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This paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimal L2 error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.
In this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.
Inspired by the behavior of the blind for hill-climbing using a stick to detect a higher place by drawing a circle, we propose a heuristic direct search method to solve the unconstrained optimization problems. Instead of searching a neighbourhood of the current point as done in the traditional hill-climbing, or along specified search directions in standard direct search methods, the new algorithm searches on a surface with radius determined by the motion of the stick. The significant feature of the proposed algorithm is that it only has one parameter, the search radius, which makes the algorithm convenient in practical implementation. The developed method can shrink the search space to a closed ball, or seek for the final optimal point by adjusting search radius. Furthermore our algorithm possesses multi-resolution feature to distinguish the local and global optimum points with different search radii. Therefore, it can be used by itself or integrated with other optimization methods flexibly as a mathematical optimization technique. A series of numerical tests, including high-dimensional problems, have been well designed to demonstrate its performance.
In this work, we investigate wave propagation through a zero index meta-material (ZIM) waveguide embedded with triangular dielectric defects. We provide a theoretical guidance on how to achieve total reflection and total transmission (i.e., cloaking) by adjusting the defect sizes and/or permittivities of the defects. Our work provides a systematical way in manipulating wave propagation through ZIM in addition to the widely studied dielectric defects with cylindrical and rectangular geometries.
A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors in L2-norm and L∞-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.
A two-grid method for solving the Cahn-Hilliard equation is proposed in this paper. This two-grid method consists of two steps. First, solve the Cahn-Hilliard equation with an implicit mixed finite element method on a coarse grid. Second, solve two Poisson equations using multigrid methods on a fine grid. This two-grid method can also be combined with local mesh refinement to further improve the efficiency. Numerical results including two and three dimensional cases with linear or quadratic elements show that this two-grid method can speed up the existing mixed finite method while keeping the same convergence rate.
An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ = (1), and a special initial guess for λ ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ = (1) and λ ≪ 1. The (m + 1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.
In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.
In this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent order O(h2+k2) is obtained for the numerical solution in a discrete L2-norm. A numerical experiment is presented to test the theoretical result.
In this paper, the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated. Existence and uniqueness of the modeling equations are proved. Two fully discrete finite element schemes are proposed, and their practical implementation and stability are discussed.
The ordered patterns formed by microphase-separated block copolymer systems demonstrate periodic symmetry, and all periodic structures belong to one of 230 space groups. Based on this fact, a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is developed. In particular, the initial period of the computational box is estimated by the Landau-Brazovskii model as well. By planting the strategy into the whole-space discrete method, several new metastable patterns are discovered in diblock copolymers.
In this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.
In this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer(PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of H(grad) variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete H(curl)-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.
We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.
In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.
In this paper, we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem. We derive robust a posteriori error estimators for them and prove their reliability and efficiency.
In this paper, we discuss the a posteriori error estimates of the mixed finite element method for quadratic optimal control problems governed by linear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.
We present a novel adaptive finite element method (AFEM) for elliptic equations which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent gradient recovery. The constructions of CVT and its dual Centroidal Voronoi Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce almost equilateral two dimensional meshes. Working with finite element solutions on such high quality triangulations, superconvergent recovery methods become particularly effective so that asymptotically exact a posteriori error estimations can be obtained. Through a seamless integration of these techniques, a convergent adaptive procedure is developed. As demonstrated by the numerical examples, the new AFEM is capable of solving a variety of model problems and has great potential in practical applications.
A glass-ceramic (GC0) with nominal composition of 51.2% CaO–12.1% MgO–36.7% SiO2 (wt%) was synthesized. Then multiphase glass-ceramics of MGC1 and MGC2 were obtained by adding 1 and 2 wt% B2O3 to GC0 followed by thermal treatment. The bending strength of MGC1 was the highest, about 89.46 MPa, and the coefficient of thermal expansion was 10.67 × 10−6 °C−1, closer to that of Ti–6Al–4V alloy (10.03 × 10−6 °C−1). X-ray diffraction analysis confirmed that MGC1 was predominantly composed of akermanite, merwinite, and small amounts of dicalcium silicate crystalline phases. The bioactivity and cytocompatibility in vitro of MGC1 were detected by investigating the bonelike apatite-formation ability in simulated body fluid (SBF) and osteoblast morphology and viability. The results showed that MGC1 possessed bonelike apatite-formation ability in SBF and could release ionic products to significantly stimulate cell growth and viability. Furthermore, osteoblasts adhered and spread well on MGC1, indicating good bioactivity and potential cytocompatibility.
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