Pointwise Error Estimates Research Articles

Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations

Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations

Metric Approximation of Set-Valued Functions of Bounded Variation by Integral Operators

AbstractWe introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of $${\mathbb {R}}^d$$ R d . All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $$L^1$$ L 1 provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.

Pointwise Error Estimates of Numerical Solutions to Linear Quadratic Optimal Control Problems

An auxiliary optimal control problem is formulated that provides with its unique solution, a continuous representation of the global error of a numerical approximation to the solution of a linear quadratic optimal control problem. The resulting error functions are characterized as the unique solutions to an optimality system that is reformulated as a boundary value problem. With this formulation, reliable pointwise error estimates can be generated utilizing well-established techniques of defect control. A novel algorithm based on defect correction and defect control is presented that generates pointwise approximations to the global error of numerical optimal control solutions on a uniform grid. It is proven and numerically validated that this algorithm can generate pointwise estimates that approximate the true global error with a prescribed accuracy.

Optimal pointwise error estimates for piecewise functions expanded with Laguerre polynomials

This paper focuses on pointwise error estimates for piecewise functions expanded in the form of Laguerre polynomial series. By utilizing the reproducing kernel of Laguerre polynomials, we derive an explicit pointwise error formula, which enables the estimation of convergence orders of pointwise values for piecewise functions by using the Hilb-type formula and generalized van der Corput-type Lemmas. The presented convergence analyses illustrate that the piecewise functions exhibit local superconvergence away from the singular points. Plenty of numerical experiments are carried out to validate the accuracy of these theoretical results.

Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation

<abstract><p>This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Numerical results verify the theoretical conclusions.</p></abstract>

An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a one-dimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

On Solution Regularity of the Two-Dimensional Radiation Transfer Equation and Its Implication on Numerical Convergence

In this paper, we deal with the differential properties of the scalar flux ϕ ( x ) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation. Estimates for the derivatives of ϕ ( x ) near the boundary of the domain are given based on Vainikko’s regularity theorem. The optimal pointwise error estimates in terms of the scalar flux are presented for the two classic finite difference methods: diamond difference (DD) and step difference (SD). Numerical results indicate the implication of the solution smoothness on the numerical convergence behavior.

Localized pointwise error estimates for hybrid finite element methods

We present localized pointwise error estimates for a numerical method designed for accurate and efficient approximations for the flux variables. The method can be considered as a reduced version of the mixed finite element method. The method uses much less degrees of freedom compared to the mixed finite element methods while produces approximation solutions for the flux variable of the same accuracy. In this paper, we present optimal pointwise error estimates for the method, and they show local dependence of the error at a point and weaker dependence on global norms.

On pointwise error estimates for Voronoï-based finite volume methods for the Poisson equation on the sphere

On pointwise error estimates for Voronoï-based finite volume methods for the Poisson equation on the sphere

Pointwise error estimates and local superconvergence of Jacobi expansions

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of | x − a | |x-a| (with | a | > 1 |a|>1 ) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95 95% range of [ − 1 , 1 ] [-1,1] except for a small neighbourhood near the singular point x = a . x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of Φ \Phi -functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired log ⁡ n \log n -factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.

Pointwise error estimates for linear finite element approximation to elliptic Dirichlet problems in smooth domains

Pointwise error estimates for linear finite element approximation to elliptic Dirichlet problems in smooth domains

Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers’ equation, from which L1-discretization formula is applied to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete L1 compact difference scheme on uniform meshes can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability of the proposed scheme are rigorously proved in the L∞-norm by the energy argument and mathematical induction. We also establish a temporal second-order compact difference scheme on graded time meshes for solving the problem with weak initial singularity. Finally, several numerical experiments are provided to test the accuracy of two numerical schemes and verify the theoretical analysis.

Optimal point-wise error estimates of two conservative finite difference schemes for the coupled Gross–Pitaevskii equations with angular momentum rotation terms

This paper is concerned with numerical analysis of two finite difference schemes for solving the coupled Gross–Pitaevskii equations with angular momentum rotation terms which model the dynamics of rotating two-component Bose–Einstein condensates. Both schemes are proved to be uniquely solvable and inherit the conservation laws of mass and energy in the discrete sense. Besides the energy method, the ‘cut-off’ function technique and inductive argument are introduced to establish the optimal error estimates of the numerical methods without any requirement on the grid-ratio or initial values. The convergence order of the numerical solutions to the exact solutions is proved to be of O(τ2+h2) with time-step τ and mesh size h. Several numerical results are reported to validate the error estimates and conservation laws.

One step forward towards the full integration of BEM and CAD software: An effective adaptive approach

The advent of Isogeometric analysis enabled advances towards the straightforward connection between geometric design and mechanical modelling phases. 3D approaches of the Isogeometric Boundary Element Method (IGABEM) stand out in this context, because it requires information only from the boundary as well as the Computer-Aided Design (CAD) models. Thus, the 3D IGABEM best fulfils the isogeometric paradigm, since the geometric representation provided by CAD packages can be interpreted directly by BEM as mesh. However, there is a lack of information regarding convergence and accurate mesh refinement for the proper mechanical fields representation in the 3D IGABEM. Although the IGA models from CAD are geometrically exact in various problems, they usually are not refined enough for the accurate mechanical fields representation. This study proposes mesh adaptivity strategies for the 3D IGABEM formulation in elastostatics, which provide accurate geometric representation and mechanical fields description and contribute towards the full coupling of BEM and CAD schemes. The proposed strategy utilises the error based upon the hypersingular residuals, which provides point-wise error estimates at the boundary. Then, errors based on displacements/tractions or strains can be assessed. The adaptive scheme utilises mesh optimality criteria for both local and global conditions. The refinement strategy applies the knot insertion process, which makes the adaptivity procedure robust and accessible since it does not require iterative communication with the CAD system. The proposed adaptive strategy based on strains error provides good convergence rates in comparison to globally uniform refinement for homogeneous and nonhomogeneous bodies. Additionally, the mesh adaptivity strategy can be applied for fibre-reinforced IGABEM formulations, for which a different error estimator is proposed accounting for the coupling 1DBEM/BEM and FEM/BEM. The strain-based error estimator identifies the required mesh refinement for minimising the oscillating adherence forces surrounding the fibre discontinuity regions. Five applications demonstrate the accuracy of the proposed adaptive schemes, in which the globally homogeneous refinement is a reference.

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.

The Hamiltonian structure and fast energy-preserving algorithms for the fractional Klein-Gordon equation

In this paper, an energy-preserving difference scheme is proposed for solving the space fractional Klein-Gordon equation based on the Hamiltonian form of the equation. First, we study some properties of the fractional Laplacian operator and reformulate the equation as an infinite-dimension canonical Hamiltonian system. Then, we use the fractional centered difference formula to discrete the equation and derive a semi-discrete Hamiltonian system that can conserve the semi-discrete energy. Subsequently, a fully-discrete scheme is obtained by applying the averaged vector field method to the semi-discrete system. The resulting scheme's unconditional point-wise error estimate is discussed by using the “cut-off” technique. Furthermore, a fast solver is presented to reduce the computational complexity of the scheme based on the fast Fourier transformation technique in practical computation. Finally, some numerical experiments are displayed to demonstrate the efficiency and conservation of the constructed scheme in long time simulation.

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

An Invariant-Preserving Scheme for the Viscous Burgers-Poisson System

We formulate and analyze a new finite difference scheme for a shallow water model in the form of viscous Burgers-Poisson system with periodic boundary conditions. The proposed scheme belongs to a family of three-level linearized finite difference methods. It is proved to preserve both momentum and energy in the discrete sense. In addition, we proved that the method converges uniformly and has second order of accuracy in space. The analysis given in this work is the first time a pointwise error estimation is done on a second-order finite difference operator applied to the Burgers-Poisson system. We validate our findings by performing various numerical simulations on both viscous and inviscous problems. These numerical examples show the efficacy of the proposed method and confirm the proven theoretical results.

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of L_{p,{1 / omega }}-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.