TY - JOUR

T1 - Global and explicit approximation of piecewise-smooth two-dimensional functions from cell-average data

AU - Amat, Sergio

AU - Levin, David

AU - Ruiz-Alvárez, Juan

AU - Yáñez, Dionisio F.

N1 - Publisher Copyright:
© 2022 The Author(s). Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

PY - 2023/7/1

Y1 - 2023/7/1

N2 - Given cell-average data values of a piecewise-smooth bivariate function within a domain, we look for a piecewise adaptive approximation to. 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 is assumed to be known, the subdivision of to subdomains on which 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. In the second stage, we simultaneously look for approximations to the different smooth segments of, where on each segment we approximate the function by a linear combination of basis functions, 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. 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. The main contributions here are improved convergence rates to the approximation of the singularity curves and the approximation of, an explicit and global formula, and, in particular, the derivation of a piecewise-smooth high-order approximation to the function.

AB - Given cell-average data values of a piecewise-smooth bivariate function within a domain, we look for a piecewise adaptive approximation to. 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 is assumed to be known, the subdivision of to subdomains on which 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. In the second stage, we simultaneously look for approximations to the different smooth segments of, where on each segment we approximate the function by a linear combination of basis functions, 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. 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. The main contributions here are improved convergence rates to the approximation of the singularity curves and the approximation of, an explicit and global formula, and, in particular, the derivation of a piecewise-smooth high-order approximation to the function.

UR - http://www.scopus.com/inward/record.url?scp=85168311225&partnerID=8YFLogxK

U2 - 10.1093/imanum/drac042

DO - 10.1093/imanum/drac042

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AN - SCOPUS:85168311225

SN - 0272-4979

VL - 43

SP - 2299

EP - 2319

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 4

ER -