research-article

Simplicial Bernstein form and positivity certificates for solutions obtained in a stationary digital twin by Bernstein Bubnov-Galerkin method

Authors:

Tareq Hamadneh,

Jochen Merker,

Willi Schimmel,

Gregor SchuldtAuthors Info & Claims

ICoMS '22: Proceedings of the 2022 5th International Conference on Mathematics and Statistics

Pages 41 - 46

https://doi.org/10.1145/3545839.3545846

Published: 10 September 2022 Publication History

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ICoMS '22: Proceedings of the 2022 5th International Conference on Mathematics and Statistics

Simplicial Bernstein form and positivity certificates for solutions obtained in a stationary digital twin by Bernstein Bubnov-Galerkin method

Pages 41 - 46

Abstract
References

Abstract

In this article, using the simplicial Bernstein form of polynomials on a simplex, we provide positivity certificates for the approximate solution of a linear elliptic PDE obtained by simplicial Bernstein Bubnov-Galerkin method. Particularly, we show how to obtain a simplicial Bernstein certificate of positivity for the approximate solution, although the discretized system does not satisfy a discrete weak maximum principle.

References

[1]

Praveen Agarwal, Jochen Merker, and Gregor Schuldt. 2021. Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs. Axioms 10, 2 (2021), 74. https://doi.org/10.3390/axioms10020074

Crossref

Google Scholar

[2]

Philippe G. Ciarlet. 1970. Discrete maximum principle for finite-difference operators. Aequationes mathematicae 4 (1970), 266–268.

Google Scholar

[3]

Andrei Drǎgǎnescu, Todd Dupont, and L.R. Scott. 2005. Failure of the discrete maximum principle for an elliptic finite element problem. Mathematics of computation 74 (2005), 1–23.

Google Scholar

[4]

D. Gilbarg and N. Trudinger. 2001. Elliptic partial differential equations of second order (2nd. ed.). Springer, New York, NY.

Google Scholar

[5]

Tareq Hamadneh, Jochen Merker, and Gregor Schuldt. 2022. Discrete Maximum Principle and Positivity Certificates for the Bernstein dual Petrov-Galerkin method. In The 7th International Arab Conference on Mathematics and Computations (IACMC 2022). to be published by Springer Proceedings in Mathematics & Statistics, Zarqa, Jordan.

Google Scholar

[6]

Robert C. Kirby. 2011. Fast simplicial finite element algorithms using Bernstein polynomials. Numer. Math. 117(2011), 631–652.

Digital Library

Google Scholar

[7]

Robert C. Kirby and Kieu Tri Thinh. 2012. Fast simplicial quadrature-based finite element operators using Bernstein polynomials. Numer. Math. 121(2012), 261–279.

Digital Library

Google Scholar

[8]

L.M. Leslie and B.J. McAveney. 1973. Comparative test of direct and iterative methods for solving Helmholtz-type equations. Mon. Wea. Rev. 101(1973), 235–239.

Crossref

Google Scholar

[9]

Jochen Merker, Benjamin Kunsch, and Gregor Schuldt. 2021. Nonlinear Compartment Models with Time-Dependent Parameters. Mathematics 9, 14 (2021), 1657. https://doi.org/10.3390/math9141657

Crossref

Google Scholar

[10]

R.J. Plemmons. 1977. M-Matrix Characterizations. I – Nonsingular M-Matrices. Linear Algebra Appl. 18(1977), 175–188.

Crossref

Google Scholar

[11]

T.E. Rosmond and F.D. Faulkner. 1976. Direct solution of elliptic equations by block cyclic reduction and factorization. Mon. Wea. Rev. 104(1976), 641–649.

Crossref

Google Scholar

[12]

Andrew P. Smith. 2012. Enclosure methods for systems of polynomial equations and inequalities. Ph. D. Dissertation. University of Konstanz, Konstanz, Germany.

Google Scholar

[13]

Richard S. Varga. 1962. Matrix Iterative Analysis. Prentice Hall, New York, NY.

Crossref

Google Scholar

[14]

Richard S. Varga. 1966. On a discrete maximum principle. SIAM J. Numer. Anal. 3(1966), 355–359.

Digital Library

Google Scholar

Cited By

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Vafaeva KKarpov DPavlov MIsmailov ABaskar SKapoor TSingh DBhardwaj NRao PGudainiyan J(2024)Solution of the Heat and Mass Transfer Problem for Soil Radiant Heating Conditions Using the Method of Finite Integral Fourier TransformE3S Web of Conferences10.1051/e3sconf/202458101041581(01041)Online publication date: 21-Oct-2024
https://doi.org/10.1051/e3sconf/202458101041
Zhu ZBao TZhu XGong JHu YZhang J(2023)An Improved Material Point Method with Aggregated and Smoothed Bernstein FunctionsMathematics10.3390/math1104090711:4(907)Online publication date: 10-Feb-2023
https://doi.org/10.3390/math11040907
Hamadneh TMerker JSchuldt G(2023)Discrete Maximum Principle and Positivity Certificates for the Bernstein Dual Petrov–Galerkin MethodMathematics and Computation10.1007/978-981-99-0447-1_16(195-211)Online publication date: 30-May-2023
https://doi.org/10.1007/978-981-99-0447-1_16

Index Terms

Simplicial Bernstein form and positivity certificates for solutions obtained in a stationary digital twin by Bernstein Bubnov-Galerkin method
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
  2. Mathematical software
    1. Solvers

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ICoMS '22: Proceedings of the 2022 5th International Conference on Mathematics and Statistics

June 2022

137 pages

ISBN:9781450396233

DOI:10.1145/3545839

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Published: 10 September 2022

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European Social Fund
Saxonian State Ministry for Higher Education, Research and the Arts

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ICoMS 2022

ICoMS 2022: 2022 5th International Conference on Mathematics and Statistics

June 17 - 19, 2022

Paris, France

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Vafaeva KKarpov DPavlov MIsmailov ABaskar SKapoor TSingh DBhardwaj NRao PGudainiyan J(2024)Solution of the Heat and Mass Transfer Problem for Soil Radiant Heating Conditions Using the Method of Finite Integral Fourier TransformE3S Web of Conferences10.1051/e3sconf/202458101041581(01041)Online publication date: 21-Oct-2024
https://doi.org/10.1051/e3sconf/202458101041
Zhu ZBao TZhu XGong JHu YZhang J(2023)An Improved Material Point Method with Aggregated and Smoothed Bernstein FunctionsMathematics10.3390/math1104090711:4(907)Online publication date: 10-Feb-2023
https://doi.org/10.3390/math11040907
Hamadneh TMerker JSchuldt G(2023)Discrete Maximum Principle and Positivity Certificates for the Bernstein Dual Petrov–Galerkin MethodMathematics and Computation10.1007/978-981-99-0447-1_16(195-211)Online publication date: 30-May-2023
https://doi.org/10.1007/978-981-99-0447-1_16

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Certificates of Positivity in the Bernstein Basis

Generalized Bernstein Polynomials

Dual generalized Bernstein basis

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