High convergence rates of digital image transformation by numerical integration using spline functions

This paper introduces an intriguing topic in image processing where accuracy of images even in details is important and adopts an intriguing methodology dealing with discrete topics by continuous mathematics and numerical approximation. The key idea is that a pixel of images at different levels can be quantified by a greyness value, which can then be regarded as the mean of an integral of continuous functions over a certain region, and evaluated by numerical integration approximately. However, contrasted to the traditional integration, the integrand has different smooth nature in different subregions due to piecewise interpolation and approximation. New treatments of approximate integration and new discrete algorithms of images have been developed.The cycle conversion T⁻¹T of image transformations is said if an image is distorted by a transformation T and then restored back to itself by the inverse transformation T⁻¹. Combined algorithms of discrete techniques proposed in [1–3] only have the convergence rates O(1/N) and O(1/N²) of sequential greyness errors, where N is the division number for a pixel split into N² subpixels. This paper reports new combination algorithms using spline functions to achieve the high convergence rates O(1/N³) and O(1/N⁴) of digital image transformations under the cycle conversion. Both error analysis and numerical experiments have been provided to verify the high convergence rates. High convergence rates of discrete algorithms are important in saving CPU time, particularly to multi-greyness images. Moreover, the computational figures for real images of 256 × 256 with 256 greyness levels, in which N = 2 is good enough for practical requirements, display validity, and effectiveness of the new algorithms in this paper.     

A priori error estimation of a four-node Reissner–Mindlin plate element for elasto-dynamics

The explicit finite element method for transient dynamics of linear elasticity by Reissner–Mindlin plate model is introduced. For clamped rectangular plate, the a priori error estimates are derived for the four-node Bathe–Dvorkin element. For fixed thickness, the convergence rates of deflection, rotation, and their velocities, measured both in H¹-norm and L²-norm, can possibly all be optimal under certain conditions. In some cases, the numerical examples show that the convergence rate stays optimal for a certain range of thickness. In other cases, however, the deterioration in rate of convergence and even locking may occur to the velocity terms.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability ∝t^−α, but independent of the dimension, where α is the anomalous diffusion exponent.     

A galerkin method for a nonlinear integro-differential wave system

A Galerkin method is introduced for the numerical solution of a system equivalent to the equation ω_tt = (1 +∝₀^π ω_x²dx)ω_xx which models the vibrations of a string when the increase in tension due to extension is taken into account. We prove the convergence of the semidiscrete approximations and present some numerical tests.     

On convergence of nonconforming convex quadrilateral finite elements AGQ6

The 4-node quadrilateral membrane elements AGQ6-I and II are two novel incompatible models formulated by the Quadrilateral Area Coordinate Method (QACM). In this paper, the sufficient conditions for their convergence are established. It is further shown theoretically that the convergence in the energy norm is linear, and the convergence in the L² norm is quadratic provided that certain geometric conditions are met requiring asymptotically parallelograms meshes to be used. The necessity of conditions is also discussed. The results of numerical examples completely confirm the theoretical findings.     

An analysis of some mixed-enhanced finite element for plane linear elasticity

The paper investigates Mixed-Enhanced Strain finite elements developed within the context of the u/p formulation for nearly incompressible linear elasticity problems. A rigorous convergence and stability analysis is detailed, providing also L²-error estimates for the displacement field. Extensive numerical tests are developed, showing in particular the accordance of the computational results with the theoretical predictions.     

Analysis of an Interior Penalty Method for Fourth Order Problems on Polygonal Domains

Error analysis for a stable C ⁰ interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H ², H ¹ and L ₂ norms. L _?? norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments.     

Covolume method for new nonconforming rectangle element for the Stokes problem

We analyze a covolume method based on the new nonconforming element introduced by Douglas et al. [1]. We show the H¹ optimal order convergence of the scheme for Stokes problem and study the hybrid domain decomposition procedure for this covolume scheme. The numerical experiment shows that the covolume scheme is somewhat better than finite element scheme in the computation of pressure.     

Legendre Spectral Collocation Methods for Pantograph Volterra Delay-Integro-Differential Equations

This paper is concerned with the convergence properties of the Legendre spectral collocation methods when used to approximate smooth solutions of Volterra integro-differential equations with proportional (vanishing) delays. We provide a vigorous error analysis for the proposed methods. Furthermore, we prove that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in L ²-norm and L ^??-norm. Some numerical experiments are given to confirm the theoretical results.     

On a numerical Liapunov-Schmidt method for operator equations

Let, for a higher singular pointx ₀ of an operator equationG(x ₀)=0 and the kernels of the respective derivativesG′(x ₀) andG′(x ₀)^*, see [1], approximations be available. We present a method to numerically compute the manifolds bifurcating atx ₀. In particular, the question of convergence of the numerical to the exact solution is studied by proving stability and convergence for solution parts of different order of magnitude. Different approaches are presented and applied to elliptic problems.     

A numerical method for exact boundary controllability problems for the wave equation

The computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied. A numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem. The uniqueness of the discrete finite-difference solutions obtained in this manner is demonstrated. The convergence properties of the method are illustrated through computational experiments. Efficient implementation strategies for the method are also discussed. It is shown that for smooth, minimum L²-norm Dirichlet controls, the method results in convergent approximations without the need to introduce regularization. Furthermore, for the generic case of nonsmooth Dirichlet controls, convergence with respect to L² norms is also numerically demonstrated. One of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria; such generalizations are discussed. In particular, the minimum H¹-norm Dirichlet controllability problem is approximated and solved, as are minimum regularized L²-norm Dirichlet controllability problems with small penalty constants. Finally, a discussion is provided about the differences between our method and existing methods; these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner.     

Locking-Free Optimal Discontinuous Galerkin Methods for a Naghdi-Type Arch Model

In this paper, we introduce and analyze discontinuous Galerkin methods for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking. We also prove that, when polynomials of degree k are used, all the numerical traces superconverge with a rate of order h ^2k+1. Numerical experiments verifying the above-mentioned theoretical results are displayed.     

Accelerazione della convergenza per metodi di risoluzione di equazioni differenziali ordinarie

In this paper a procedure for the acceleration of the convergence is given. It allows the doubling of the order of the multistep methods for the numerical solution of the ordinary differential equation $$y' = f(x,y),y_0 = y(x_0 );{}_{x_0 }^x \in [a,b].$$ This acceleration is applicable to any method of orderp≥1 whatsoever, and it requires the evaluation of the globalp-th derivate of the functionf(x, y). Special attention is confined to the 2⁰ and 3⁰ order methods, and a numerical exemple is provided.     

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

We numerically verify that the non-symmetric interior penalty Galerkin method and the Oden-Babus?ka-Baumann method have sub-optimal convergence properties when measured in the L ²-norm for odd polynomial approximations. We provide numerical examples that use piece-wise linear and cubic polynomials to approximate a second-order elliptic problem in one and two dimensions.     

Discontinuous bubble scheme for elliptic problems with jumps in the solution

We propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface. By considering a suitable function which has the same jumps as the solution, we transform the problem into one without jumps. Then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in [1], [2], [3]. Some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested. Our method can also handle the case when the interface passes through grid points. We believe this paper presents the first resolution of such cases. Numerical experiments for various problems show second-order convergence in L² and first order in H¹-norms. Moreover, the convergence order is very robust for all problems tested.     

Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p=4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et al. (Numer. Math. 116:317–356, 2010), we add a vanishing numerical viscosity term to a semi-discrete scheme so as to preserve similar properties of stability, and thus able to prove the convergence in L ⁴-strong. The semi-discretization of the spatial structure via central finite difference method yields a stiff system of ODE. Hence, for the temporal discretization, we resort to the two-stage implicit Runge-Kutta scheme of the Gauss-Legendre type. The resulting system is unconditionally stable and possesses favorable nonlinear properties. On the other hand, despite the formation of blow up for the critical case of GKdV, it is known that a localized damping term added to the GKdV-4 equation leads to the exponential decay of the energy for small enough initial conditions, which is interesting from the standpoint of the Control Theory. Then, combining the result of convergence in L ⁴-strong with discrete multipliers and a contradiction argument, we show that the presence of the vanishing numerical viscosity term allows the uniform (with respect to the mesh size) exponential decay of the total energy associated to the semi-discrete scheme of higher-order in space with the localized damping term. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.     

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form Δt∝²(Δx) for explicit calculations. A new semi-implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It is based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.     

New high-order convergence iteration methods without employing derivatives for solving nonlinear equations

A family of new iteration methods without employing derivatives is proposed in this paper. We have proved that these new methods are quadratic convergence. Their efficiency is demonstrated by numerical experiments. The numerical experiments show that our algorithms are comparable to well-known methods of Newton and Steffensen. Furthermore, combining the new method with bisection method we construct another new high-order iteration method with nice asymptotic convergence properties of the diameters (b_n − a_n).     

A comparison of point and ball iterations in the contractive mapping case

In applications, one of the basic problems is to solve the fixed point equationx=Tx withT a contractive mapping. Two theorems which can be implemented computationally to verify the existence of a solutionx ^* to the equation and to obtain a convergent approximate solution sequence {x _n } are the classical Banach contraction mapping theorem and the newly established global convergence theorem of the ball algorithms in You, Xu and Liu [16]. These two theorems are compared on the basis of sensitivity, precision, computational complexity and efficiency. The comparison shows that except for computational complexity, the latter theorem is of far greater sensivity, precision and computational efficiency. This conclusion is supported by a number of numerical examples.