scholarly journals Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ruifeng Wu
Keyword(s):  
Error Estimates ◽  
Error Bounds ◽  
Nonnegative Integer ◽  
Convergence Rates ◽  
Approximation Order ◽  
Degree Polynomial ◽  
Shape Preserving ◽  
Interpolation Operators ◽  
Interpolation Polynomials ◽  
Polynomial Reproducing

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

scholarly journals Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

2017 ◽  
Vol 21 (3) ◽  
pp. 835-866 ◽  
Author(s):  
Meng Wu ◽  
Bernard Mourrain ◽  
André Galligo ◽  
Boniface Nkonga
Keyword(s):  
Isogeometric Analysis ◽  
Convergence Rates ◽  
Approximation Order ◽  
Basis Functions ◽  
Physical Domain ◽  
Piecewise Polynomial ◽  
Numerical Convergence ◽  
Rectangular Mesh ◽  
Piecewise Polynomial Functions ◽  
Potential Applications

AbstractMotivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

Approximation based on elliptic splines

2014 ◽  
Vol 12 (05) ◽  
pp. 1461014 ◽  
Author(s):  
Zhuyuan Yang ◽  
Zongwen Yang
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Fourier Transforms ◽  
Projection Operators ◽  
Order Of Approximation ◽  
B Splines ◽  
Interpolation Operators ◽  
Polynomial Reproducing

In this paper, we study the elliptic splines which include the well-known polyharmonic B-splines. We analyze their Fourier transforms, decay behaviors and polynomial reproducing properties. We also study the order of approximation in Sobolev spaces and consider their characterizations of Besov spaces by the scale projection operators, quasi-interpolation operators and wavelet operators.

scholarly journals Two-scale method for the Monge–Ampère equation: pointwise error estimates

2018 ◽  
Vol 39 (3) ◽  
pp. 1085-1109 ◽  
Author(s):  
R H Nochetto ◽  
D Ntogkas ◽  
W Zhang
Keyword(s):  
Error Estimates ◽  
Viscosity Solution ◽  
Continuous Dependence ◽  
Convergence Rates ◽  
Classical Solutions ◽  
Discrete Version ◽  
Scale Method ◽  
Pointwise Error ◽  
Pointwise Error Estimates ◽  
Monge Ampere

Abstract In this paper we continue the analysis of the two-scale method for the Monge–Ampère equation for dimension d ≥ 2 introduced in the study by Nochetto et al. (2017, Two-scale method for the Monge–Ampère equation: convergence to the viscosity solution. Math. Comput., in press). We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.

scholarly journals Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

2019 ◽  
Vol 27 (3) ◽  
pp. 155-182 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Convergence Rates ◽  
Time Discretization ◽  
Linear Constraints ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems ◽  
Discretization Methods ◽  
Optimal Convergence

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

scholarly journals Hermite-Fejér type interpolation and Korovkin's theorem

1990 ◽  
Vol 42 (3) ◽  
pp. 383-390
Author(s):  
H.-B. Knoop ◽  
F. Locher
Keyword(s):  
Uniform Convergence ◽  
Jacobi Polynomials ◽  
Korovkin Theorem ◽  
Type Interpolation ◽  
Special Values ◽  
Convergence Results ◽  
Interpolation Operators ◽  
Positive Functionals ◽  
Interpolation Polynomials ◽  
Korovkin’S Theorem

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

scholarly journals A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Ruifeng Wu ◽  
Huilai Li ◽  
Tieru Wu
Keyword(s):  
Initial Value Problems ◽  
Bernoulli Polynomials ◽  
Interpolation Scheme ◽  
Degree Polynomial ◽  
Initial Value ◽  
Linear Combinations ◽  
Shape Preserving ◽  
Runge Kutta Method ◽  
Even Order ◽  
Derivatives Of

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the functionf(x)(x∈ℝ)to approximate the derivatives off(x), we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parametercand a nonnegative integerm. Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.’s quasi-interpolation scheme.

scholarly journals A study of defect-based error estimates for the Krylov approximation of φ-functions

2021 ◽  
Author(s):  
Tobias Jawecki
Keyword(s):  
Error Estimates ◽  
Error Bounds ◽  
A Priori ◽  
Posteriori Error ◽  
Finite Precision ◽  
Ritz Values ◽  
A Posteriori Error Bounds ◽  
The Matrix ◽  
And Performance ◽  
Point Arithmetic

AbstractPrior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

scholarly journals Approximation and shape preserving properties of the nonlinear Favardsza-Szász-Mirakjan operator of max-product kind

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 55-72 ◽  
Author(s):  
Barnabás Bede ◽  
Lucian Coroianu ◽  
Sorin Gal
Keyword(s):  
Nonlinear Operator ◽  
Approximation Error ◽  
Pointwise Estimate ◽  
Approximation Order ◽  
Order Of Approximation ◽  
Bernstein Type ◽  
Shape Preserving ◽  
Explicit Constant ◽  
Open Question ◽  
Shape Preserving Properties

Starting from the study of the Shepard nonlinear operator of max-prod type in [6], [7], in the book [8], Open Problem 5.5.4, pp. 324-326, the Favard-Sz?sz-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the recent paper [1], by using a pretty complicated method to this open question an answer is given by obtaining an upper pointwise estimate of the approximation error of the form C?1(f;?x/?n) (with an unexplicit absolute constant C>0) and the question of improving the order of approximation ?1(f;?x/?n) is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages : it produces an explicit constant in front of ?1(f;?x/?n) and it can easily be extended to other max-prod operators of Bernstein type. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ?1(f;?) cannot be improved. However, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order ?1 (f;1/n) is obtained. Finally, some shape preserving properties are obtained.

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