Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.     

Comparison of Taylor finite difference and window finite difference and their application in FDTD

The finite difference time domain (FDTD) method is an important tool in numerical electromagnetic simulation. There are many ways to construct a finite difference approximation such as the Taylor series expansion theorem, the filtering theory, etc. This paper aims to provide the comparison between the Taylor finite difference (TFD) scheme based on the Taylor series expansion theorem and the window finite difference (WFD) scheme based on the filtering theory. Their properties have been examined in detail, separately. In addition, the formula of the generalized finite difference (GFD) scheme is presented, which can include both the TFD scheme and the WFD scheme. Furthermore, their application in the numerical solution of Maxwell's equations is presented. The formulas for the stability criterion and the numerical dispersion relation are derived and analyzed. In order to evaluate their performance more accurately, a new definition of error is presented. Upon it, the effect of several factors including the grid resolution, the Courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined.     

High-order nonreflecting boundary conditions for the dispersive shallow water equations

Time-dependent dispersive shallow water waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary $\text{B}$, and a high-order non-reflecting boundary condition (NRBC) is imposed on $\text{B}$. Then the problem is solved by a finite difference scheme in the finite domain bounded by $\text{B}$. The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original low-order implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdon-type NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via a numerical example.     

Some useful filtering techniques for illposed problems

Some useful filtering techniques for computing approximate solutions of illposed are presented. Special attention is given to the role of smoothness of the filters and the choice of time-dependent parameters used in these filtering techniques. Smooth filters and proper choice of time-dependent parameters in these filtering techniques allow numerical construction of more accurate approximate solutions of illposed problems. In order to illustrate this and the filtering techniques, a severely illposed fourth-order nonlinear wave equation is numercally solved using a three time-level finite difference scheme. Numerical examples are given showing the merits of the filtering techniques.     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

Adaptive shock-diffusion model for restoration of degraded document images

This paper presents an adaptive enhancement model with selective smoothing for restoration of degraded document images with blur, noise and bleed-through, which involves adaptive shock filtering and selective diffusion processes. A novel hybrid scheme is developed to solve the proposed model numerically, which combines explicit finite difference and exponential smoothing. Numerical experiments show that the proposed model is very effective for restoration of degraded document images with blur, noise and bleed-through, and has averagely the best performance on the DIBCO (Document Image Binarization Competition) series datasets, compared to five PDE (partial differential equation)-based models for restoration of degraded document images.     

奇异摄动反应扩散方程的后验误差估计及自适应算法

研究了一类奇异摄动半线性反应扩散方程的自适应网格方法.在任意非均匀网格上建立迎风有限差分离散格式,并推导出离散格式的后验误差界,然后用该误差界设计自适应网格移动算法.数值实验结果证明了所提出的自适应网格方法的有效性.     

High order Hybrid central—WENO finite difference scheme for conservation laws

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.     

Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical chaotic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in finite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the effectiveness of the proposed control scheme.     

Adaptive solution of infinite linear systems by Krylov subspace methods

In this paper we consider the problem of approximating the solution of infinite linear systems, finitely expressed by a sparse coefficient matrix. We analyse an algorithm based on Krylov subspace methods embedded in an adaptive enlargement scheme. The management of the algorithm is not trivial, due to the irregular convergence behaviour frequently displayed by Krylov subspace methods for nonsymmetric systems. Numerical experiments, carried out on several test problems, indicate that the more robust methods, such as GMRES and QMR, embedded in the adaptive enlargement scheme, exhibit good performances.     

A comparative analysis of adaptive algorithms in the finite element method for solving the boundary value problem for a stationary reaction-diffusion equation

A new adaptive algorithm is proposed for constructing grids in the hp-version of the finite element method with piecewise polynomial basis functions. This algorithm allows us to find a solution (with local singularities) to the boundary value problem for a one-dimensional reaction-diffusion equation and smooth the grid solution via the adaptive elimination and addition of grid nodes. This algorithm is compared to one proposed earlier that adaptively refines the grid and deletes nodes with the help of an estimate for the local effect of trial addition of new basis functions and the removal of old ones. Results are presented from numerical experiments aimed at assessing the performance of the proposed algorithm on a singularly perturbed model problem with a smooth solution.     

Adaptive meshless refinement schemes for RBF-PUM collocation

In this paper we present an adaptive discretization technique for solving elliptic partial differential equations via a collocation radial basis function partition of unity method. In particular, we propose a new adaptive scheme based on the construction of an error indicator and a refinement algorithm, which used together turn out to be ad-hoc strategies within this framework. The performance of the adaptive meshless refinement scheme is assessed by numerical tests.     

Exact Solutions of Some Discrete Deconvolution Problems

In this work, we study the existence of solutions of the deconvolution problems in the discrete setting. More precisely, we prove the existence of solutions of the discrete multichannel deconvolution problems DMDP with convolvers being the characteristic functions of finite sets of positive integers. Also, we provide the reader with a simple method and a fast algorithm for finding the closed forms of the discrete deconvolvers with minimal supports that constitute exact solutions of the DMDP. Moreover, we show that unlike the singular value decomposition scheme, the multichannel deconvolution scheme based on the use of these discrete deconvolvers is not very sensitive to small 2-norm perturbation of the data. Finally, we show how to generalize our method for solving the 2-D version of the DMDP.     

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

Minimum-Time Travel for a Vehicle with Acceleration Limits: Theoretical Analysis and Receding-Horizon Implementation

A methodology is proposed to generate minimum-time optimal velocity profiles for a vehicle with prescribed acceleration limits along a specified path. The necessary optimality conditions are explicitly derived, allowing the construction of the optimal solution semianalytically. A receding horizon implementation is also proposed for the on-line implementation of the velocity optimizer. Robustness of the receding horizon algorithm is guaranteed by the use of an adaptive scheme that determines the planning and execution horizons. Application to a real-life scenario with a comparison between the infinite and finite receding horizon schemes provides a validation of the proposed methodology. This work has been supported in part by the US Army Research Office, Awards DAAD19-00-1-0473 and W911NF-05-1-0331. The authors thank an anonymous reviewer for his insightful comments regarding the results in Sect. 5.     

Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex Minimization

In this paper, we propose and analyze an algorithm that couples the gradient method with a general exterior penalization scheme for constrained or hierarchical minimization of convex functions in Hilbert spaces. We prove that a proper but simple choice of the step sizes and penalization parameters guarantees the convergence of the algorithm to solutions for the optimization problem. We also establish robustness and stability results that account for numerical approximation errors, discuss implementation issues and provide examples in finite and infinite dimension.     

Adaptive Wavelet Methods II—Beyond the Elliptic Case

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.     

A linear eigenvalue algorithm for the nonlinear eigenvalue problem

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted ${\mathcal {B}}$ . We consider the Arnoldi method for the operator ${\mathcal {B}}$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator ${\mathcal {B}}$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.     

Steklov eigenvalue problem and its applications An accurate a posteriori error estimator for the

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.