A Prolate-Element Method for Nonlinear PDEs on the Sphere

A p-type spectral-element method using prolate spheroidal wave functions (PSWFs) as basis functions, termed as the prolate-element method, is developed for solving partial differential equations (PDEs) on the sphere. The gridding on the sphere is based on a projection of the prolate-Gauss-Lobatto points by using the cube-sphere transform, which is free of singularity and leads to quasi-uniform grids. Various numerical results demonstrate that the proposed prolate-element method enjoys some remarkable advantages over the polynomial-based element method: (i) it can significantly relax the time step size constraint of an explicit time-marching scheme, and (ii) it can increase the accuracy and enhance the resolution.     

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r≥1 for a class of quasi-linear elliptic problems in Ω⊂ℝ². We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H ¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ ^d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Ω⊂ℝ³.     

A Parallel Method for Earth Mover’s Distance

We propose a new algorithm to approximate the Earth Mover’s distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar \(L_1\) type minimization. We use a regularization which gives us a unique solution for this \(L_1\) type problem. The new regularized minimization is very similar to problems which have been solved in the fields of compressed sensing and image processing, where several fast methods are available. In this paper, we adopt a primal-dual algorithm designed there, which uses very simple updates at each iteration and is shown to converge very rapidly. Several numerical examples are provided.     

A maximal monotone impact law for the 3-ball Newton’s cradle

The 3-ball Newton’s cradle is used as a stepping stone to divulge the structure of impact laws. A continuous conewise linear impact law that maps the preimpact contact velocities to the postimpact contact velocities is proposed for the 3-ball Newton’s cradle. The proposed impact law is kinematically, kinetically, and energetically consistent. It reproduces the outcomes of experimental observation. Moreover, it is in accordance with the outcome of the collision of three identical linear-elastic thin rods for which the impact process is governed by the one-dimensional wave equation. The proposed impact law is shown to be nonexpansive. Therefore, the relationship between the mean contact velocity and its dual, the impulsive force, is maximal monotone. A counterexample to maximal cyclical monotonicity of this relationship allows us to conclude that no dissipation function exists for the proposed impact law.     

A Newton method for Bernoulli’s free boundary problem in three dimensions

Summary The present paper is dedicated to the numerical solution of Bernoulli’s free boundary problem in three dimensions. We reformulate the given free boundary problem as a shape optimization problem and compute the shape gradient and Hessian of the given shape functional. To approximate the shape problem we apply a Ritz–Galerkin discretization. The necessary optimality condition is resolved by Newton’s method. All information of the state equation, required for the optimization algorithm, are derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results confirm that the proposed Newton method yields an efficient algorithm to treat the considered class of problems.      

An Asymptotic Preserving Method for Linear Systems of Balance Laws Based on Galerkin’s Method

We apply the concept of asymptotic preserving schemes (SIAM J Sci Comput 21:441–454, 1999) to the linearized \(p\) -system and discretize the resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. The fully discrete method is analyzed with respect to consistency, and we compare it numerically with more traditional methods such as Implicit Euler’s method.     

A Method for Minimization Design of Two-Level Logic Networks Using Multiplexer Universal Logic Modules

A decomposition approach of the combinational functions is discussed.A design method,by which the minimization or near minimization of two-level combinational network can be obtained,is presented for a combinational function realized by using multiplexer universal logic modules.Using the method,the automated synthesis of the combinational functions can be accomplished on a computer.     

Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.     

A Rectangular Finite Volume Element Method for a Semilinear Elliptic Equation

In this paper we extend the idea of interpolated coefficients for semilinear problems to the finite volume element method based on rectangular partition. At first we introduce bilinear finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. Next we derive convergence estimate in H ¹-norm and superconvergence of derivative. Finally an example is given to illustrate the effectiveness of the proposed method. This work is supported by Program for New Century Excellent Talents in University of China State Education Ministry, National Science Foundation of China, the National Basic Research Program under the Grant (2005CB321703), the key project of China State Education Ministry (204098), Scientific Research Fund of Hunan Provincial Education Department, China Postdoctoral Science Foundation (No. 20060390894) and China Postdoctoral Science Foundation (No. 20060390894).     

An Adaptive P 1 Finite Element Method for Two-Dimensional Maxwell’s Equations

Recently a new numerical approach for two-dimensional Maxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper we present an adaptive P ₁ finite element method for two-dimensional Maxwell’s equations that is based on this new approach. The reliability and efficiency of a posteriori error estimators based on the residual and the dual weighted-residual are verified numerically. The performance of the new approach is shown to be competitive with the lowest order edge element of Nédélec’s first family.     

Energy Stable Nodal Discontinuous Galerkin Methods for Nonlinear Maxwell’s Equations in Multi-dimensions

Journal of Scientific Computing - The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method...     

A Penalized Crouzeix–Raviart Element Method for Second Order Elliptic Eigenvalue Problems

In this paper we propose a penalized Crouzeix–Raviart element method for eigenvalue problems of second order elliptic operators. The key idea is to add a penalty term to tune the local approximation property and the global continuity property of discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, some of the resulted discrete eigenvalues are upper bounds of exact ones, and the others are lower bounds, and consequently a large portion of them can be reliable and approximate eigenvalues with high accuracy. Furthermore, we design an algorithm to select a penalty parameter which meets the condition. Finally we provide numerical tests to demonstrate the performance of the proposed method.     

Stochastic Gradient Method with Barzilai–Borwein Step for Unconstrained Nonlinear Optimization

Journal of Computer and Systems Sciences International - The use of stochastic gradient algorithms for nonlinear optimization is of considerable interest, especially in the case of high dimensions....     

A Nonlinear H∞ Set-point Control Method for Turbofan Engines with Disturbances

International Journal of Control, Automation and Systems - To provide desired control performance for turbofan engines subject to external disturbances, a nonlinear H∞ set-point control...     

EFAST Method for Global Sensitivity Analysisof Remote Sensing Model’s Parameters

Sensitivity analysis for parameters of remote sensing physical models is a prerequisite for inversion.The EFAST(Extended Fourier Amplitude Sensitivity Test)as a global sensitivity analysis method,can analyze not only a single parameter’s sensitivity but also the coupling effects among parameters.It is usually applied to analyse parameters’ sensitivity of the high-dimensional nonlinear models.In this paper,the SAIL model is taken as an example,the EFAST method and the field measured data of winter wheat in Shunyi district in 2001 were applied to analyze the model parameters’ sensitivity throughout the growing season and in different growth stages respectively.The results are compared with those of the USM (Uncertainty and Sensitivity Matrix) method.The results show that either the EFAST or the USM method for parameters’ sensitivity analysis of the SAIL model is feasible;but the EFAST method,which takes into account of the coupling effects among all the parameters and the analysis result is global,compared to the USM method,is more objective and comprehensive.     

On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.     

A Level-Set Method for Computing the Eigenvalues of Elliptic Operators Defined on Compact Hypersurfaces

We demonstrate, through separation of variables and estimates from the semi-classical analysis of the Schrödinger operator, that the eigenvalues of an elliptic operator defined on a compact hypersurface in ? ⁿ can be found by solving an elliptic eigenvalue problem in a bounded domain Ω?? ⁿ . The latter problem is solved using standard finite element methods on the Cartesian grid. We also discuss the application of these ideas to solving evolution equations on surfaces, including a new proof of a result due to Greer (J. Sci. Comput. 29(3):321–351, 2006).