Global and explicit approximation of piecewise-smooth two-dimensional functions from cell-average data

Author:

Amat Sergio1,Levin David2,Ruiz-Alvárez Juan1,Yáñez Dionisio F3

Affiliation:

1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena , Cartagena 30202, Spain

2. School of Mathematical Sciences, Tel-Aviv University , Tel-Aviv 6997801, Israel

3. Departamento de Matemáticas, Universidad de Valencia , Valencia 46010, Spain

Abstract

AbstractGiven cell-average data values of a piecewise-smooth bivariate function $f$ within a domain $\varOmega $, we look for a piecewise adaptive approximation to $f$. We are interested in an explicit and global (smooth) approach. Bivariate approximation techniques, as trigonometric or splines approximations, achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. Whereas the boundary of $\varOmega $ is assumed to be known, the subdivision of $\varOmega $ to subdomains on which $f$ is smooth is unknown. The first challenge of the proposed approximation algorithm would be to find a good approximation to the curves separating the smooth subdomains of $f$. In the second stage, we simultaneously look for approximations to the different smooth segments of $f$, where on each segment we approximate the function by a linear combination of basis functions $\{p_i\}_{i=1}^M$, considering the corresponding cell averages. A discrete Laplacian operator applied to the given cell-average data intensifies the structure of the singularity of the data across the curves separating the smooth subdomains of $f$. We refer to these derived values as the signature of the data, and we use it for both approximating the singularity curves separating the different smooth regions of $f$. The main contributions here are improved convergence rates to the approximation of the singularity curves and the approximation of $f$, an explicit and global formula, and, in particular, the derivation of a piecewise-smooth high-order approximation to the function.

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Mathematics,General Mathematics

Reference15 articles.

1. A two-stage approximation strategy for piecewise smooth functions in two and three dimensions;Amat;IMA J. Numer. Anal,2021

2. Corrected subdivision approximation of piecewise smooth functions;Amat,2022

3. Explicit multivariate approximations from cell-average data;Amat,2022

4. High order approximation to non-smooth multivariate functions;Amir;Comput. Aided Geom. Des.,2018

5. Interpolation and approximation of piecewise smooth functions;Aràndiga;SIAM J. Numer. Anal.,2005

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