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On the improvement of the triangular shepard method by non conformal polynomial elements

APPLIED NUMERICAL MATHEMATICS 2023年 184卷 446-460页

作者： Dell'Accio, Francesco Di Tommaso, Filomena Guessab, Allal Nudo, Federico Univ Calabria Dept Math & Comp Sci Arcavacata Di Rende CS Italy Univ Pau & Pays Adour UPPA Lab Math Appl Pau LMAP UMR 5142 E2S UPPA Ave Univ F-64000 Pau France

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of the proposed method. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Scattered data interpolation triangular shepard method Enriched finite element method Nonconforming finite element

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Increasing the approximation order of the triangular shepard method

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APPLIED NUMERICAL MATHEMATICS 2018年 126卷 78-91页

作者： Dell'Accio, F. Di Tommaso, F. Nouisser, O. Zerroudi, B. Univ Calabria Dept Math & Comp Sci Arcavacata Di Rende Italy Univ Ibn Tofail Dept Math Kenitra Morocco

In this paper we discuss an improvement of the triangular shepard operator proposed by Little to extend the shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and derivative data, has cubic approximation order and a good accuracy of approximation. Suggestions on how to avoid the use of derivative data, without losing both order and accuracy of approximation, are given. Published by Elsevier B.V. on behalf of IMACS.

关键词： Multivariate polynomial interpolation Scattered data interpolation triangular shepard method

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Modified shepard's method by six-points local interpolant

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2021年第1-2期65卷 651-667页

作者： Nouisser, Otheman Zerroudi, Benaissa Ibn Tofail Univ Dept Math Kenitra Morocco

In this paper, we present an improvement of the Hexagonal shepard method which uses functional and first order derivative data. More in details, we use six-point basis functions in combination with the modified local interpolant on six-points. The resulting operator reproduces polynomials up to degree 3 and has quartic approximation order. Several numerical results show the good accuracy of approximation of the proposed operator.

关键词： Scatterd data interpolation Multivariate interpolation triangular shepard method Hexagonal shepard method

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On the hexagonal shepard method

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APPLIED NUMERICAL MATHEMATICS 2020年 150卷 51-64页

作者： Dell'Accio, Francesco Di Tommaso, Filomena Univ Calabria Dept Math & Comp Sci Via P Bucci Cubo 30 A I-87036 Arcavacata Di Rende CS Italy

The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular shepard method, we introduce an hexagonal shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Scattered data interpolation triangular shepard method Multivariate Lagrange interpolation

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Fast and accurate scattered Hermite interpolation by triangular shepard operators

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 382卷 113092-113092页

作者： Dell'Accio, Francesco Di Tommaso, Filomena Nouisser, Otheman Zerroudi, Benaissa Univ Calabria Dept Math & Comp Sci Arcavacata Di Rende Italy Natl Res Council Italy CNR Ist Applicaz Calcolo Mauro Picone Naples Branch Naples Italy Univ Ibn Tofail Dept Math Kenitra Morocco

The triangular shepard method is a fast and accurate scheme for interpolating scattered data. In this paper, we introduce an improvement of the triangular shepard method for interpolating functional and first order derivatives values at the scattered points. Theoretical and numerical results show that the proposed method reaches at least cubic approximation order. Its fastness, accuracy and simplicity make it usable in real world applications. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Scattered data interpolation Hermite interpolation triangular shepard method

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Fast computation of triangular shepard interpolants

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 354卷 457-470页

作者： Cavoretto, Roberto De Rossi, Alessandra Dell'Accio, Francesco Di Tommaso, Filomena Univ Torino Dept Math Giuseppe Peano Via Carlo Alberto 10 I-10123 Turin Italy Univ Calabria Dept Math & Comp Sci Via P Bucci Cubo 30 A I-87036 Arcavacata Di Rende CS Italy

In this paper, we present an efficient algorithm for the computation of triangular shepard interpolation method. More precisely, it is well known that the triangular shepard method reaches an approximation order better than the shepard one (DellAccio et al., 2016), but it needs to identify useful general triangulation of the node set. Here we propose a searching technique used to detect and select the nearest neighbor points in the interpolation scheme (Cavoretto et al., 2016, 2017). It consists in determining the closest points belonging to the different neighborhoods and subsequently applies to the triangulation -based approach. Numerical experiments and some geological applications show efficiency and accuracy of the interpolation procedure. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Scattered data interpolation triangular shepard method Fast computation Approximation algorithms

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