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Space-time least-squares isogeometric method and efficient solver for parabolic problems


Authors: Monica Montardini, Matteo Negri, Giancarlo Sangalli and Mattia Tani
Journal: Math. Comp. 89 (2020), 1193-1227
MSC (2010): Primary 65F08, 65M60, 65D07; Secondary 35K05
DOI: https://doi.org/10.1090/mcom/3471
Published electronically: September 24, 2019
MathSciNet review: 4063316
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Abstract: In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.


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Additional Information

Monica Montardini
Affiliation: Università di Pavia, Dipartimento di Matematica “F. Casorati”, Via A. Ferrata 5, 27100 Pavia, Italy
Email: monica.montardini01@universitadipavia.it

Matteo Negri
Affiliation: Università di Pavia, Dipartimento di Matematica “F. Casorati”, Via A. Ferrata 5, 27100 Pavia, Italy
Email: matteo.negri@unipv.it

Giancarlo Sangalli
Affiliation: Università di Pavia, Dipartimento di Matematica “F. Casorati”, Via A. Ferrata 5, 27100 Pavia, Italy; and IMATI-CNR “Enrico Magenes”, Via A. Ferrata 1, 27100 Pavia, Italy
Email: giancarlo.sangalli@unipv.it

Mattia Tani
Affiliation: Università di Pavia, Dipartimento di Matematica “F. Casorati”, Via A. Ferrata 5, 27100 Pavia, Italy
Email: mattia.tani@imati.cnr.it

DOI: https://doi.org/10.1090/mcom/3471
Keywords: Isogeometric analysis, parabolic problem, space-time method, $k$-method, splines, least-squares, Sylvester equation
Received by editor(s): September 27, 2018
Received by editor(s) in revised form: April 20, 2019, and June 5, 2019
Published electronically: September 24, 2019
Additional Notes: The first, third and fourth authors are members of the Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INDAM)
The second author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni-Istituto Nazionale di Alta Matematica (GNAMPA-INDAM). This support is gratefully acknowledged.
This research was partially supported by the European Research Council through the FP7 Ideas Consolidator Grant HIGEOM n.616563, and by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Mathematics, University of Pavia”.
Article copyright: © Copyright 2019 American Mathematical Society