior of the system as the size of the nonlinear material vanishes: error estimates and asymptotic expansion of the solution are derived for both TE and TM approximations.

We present the Crouzeix-Raviart linear nonconforming finite element approximation of the variational inequality resulting from Signorini problem. We show if the displacement field is of $H^2$ regularity, then the convergence rate can be improved from $\mathcal{O}(h^{3/4})$ to quasi-optimal $\mathcal{O}(h|\log h|^{1/4})$ with respect to the energy norm as that of the continuous linear finite element approximation. If stronger but reasonable regularity is available, the convergence rate can be improved to the optimal $\mathcal{O}(h)$ as expected by the linear approximation.

The analytic and discretized dissipativity of nonlinear infinite-delay systems of the form $x'(t)=g(x(t),x(qt)) (q\in (0,1),t›0)$ is investigated. A sufficient condition is presented to ensure that the above nonlinear system is dissipative. It is proved that the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.

In this paper, we propose an algorithm for solving nonlinear monotone equations by combining the limited memory BFGS method (L-BFGS) with a projection method. We show that the method is globally convergent if the equation involves a Lipschitz continuous monotone function. We also present some preliminary numerical results.

In this paper, we propose a new approach to solve the approximate implicitization problem based on RBF networks and MQ quasi-interpolation. This approach possesses the advantages of shape preserving, better smoothness, good approximation behavior and relatively less data etc. Several numerical examples are provided to demonstrate the effectiveness and flexibility of the proposed method.

In this paper, a nonmonotone method based on McCormick's second-order Armijo's step-size rule [7] for unconstrained optimization problems is proposed. Every limit point of the sequence generated by using this procedure is proved to be a stationary point with the second-order optimality conditions. Numerical tests on a set of standard test problems are presented and show that the new algorithm is efficient and robust.