478 results
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2. Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems
- Author
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Ke Li, Yunxiang Zhao, and Dali Guo
- Subjects
Article Subject ,Discretization ,Differential equation ,General Mathematics ,General Engineering ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Convolution ,Fractional calculus ,Quadrature (mathematics) ,010101 applied mathematics ,Rate of convergence ,QA1-939 ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Temporal discretization ,Mathematics - Abstract
Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.
- Published
- 2021
3. Analysis of backward Euler projection FEM for the Landau–Lifshitz equation
- Author
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Weiwei Sun and Rong An
- Subjects
010101 applied mathematics ,Computational Mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,Projection (set theory) ,01 natural sciences ,Backward Euler method ,Landau–Lifshitz–Gilbert equation ,Finite element method ,Mathematics - Abstract
The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.
- Published
- 2021
4. Minimization arguments in analysis of variational-hemivariational inequalities
- Author
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Weimin Han and Mircea Sofonea
- Subjects
Applied Mathematics ,General Mathematics ,Hilbert space ,Structure (category theory) ,General Physics and Astronomy ,Contrast (statistics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Contact mechanics ,Compact space ,symbols ,Applied mathematics ,Minification ,0101 mathematics ,Mathematics - Abstract
In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.
- Published
- 2022
- Full Text
- View/download PDF
5. Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations
- Author
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Huadong Gao, Chengda Wu, and Weiwei Sun
- Subjects
010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Applied Mathematics ,General Mathematics ,Thermistor ,Applied mathematics ,010103 numerical & computational mathematics ,Mixed finite element method ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.
- Published
- 2020
6. A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation
- Author
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Yueyue Pan, Lifei Wu, and Xiaozhong Yang
- Subjects
Article Subject ,General Mathematics ,General Engineering ,Fisher equation ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Simple (abstract algebra) ,Scheme (mathematics) ,Convergence (routing) ,QA1-939 ,Parallelism (grammar) ,Applied mathematics ,Uniqueness ,TA1-2040 ,0101 mathematics ,Absolute stability ,Mathematics - Abstract
This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.
- Published
- 2020
7. Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra
- Author
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Vitoriano Ruas
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Finite element solution ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Tetrahedron ,symbols ,0101 mathematics ,Mathematics - Abstract
In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.
- Published
- 2020
8. A priori analysis of a higher-order nonlinear elasticity model for an atomistic chain with periodic boundary condition
- Author
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Lei Zhang, Hao Wang, and Yangshuai Wang
- Subjects
Applied Mathematics ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Chain (algebraic topology) ,Periodic boundary conditions ,Order (group theory) ,A priori and a posteriori ,Applied mathematics ,0101 mathematics ,Nonlinear elasticity ,Mathematics - Abstract
Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy–Born model. While the Cauchy–Born model only depends on the strain, effects of higher-order strain gradients are significant and higher-order continuum models are preferred in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper we rigorously derive a higher-order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second-order accuracy of the Cauchy–Born model, the higher-order continuum model in this paper is of fourth-order accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition we discuss the key issues for the derivation of higher-order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.
- Published
- 2020
9. Trace finite element methods for surface vector-Laplace equations
- Author
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Thomas Jankuhn and Arnold Reusken
- Subjects
Partial differential equation ,Discretization ,Applied Mathematics ,General Mathematics ,Tangent ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Lagrange multiplier ,Norm (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,symbols ,65N30, 65N12, 65N15 ,Applied mathematics ,Vector field ,Penalty method ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.
- Published
- 2020
10. On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation
- Author
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Qiuyan Xu and Zhiyong Liu
- Subjects
Collocation ,Article Subject ,General Mathematics ,Direct method ,General Engineering ,Boundary (topology) ,Monge–Ampère equation ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Discrete system ,symbols.namesake ,Nonlinear system ,QA1-939 ,symbols ,Applied mathematics ,Radial basis function ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.
- Published
- 2020
11. An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications
- Author
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Fei Xu and Qiumei Huang
- Subjects
Series (mathematics) ,General Mathematics ,Estimator ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,Space (mathematics) ,01 natural sciences ,Finite element method ,010101 applied mathematics ,A priori and a posteriori ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Smoothing ,Mathematics - Abstract
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.
- Published
- 2019
12. On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros
- Author
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Petko D. Proinov and Milena Petkova
- Subjects
Polynomial ,iteration functions ,Iterative method ,General Mathematics ,010103 numerical & computational mathematics ,Construct (python library) ,multi-point iterative methods ,Type (model theory) ,01 natural sciences ,Local convergence ,010101 applied mathematics ,error estimates ,Convergence (routing) ,semilocal convergence ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,local convergence ,0101 mathematics ,polynomial zeros ,Engineering (miscellaneous) ,Multi point ,Mathematics - Abstract
In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.
- Published
- 2021
13. Error Estimations for Total Variation Type Regularization
- Author
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Chun Huang, Ziyang Yuan, and Kuan Li
- Subjects
Series (mathematics) ,General Mathematics ,Stability (learning theory) ,010103 numerical & computational mathematics ,Inverse problem ,Type (model theory) ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,regularization ,total variation ,Rate of convergence ,Consistency (statistics) ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,A priori and a posteriori ,inverse problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed, the results of the corresponding convergence rate are also presented under certain mild conditions.
- Published
- 2021
14. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?
- Author
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B. Cano and Nuria Reguera
- Subjects
Order reduction ,General Mathematics ,Krylov methods ,010103 numerical & computational mathematics ,Exponential integrator ,01 natural sciences ,Exponential function ,010101 applied mathematics ,efficiency ,QA1-939 ,Computer Science (miscellaneous) ,Spite ,Applied mathematics ,avoiding order reduction ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.
- Published
- 2021
- Full Text
- View/download PDF
15. A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing
- Author
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Can Huang, Kim Ngan Le, and Martin Stynes
- Subjects
Forcing (recursion theory) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Gronwall's inequality ,Applied mathematics ,Fokker–Planck equation ,0101 mathematics ,Mathematics - Abstract
First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.
- Published
- 2019
16. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- Author
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Juan Carlos Cortés, Marc Jornet, and Julia Calatayud
- Subjects
Non-autonomous and random dynamical systems ,Adaptive generalized Polynomial Chaos ,General Mathematics ,Comparative case ,Monte Carlo method ,random Fröbenius method ,Random Frobenius method ,010103 numerical & computational mathematics ,01 natural sciences ,93e03 ,non-autonomous and random dynamical systems ,computational uncertainty quantification ,Stochastic Galerkin projection technique ,Linear differential equation ,34f05 ,QA1-939 ,Applied mathematics ,Order (group theory) ,60h35 ,0101 mathematics ,Uncertainty quantification ,Mathematics ,Final version ,Computational uncertainty quantification ,random fröbenius method ,93E03 ,adaptive generalized Polynomial Chaos ,stochastic Galerkin projection technique ,010101 applied mathematics ,Frobenius method ,34F05 ,60H35 ,MATEMATICA APLICADA - Abstract
[EN] This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Frobenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Frobenius method, in which the random input coefficients may be expressed via a Karhunen-Loeve expansion., This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewer, which have improved the final version of the paper.
- Published
- 2018
17. Superconvergence of kernel-based interpolation
- Author
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Robert Schaback
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Open problem ,Hilbert space ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Positive-definite matrix ,Superconvergence ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Spline (mathematics) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Spline interpolation ,Analysis ,Mathematics - Abstract
From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.
- Published
- 2018
18. Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation
- Author
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Boling Guo and Tingchun Wang
- Subjects
Conservation law ,Angular momentum ,General Mathematics ,Finite difference ,Finite difference method ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Applied mathematics ,Unconditional convergence ,0101 mathematics ,Energy functional ,Mathematics - Abstract
This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.
- Published
- 2018
19. Stream function formulation of surface Stokes equations
- Author
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Arnold Reusken
- Subjects
010101 applied mathematics ,Surface (mathematics) ,Computational Mathematics ,Applied Mathematics ,General Mathematics ,Stream function ,010103 numerical & computational mathematics ,Mechanics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a $C^2$ connected (not necessarily simply connected) oriented hypersurface $\varGamma \subset \mathbb{R}^3$ without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space $\mathbb{R}^3$. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. We introduce surface $\mathbf H({\mathop{\rm div}}_{\varGamma})$ and $\mathbf H({\mathop{\rm curl}}_{\varGamma})$ spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. As a corollary of this decomposition we obtain that for a simply connected surface to every tangential divergence-free velocity field there corresponds a unique scalar stream function. Using this result the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth-order equation for the stream function. The latter can be rewritten as two coupled second-order equations, which form the basis for a finite element discretization. A particular finite element method is explained and the results of a numerical experiment with this method are presented.
- Published
- 2018
20. An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems
- Author
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Hala A. Omar
- Subjects
Article Subject ,Differential equation ,General Mathematics ,Homotopy ,General Engineering ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Square (algebra) ,010101 applied mathematics ,Linear map ,Nonlinear system ,Genetic algorithm ,Convergence (routing) ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Homotopy analysis method ,Mathematics - Abstract
Solving nonlinear equation systems for engineering applications is one of the broadest and most essential numerical studies. Several methods and combinations were developed to solve such problems by either finding their roots mathematically or formalizing such problems as an optimization task to obtain the optimal solution using a predetermined objective function. This paper proposes a new algorithm for solving square and nonsquare nonlinear systems combining the genetic algorithm (GA) and the homotopy analysis method (HAM). First, the GA is applied to find out the solution. If it is realized, the algorithm is terminated at this stage as the target solution is determined. Otherwise, the HAM is initiated based on the GA stage’s computed initial guess and linear operator. Moreover, the GA is utilized to calculate the optimum value of the convergence control parameter (h) algebraically without plotting the h-curves or identifying the valid region. Four test functions are examined in this paper to verify the proposed algorithm’s accuracy and efficiency. The results are compared to the Newton HAM (NHAM) and Newton homotopy differential equation (NHDE). The results corroborated the superiority of the proposed algorithm in solving nonlinear equation systems efficiently.
- Published
- 2021
21. Neural Network Method for Solving Time-Fractional Telegraph Equation
- Author
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Lelisa Kebena Bijiga and Wubshet Ibrahim
- Subjects
Optimization problem ,Artificial neural network ,Article Subject ,Differential equation ,General Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Function (mathematics) ,Telegrapher's equations ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,QA1-939 ,Applied mathematics ,Development (differential geometry) ,Boundary value problem ,0101 mathematics ,Fractional differential ,TA1-2040 ,Mathematics - Abstract
Recently, the development of neural network method for solving differential equations has made a remarkable progress for solving fractional differential equations. In this paper, a neural network method is employed to solve time-fractional telegraph equation. The loss function containing initial/boundary conditions with adjustable parameters (weights and biases) is constructed. Also, in this paper, a time-fractional telegraph equation was formulated as an optimization problem. Numerical examples with known analytic solutions including numerical results, their graphs, weights, and biases were also discussed to confirm the accuracy of the method used. Also, the graphical and tabular results were analyzed thoroughly. The mean square errors for different choices of neurons and epochs have been presented in tables along with graphical presentations.
- Published
- 2021
22. A Numerical Method for Compressible Model of Contamination from Nuclear Waste in Porous Media
- Author
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Zhifeng Wang
- Subjects
Article Subject ,General Mathematics ,Numerical analysis ,Linear system ,General Engineering ,010103 numerical & computational mathematics ,Mixed finite element method ,Grid ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Asymptotically optimal algorithm ,Compressibility ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Porous medium ,Mathematics - Abstract
This paper studies and analyzes a model describing the flow of contaminated brines through the porous media under severe thermal conditions caused by the radioactive contaminants. The problem is approximated based on combining the mixed finite element method with the modified method of characteristics. In order to solve the resulting algebraic nonlinear equations efficiently, a two-grid method is presented and discussed in this paper. This approach includes a small nonlinear system on a coarse grid with size H and a linear system on a fine grid with size h . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy H = O h 1 / 3 .
- Published
- 2021
23. On a System of k-Difference Equations of Order Three
- Author
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Ibrahim Yalcinkaya, Yong-Min Li, Hijaz Ahmad, and Durhasan Turgut Tollu
- Subjects
Article Subject ,General Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Order (business) ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Mathematics - Abstract
In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.
- Published
- 2020
24. A convergent adaptive finite element method for elliptic Dirichlet boundary control problems
- Author
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Zhiyu Tan, Ningning Yan, Wenbin Liu, and Wei Gong
- Subjects
Applied Mathematics ,General Mathematics ,Estimator ,Finite element approximations ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Dirichlet distribution ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Norm (mathematics) ,symbols ,Partial derivative ,Applied mathematics ,A priori and a posteriori ,0101 mathematics ,Mathematics - Abstract
This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space. The contribution of this paper is twofold. First, we rigorously derive efficient and reliable a posteriori error estimates for finite element approximations of Dirichlet boundary control problems. As a by-product, a priori error estimates are derived in a simple way by introducing appropriate auxiliary problems and establishing certain norm equivalence. Secondly, for the coupled elliptic partial differential system that resulted from the first-order optimality system, we prove that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by our newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators. We give some numerical results to confirm our theoretical findings.
- Published
- 2018
25. Unified error analysis for nonconforming space discretizations of wave-type equations
- Author
-
Marlis Hochbruck, David Hipp, and Christian Stohrer
- Subjects
010101 applied mathematics ,Computational Mathematics ,Error analysis ,Applied Mathematics ,General Mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Mathematics - Abstract
This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.
- Published
- 2018
26. The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations
- Author
-
Xinyuan Wu and Bin Wang
- Subjects
Applied Mathematics ,General Mathematics ,010103 numerical & computational mathematics ,High dimensional ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,symbols ,Applied mathematics ,0101 mathematics ,Klein–Gordon equation ,Energy (signal processing) ,Mathematics - Abstract
In this paper we focus on the analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations. A novel energy-preserving scheme is developed based on the discrete gradient method and the Duhamel principle. The local error, global convergence and nonlinear stability of the new scheme are analysed in detail. Numerical experiments are implemented to compare with existing numerical methods in the literature, and the numerical results show the remarkable efficiency of the new energy-preserving scheme presented in this paper.
- Published
- 2018
27. Two low-order nonconforming finite element methods for the Stokes flow in three dimensions
- Author
-
Jun Hu and Mira Schedensack
- Subjects
010101 applied mathematics ,Computational Mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Order (group theory) ,010103 numerical & computational mathematics ,0101 mathematics ,Stokes flow ,01 natural sciences ,Finite element method ,Mathematics::Numerical Analysis ,Mathematics - Abstract
In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.
- Published
- 2018
28. Increasing the smoothness of vector and Hermite subdivision schemes
- Author
-
Nira Dyn and Caroline Moosmüller
- Subjects
Limit of a function ,Discrete mathematics ,Hermite polynomials ,65D17, 65D05, 40A99 ,business.industry ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Scalar (mathematics) ,MathematicsofComputing_NUMERICALANALYSIS ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,business ,Smoothing ,Mathematics ,Subdivision - Abstract
In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component)., 28 pages, 4 figures. Corrected typos, updated contact information
- Published
- 2018
29. Total variation diminishing schemes in optimal control of scalar conservation laws
- Author
-
Michael Hintermüller, Stefan Ulbrich, and Soheil Hajian
- Subjects
Conservation law ,65K10 ,Applied Mathematics ,General Mathematics ,65M12 ,Scalar (mathematics) ,010103 numerical & computational mathematics ,TVD Runge-Kutta methods ,Optimal control ,01 natural sciences ,scalar conservation laws ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Total variation diminishing ,Applied mathematics ,adjoint equation ,0101 mathematics ,optimal control of PDEs ,49J20 ,Mathematics - Abstract
n this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented.
- Published
- 2017
30. Infinite-Dimensional $$\ell ^1$$ ℓ 1 Minimization and Function Approximation from Pointwise Data
- Author
-
Ben Adcock
- Subjects
Pointwise ,Truncation ,General Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Function approximation ,symbols ,A priori and a posteriori ,Jacobi polynomials ,Applied mathematics ,Minification ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider the problem of approximating a smooth function from finitely many pointwise samples using $$\ell ^1$$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require a priori bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization, namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper, we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $$\ell ^1$$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.
- Published
- 2017
31. Strong convergence result of split feasibility problems in Banach spaces
- Author
-
Yekini Shehu
- Subjects
010101 applied mathematics ,General Mathematics ,Scheme (mathematics) ,Convergence (routing) ,Calculus ,Regular polygon ,Banach space ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics ,Volume (compression) - Abstract
The purpose of this paper is to introduce and study an iterative scheme for solving the split feasibility problems in the setting of $p$-uniformly convex and uniformly smooth Banach spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper extends some recent results done by Jitsupa Deepho and Poom Kumam [Jitsupa Deepho and Poom Kumam, A Modified Halpern’s Iterative Scheme for Solving Split Feasibility Problems, Abstract and Applied Analysis, Volume 2012, Article ID 876069, 8 pages] and some others.
- Published
- 2017
32. Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2]
- Author
-
Marija Milošević and Maja Obradović
- Subjects
010101 applied mathematics ,Stochastic differential equation ,Exponential stability ,General Mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Euler–Maruyama method ,Mathematics - Abstract
This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.
- Published
- 2017
33. On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations
- Author
-
Sanda Micula
- Subjects
fixed-point theory ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Numerical analysis ,lcsh:Mathematics ,Fixed-point theorem ,010103 numerical & computational mathematics ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Integral equation ,010101 applied mathematics ,mixed Volterra–Fredholm integral equations ,cubature formulas ,Chemistry (miscellaneous) ,Fixed-point iteration ,Convergence (routing) ,Picard iteration ,Computer Science (miscellaneous) ,Applied mathematics ,Uniqueness ,0101 mathematics ,Trapezoidal rule ,numerical approximation ,Mathematics - Abstract
In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra&ndash, Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.
- Published
- 2019
34. Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells
- Author
-
Maysaa Al-Qurashi, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Muhammad Nawaz Naeem, Dumitru Baleanu, Zafar Ullah, and Aasma Khalid
- Subjects
Timoshenko beam theory ,system of linear algebraic equations ,boundary value problems ,General Mathematics ,cubic B-spline ,010103 numerical & computational mathematics ,central finite difference approximations ,System of linear equations ,01 natural sciences ,absolute error ,Approximation error ,Computer Science (miscellaneous) ,Fluid dynamics ,Applied mathematics ,8th order ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,numerical solution ,lcsh:Mathematics ,Boundary problem ,lcsh:QA1-939 ,010101 applied mathematics ,Spline (mathematics) ,Exact solutions in general relativity - Abstract
This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.
- Published
- 2019
- Full Text
- View/download PDF
35. The Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings in Banach Space
- Author
-
Yunhua Qu, Huancheng Zhang, and Yongfu Su
- Subjects
Banach space ,General Mathematics ,lcsh:Mathematics ,Field (mathematics) ,generalized viscosity implicit midpoint rule ,010103 numerical & computational mathematics ,nonexpansive mapping ,lcsh:QA1-939 ,01 natural sciences ,010101 applied mathematics ,strong convergence ,Viscosity (programming) ,Convergence (routing) ,Computer Science (miscellaneous) ,Applied mathematics ,0101 mathematics ,Midpoint method ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper constructs the generalized viscosity implicit midpoint rule for nonexpansive mappings in Banach space. It obtains strong convergence conclusions for the proposed algorithm and promotes the related results in this field. Moreover, this paper gives some applications. Finally, the paper gives six numerical examples to support the main results.
- Published
- 2019
36. Pointwise and uniform convergence of Fourier extensions
- Author
-
Daan Huybrechs, Vincent Coppé, and Marcus Webb
- Subjects
Computer Science::Machine Learning ,Fourier extension ,General Mathematics ,Uniform convergence ,010103 numerical & computational mathematics ,Computer Science::Digital Libraries ,01 natural sciences ,Gibbs phenomenon ,Statistics::Machine Learning ,symbols.namesake ,Lebesgue function ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Fourier series ,Mathematics ,Pointwise ,Pointwise convergence ,42A10, 41A17, 65T40, 42C15 ,constructive approximation ,Cantor function ,Numerical Analysis (math.NA) ,010101 applied mathematics ,Computational Mathematics ,Fourier transform ,Norm (mathematics) ,Computer Science::Mathematical Software ,symbols ,Legendre polynomials on a circular arc ,Analysis - Abstract
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.
- Published
- 2018
37. Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method
- Author
-
Weam Alharbi and Sergei Petrovskii
- Subjects
Power series ,Mittag-Leffler function ,lcsh:Mathematics ,General Mathematics ,Numerical analysis ,Homotopy ,fractional derivative ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,Ambartsumian equation ,Domain (mathematical analysis) ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Convergence (routing) ,Computer Science (miscellaneous) ,symbols ,Applied mathematics ,0101 mathematics ,Homotopy perturbation method ,homotopy perturbation method ,Engineering (miscellaneous) ,Mathematics - Abstract
The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.
- Published
- 2020
38. Solutions of Sturm-Liouville Problems
- Author
-
Christine Böckmann and Upeksha Perera
- Subjects
Work (thermodynamics) ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,inverse Sturm–Liouville problems ,Inverse ,Sturm–Liouville theory ,010103 numerical & computational mathematics ,01 natural sciences ,Magnus expansion ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Computer Science (miscellaneous) ,Applied mathematics ,Order (group theory) ,Boundary value problem ,ddc:510 ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,lcsh:Mathematics ,Institut für Mathematik ,Lie group ,Mathematics::Spectral Theory ,lcsh:QA1-939 ,010101 applied mathematics ,Sturm–Liouville problems of higher order ,Noise ,singular Sturm–Liouville problems - Abstract
This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm&ndash, Liouville problems. Next, a concrete implementation to the inverse Sturm&ndash, Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm&ndash, Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm&ndash, Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.
- Published
- 2020
39. Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously
- Author
-
Maria T. Vasileva and Petko D. Proinov
- Subjects
Polynomial ,Physics and Astronomy (miscellaneous) ,Iterative method ,Generalization ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Nourein’s method ,Convergence (routing) ,Computer Science (miscellaneous) ,Initial value problem ,Applied mathematics ,polynomial zeros ,0101 mathematics ,Mathematics ,lcsh:Mathematics ,lcsh:QA1-939 ,Local convergence ,010101 applied mathematics ,error estimates ,Chemistry (miscellaneous) ,semilocal convergence ,iterative methods ,local convergence ,A priori and a posteriori ,Verifiable secret sharing - Abstract
In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich&rsquo, s method with Newton&rsquo, s correction because it is obtained by combining Ehrlich&rsquo, s method (Commun. ACM 10:2, 1967) and the classical Newton&rsquo, s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein&rsquo, s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein&rsquo, s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.
- Published
- 2020
40. Numerical integration over implicitly defined domains for higher order unfitted finite element methods
- Author
-
Maxim A. Olshanskii and Danil Safin
- Subjects
Curvilinear coordinates ,Level set method ,Discretization ,business.industry ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Numerical integration ,010101 applied mathematics ,Indicator function ,Applied mathematics ,0101 mathematics ,business ,ComputingMethodologies_COMPUTERGRAPHICS ,Subdivision ,Extended finite element method ,Mathematics - Abstract
The paper studies several approaches to numerical integration over a domain defined implicitly by an indicator function such as the level set function. The integration methods are based on subdivision, moment–fitting, local quasi-parametrization and Monte-Carlo techniques. As an application of these techniques, the paper addresses numerical solution of elliptic PDEs posed on domains and manifolds defined implicitly. A higher order unfitted finite element method (FEM) is assumed for the discretization. In such a method the underlying mesh is not fitted to the geometry, and hence the errors of numerical integration over curvilinear elements affect the accuracy of the finite element solution together with approximation errors. The paper studies the numerical complexity of the integration procedures and the performance of unfitted FEMs which employ these tools.
- Published
- 2016
41. Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity
- Author
-
Alberto Castejón, Alicia Cachafeiro, J. García-Amor, and Elías Berriochoa
- Subjects
Polynomial ,1206.07 Interpolación, Aproximación y Ajuste de Curvas ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Convergence (routing) ,Computer Science (miscellaneous) ,Applied mathematics ,unit circle ,perturbed roots of the unity ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,convergence ,lcsh:Mathematics ,Lagrange polynomial ,1202.02 Teoría de la Aproximación ,lcsh:QA1-939 ,010101 applied mathematics ,Cardinal point ,Unit circle ,Rate of convergence ,Bounded function ,symbols ,nodal systems ,separation properties ,lagrange interpolation ,Interpolation - Abstract
The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.
- Published
- 2020
42. A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations
- Author
-
Gouranga Mallik, Neela Nataraj, and Carsten Carstensen
- Subjects
Banach fixed-point theorem ,Applied Mathematics ,General Mathematics ,Regular solution ,Estimator ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Discontinuous Galerkin method ,Applied mathematics ,Penalty method ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.
- Published
- 2018
43. The Derivative-Free Double Newton Step Methods for Solving System of Nonlinear Equations
- Author
-
Na Huang, Changfeng Ma, and Ya-Jun Xie
- Subjects
Iterative method ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Derivative ,01 natural sciences ,Newton's method in optimization ,Local convergence ,010101 applied mathematics ,Nonlinear system ,Calculus ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose two classes of derivative-free Newton-like methods for solving system of nonlinear equations based on double Newton step. We also give the local convergence analysis of the iterative methods. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.
- Published
- 2015
44. Truncated Nonsmooth Newton Multigrid Methods for Block-Separable Minimization Problems
- Author
-
Carsten Gräser and Oliver Sander
- Subjects
Partial differential equation ,Applied Mathematics ,General Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Separable space ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Range (mathematics) ,Multigrid method ,Convex optimization ,Convergence (routing) ,FOS: Mathematics ,65K15, 90C25, 49M20 ,Applied mathematics ,Minification ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional, and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere., Dedicate the paper to Elias Pipping
- Published
- 2017
45. Convection and total variation flow—erratum and improvement
- Author
-
Robert Eymard, François Bouchut, David Doyen, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
- Subjects
Convection ,Applied Mathematics ,General Mathematics ,entropy formulation ,010103 numerical & computational mathematics ,Mechanics ,Hyperbolic scalar conservation law ,01 natural sciences ,1-Laplacian ,010101 applied mathematics ,Computational Mathematics ,Flow (mathematics) ,total variation flow ,finite elements ,0101 mathematics ,Variation (astronomy) ,finite volumes ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics - Abstract
International audience; This paper includes an erratum to (Bouchut et al. (2014) Convection and total variation flow. IMA J. Numer. Anal.,34, 1037–1071.) which deals with a nonlinear hyperbolic scalar conservation law, regularized by the total variation flow operator (or 1-Laplacian), and in which a mistake occurred in the convergence proof of the numerical scheme to the continuous entropy solution. For correcting the proof, it is necessary to introduce an additional vanishing viscous term in the scheme. This modification requires casting the whole paper in the framework of discrete and continuous solutions with unbounded support. This new version, nevertheless, leads to a better result than the previous one, since the bounded variation regularity and the compactness of the support of the initial data are no longer assumed.
- Published
- 2017
46. Homotopy Approach for Integrodifferential Equations
- Author
-
Tomasz Trawiński, Edyta Hetmaniok, Krzysztof Gromysz, Roman Wituła, and Damian Słota
- Subjects
integrodifferential equation ,electromagnet jumper ,convergence ,Series (mathematics) ,lcsh:Mathematics ,General Mathematics ,Homotopy ,homotopy analysis method ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,vibrations ,010101 applied mathematics ,Mechanical system ,Vibration ,error estimation ,Convergence (routing) ,Computer Science (miscellaneous) ,Applied mathematics ,0101 mathematics ,Element (category theory) ,Engineering (miscellaneous) ,Homotopy analysis method ,Beam (structure) ,Mathematics - Abstract
In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we prove that, if this series is convergent, then its sum is the solution of the objective equation. We formulate and prove the sufficient condition of this convergence, and we give also the estimation of error of an approximate solution obtained by taking the partial sum of the considered series. Moreover, we present in this paper the example of using the investigated method for determining the vibrations of the freely supported reinforced concrete beam as well as for solving the equation of movement of the electromagnet jumper mechanical system.
- Published
- 2019
47. Exact and Inexact Subsampled Newton Methods for Optimization
- Author
-
Richard H. Byrd, Raghu Bollapragada, and Jorge Nocedal
- Subjects
Hessian matrix ,FOS: Computer and information sciences ,Applied Mathematics ,General Mathematics ,Linear system ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,MathematicsofComputing_NUMERICALANALYSIS ,Machine Learning (stat.ML) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Rate of convergence ,Optimization and Control (math.OC) ,Statistics - Machine Learning ,Conjugate gradient method ,symbols ,FOS: Mathematics ,Applied mathematics ,Stochastic optimization ,0101 mathematics ,Newton's method ,Mathematics - Optimization and Control ,Mathematics - Abstract
The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression., 37 pages
- Published
- 2016
48. On the zeros of Dirichlet -functions
- Author
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Raouf Ouni, Kamel Mazhouda, and Sami Omar
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Mathematical society ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Riemann hypothesis ,symbols.namesake ,Number theory ,Computational Theory and Mathematics ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)
- Published
- 2011
49. Generalized tractability for multivariate problems Part I
- Author
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Henryk Woźniakowski and Michael Gnewuch
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Discrete mathematics ,Statistics and Probability ,Polynomial ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Information-based complexity ,General Mathematics ,media_common.quotation_subject ,Applied Mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Space (mathematics) ,Infinity ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Set (abstract data type) ,Tensor product ,Bounded function ,0101 mathematics ,Mathematics ,media_common - Abstract
Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.
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- 2007
- Full Text
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50. Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions
- Author
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Xueqin Yang, XiangYun Meng, and Shuo Zhang
- Subjects
010101 applied mathematics ,Rate of convergence ,General Mathematics ,Norm (mathematics) ,FOS: Mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Mathematics - Numerical Analysis ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of $\mathcal{O}(h)$ order in energy norm and of $\mathcal{O}(h^2)$ order in $L^2$ norm on general $d$-rectangular grids. Moreover, when the grid is uniform, the convergence rate can be of $\mathcal{O}(h^2)$ order in energy norm, and the convergence rate in $L^2$ norm is still of $\mathcal{O}(h^2)$ order, which can not be improved. Numerical examples are presented to demonstrate our theoretical results., This paper has been withdrawn by the author due to some rewrittings of the proof
- Published
- 2015
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