Advances in Numerical Analysis

Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems

Bishnu P. Lamichhane — Thu, 11 Apr 2013 17:54:10 +0000

We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efficient numerical schemes are proposed based on using a pair of bases for the gradient of the solution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems, respectively. We also establish an optimal a priori error estimate for both finite element approximations.

Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

Gustavo Fernández-Torres and Juan Vásquez-Aquino — Sun, 24 Mar 2013 09:02:14 +0000

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

Arnak Poghosyan — Thu, 21 Mar 2013 18:51:45 +0000

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

Ababu Teklemariam Tiruneh, W. N. Ndlela, and S. J. Nkambule — Tue, 19 Mar 2013 13:53:34 +0000

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.

Modified Bézier Curves with Shape-Preserving Characteristics Using Differential Evolution Optimization Algorithm

Mohammad Asif Zaman and Shuvro Chowdhury — Wed, 13 Mar 2013 17:54:31 +0000

A parametric equation for a modified Bézier curve is proposed for curve fitting applications. The proposed equation contains shaping parameters to adjust the shape of the fitted curve. This flexibility of shape control is expected to produce a curve which is capable of following any sets of discrete data points. A Differential Evolution (DE) optimization based technique is proposed to find the optimum value of these shaping parameters. The optimality of the fitted curve is defined in terms of some proposed cost parameters. These parameters are defined based on sum of squares errors. Numerical results are presented highlighting the effectiveness of the proposed curves compared with conventional Bézier curves. From the obtained results, it is observed that the proposed method produces a curve that fits the data points more accurately.

Nonlinear Finite Element Analysis of Sloshing

Siva Srinivas Kolukula and P. Chellapandi — Wed, 27 Feb 2013 13:31:19 +0000

The disturbance on the free surface of the liquid when the liquid-filled tanks are excited is called sloshing. This paper examines the nonlinear sloshing response of the liquid free surface in partially filled two-dimensional rectangular tanks using finite element method. The liquid is assumed to be inviscid, irrotational, and incompressible; fully nonlinear potential wave theory is considered and mixed Eulerian-Lagrangian scheme is adopted. The velocities are obtained from potential using least square method for accurate evaluation. The fourth-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is employed to avoid numerical instabilities. Regular harmonic excitations and random excitations are used as the external disturbance to the container. The results obtained are compared with published results to validate the numerical method developed.

Modular Analysis of Sequential Solution Methods for Almost Block Diagonal Systems of Equations

Tarek M. A. El-Mistikawy — Sun, 24 Feb 2013 07:56:20 +0000

Almost block diagonal linear systems of equations can be exemplified by two modules. This makes it possible to construct all sequential forms of band and/or block elimination methods. It also allows easy assessment of the methods on the basis of their operation counts, storage needs, and admissibility of partial pivoting. The outcome of the analysis and implementation is to discover new methods that outperform a well-known method, a modification of which is, therefore, advocated.

Parallel Nonoverlapping DDM Combined with the Characteristic Method for Incompressible Miscible Displacements in Porous Media

Keying Ma and Tongjun Sun — Tue, 19 Feb 2013 09:51:26 +0000

Two types of approximation schemes are established for incompressible miscible displacements in porous media. First, standard mixed finite element method is used to approximate the velocity and pressure. And then parallel non-overlapping domain decomposition methods combined with the characteristics method are presented for the concentration. These methods use the characteristic method to handle the material derivative term of the concentration equation in the subdomains and explicit flux calculations on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the velocity and pressure can be approximated simultaneously, and the parallelism can be achieved for the concentration equation. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability. These schemes hold the advantages of nonoverlapping domain decomposition methods and the characteristic method. Optimal error estimates in -norm are derived for these two schemes, respectively.

New Approach for Solving a Class of Doubly Singular Two-Point Boundary Value Problems Using Adomian Decomposition Method

Randhir Singh, Jitendra Kumar, and Gnaneshwar Nelakanti — Wed, 26 Dec 2012 14:35:44 +0000

We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM) and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results.

Solution of Nonlinear Volterra-Fredholm Integrodifferential Equations via Hybrid of Block-Pulse Functions and Lagrange Interpolating Polynomials

Hamid Reza Marzban and Sayyed Mohammad Hoseini — Mon, 10 Dec 2012 12:42:49 +0000

An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results.

A Note on Fourth Order Method for Doubly Singular Boundary Value Problems

R. K. Pandey and G. K. Gupta — Wed, 05 Dec 2012 17:29:24 +0000

We present a fourth order finite difference method for doubly singular boundary value problem with boundary conditions and , where , , and are finite constants. Here and is allowed to be discontinuous at the singular point . The method is based on uniform mesh. The accuracy of the method is established under quite general conditions and also corroborated through one numerical example.

Discrete Gamma (Factorial) Function and Its Series in Terms of a Generalized Difference Operator

G. Britto Antony Xavier, V. Chandrasekar, S. U. Vasanthakumar, and B. Govindan — Tue, 13 Nov 2012 15:01:55 +0000

The recent theory and applications of difference operator introduced in (M. Maria Susai Manuel et al., 2012) are enriched and extended with a useful tool for finding the values of various series of discrete gamma functions in number theory. Illustrative examples show the effectiveness of the obtained results in finding the values of various gamma series.

The Optimal Error Estimate of Stabilized Finite Volume Method for the Stationary Navier-Stokes Problem

Guoliang He, Jian Su, and Wenqiang Dai — Sun, 11 Nov 2012 07:27:54 +0000

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier-Stokes equations is analyzed. For the element, we obtain the optimal error estimates of the finite volume solution and . We also provide some numerical examples to confirm the efficiency of the FVM. Furthermore, the effect of initial value for iterative method is analyzed carefully.

Preservation of Fine Structures in PDE-Based Image Denoising

Hakran Kim, Velinda R. Calvert, and Seongjai Kim — Sun, 21 Oct 2012 11:48:16 +0000

Image denoising processes often lead to significant loss of fine structures such as edges and textures. This paper studies various innovative mathematical and numerical methods applicable for conventional PDE-based denoising models. The method of diffusion modulation is considered to effectively minimize regions of undesired excessive dissipation. Then we introduce a novel numerical technique for residual-driven constraint parameterization, in order for the resulting algorithm to produce clear images whose corresponding residual is as free of image textures as possible. A linearized Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to simulate the resulting model efficiently. Various examples are presented to show efficiency and reliability of the suggested methods in image denoising.

An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

Rajni Sharma and Janak Raj Sharma — Thu, 18 Oct 2012 09:13:49 +0000

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

Convergence of an Eighth-Order Compact Difference Scheme for the Nonlinear Schrödinger Equation

Tingchun Wang — Wed, 17 Oct 2012 14:43:42 +0000

A new compact difference scheme is proposed for solving the nonlinear Schrödinger equation. The scheme is proved to conserve the total mass and the total energy and the optimal convergent rate, without any restriction on the grid ratio, at the order of in the discrete -norm with time step τ and mesh size h. In numerical analysis, beside the standard techniques of the energy method, a new technique named “regression of compactness” and some lemmas are proposed to prove the high-order convergence. For computing the nonlinear algebraical systems generated by the nonlinear compact scheme, an efficient iterative algorithm is constructed. Numerical examples are given to support the theoretical analysis.

Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

R. Ingram, C. C. Manica, N. Mays, and I. Stanculescu — Thu, 27 Sep 2012 08:34:51 +0000

We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above.

Further Development of Jarratt Method for Solving Nonlinear Equations

R. Thukral — Mon, 27 Aug 2012 09:27:27 +0000

We present two new families of Jarratt-type methods for solving nonlinear equations. It is proved that the order of convergence of each family member is improved from four to six by the addition of one function evaluation. Per iteration, these new methods require two evaluations of the function and two evaluations of the first-order derivatives. In fact, the efficiency index of these methods is 1.565. Numerical comparisons are made with other existing methods to show the performance of the presented methods.

Interpreting the Phase Spectrum in Fourier Analysis of Partial Ranking Data

Ramakrishna Kakarala — Wed, 27 Jun 2012 10:09:57 +0000

Whenever ranking data are collected, such as in elections, surveys, and database searches, it is frequently the case that partial rankings are available instead of, or sometimes in addition to, full rankings. Statistical methods for partial rankings have been discussed in the literature. However, there has been relatively little published on their Fourier analysis, perhaps because the abstract nature of the transforms involved impede insight. This paper provides as its novel contributions an analysis of the Fourier transform for partial rankings, with particular attention to the first three ranks, while emphasizing on basic signal processing properties of transform magnitude and phase. It shows that the transform and its magnitude satisfy a projection invariance and analyzes the reconstruction of data from either magnitude or phase alone. The analysis is motivated by appealing to corresponding properties of the familiar DFT and by application to two real-world data sets.

Signorini Cylindrical Waves and Shannon Wavelets

Carlo Cattani — Tue, 26 Jun 2012 11:07:30 +0000

Hyperelastic materials based on Signorini’s strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves’ solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis.

Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations

Anas Rachid, Mohamed Bahaj, and Noureddine Ayoub — Mon, 07 May 2012 14:54:15 +0000

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.

A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential Equations in Polar Coordinates

Jyoti Talwar and R. K. Mohanty — Thu, 15 Mar 2012 08:34:39 +0000

In this piece of work using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of fourth-order ordinary differential equation 𝑢𝑖𝑣(𝑥)=𝑓(𝑥,𝑢(𝑥),𝑢(𝑥),𝑢(𝑥),𝑢(𝑥)), 𝑎<𝑥<𝑏, subject to boundary conditions 𝑢(𝑎)=𝐴0, 𝑢(𝑎)=𝐴1, 𝑢(𝑏)=𝐵0, and 𝑢(𝑏)=𝐵1, where 𝐴0, 𝐴1, 𝐵0, and 𝐵1 are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods.

The Exponential Dichotomy under Discretization on General Approximation Scheme

Javier Pastor and Sergey Piskarev — Tue, 14 Feb 2012 16:27:03 +0000

This paper is devoted to the numerical analysis of abstract parabolic problem 𝑢(𝑡)=𝐴𝑢(𝑡); 𝑢(0)=𝑢0, with hyperbolic generator 𝐴. We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential decaying solutions in opposite time direction. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results is naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite element as well as finite difference methods.

Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

Guillaume Jouvet and Jacques Rappaz — Thu, 12 Jan 2012 11:49:52 +0000

The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.

Approximation Solution of Fractional Partial Differential Equations by Neural Networks

Adel A. S. Almarashi — Tue, 27 Dec 2011 11:30:39 +0000

Neural networks with radial basis functions method are used to solve a class of initial boundary value of fractional partial differential equations with variable coefficients on a finite domain. It takes the case where a left-handed or right-handed fractional spatial derivative may be present in the partial differential equations. Convergence of this method will be discussed in the paper. A numerical example using neural networks RBF method for a two-sided fractional PDE also will be presented and compared with other methods.

Different Versions of ILU and IUL Factorizations Obtained from Forward and Backward Factored Approximate Inverse Processes—Part I

Amin Rafiei and Fatemeh Shahlaei — Sun, 25 Dec 2011 10:45:52 +0000

We present an incomplete UL (IUL) decomposition of matrix 𝐴 which is extracted as a by-product of BFAPINV (backward factored approximate inverse) process. We term this IUL factorization as IULBF. We have used ILUFF [3] and IULBF as left preconditioner for linear systems. Different versions of ILUFF and IULBF preconditioners are computed by using different dropping techniques. In this paper, we compare quality of different versions of ILUFF and IULBF preconditioners.

Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

N. Akhondi and F. Toutounian — Sun, 11 Dec 2011 11:22:10 +0000

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix 𝐴. In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.

Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

F. Soleymani — Thu, 08 Dec 2011 14:37:03 +0000

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

A Generalization of a Class of Matrices: Analytic Inverse and Determinant

F. N. Valvi — Thu, 01 Dec 2011 13:34:37 +0000

The aim of this paper is to present the structure of a class of matrices that enables explicit inverse to be obtained. Starting from an already known class of matrices, we construct a Hadamard product that derives the class under consideration. The latter are defined by 4𝑛−2 parameters, analytic expressions of which provide the elements of the lower Hessenberg form inverse. Recursion formulae of these expressions reduce the arithmetic operations in evaluating the inverse to 𝒪(𝑛2).

Refinement Methods for State Estimation via Sylvester-Observer Equation

H. Saberi Najafi and A. H. Refahi Sheikhani — Mon, 12 Sep 2011 08:11:00 +0000

We present new iterative methods based on refinement process for solving large sparse Sylvester-observer equations applied in state estimation of a continuous-time system. These methods use projection methods to produce low-dimensional Sylvester-observer matrix equations that are solved by the direct methods. Moreover, the refinement process described in this paper has the capability of improving the results obtained by any other methods. Some numerical results will be reported to illustrate the efficiency of the proposed methods.