Abstract
In many applications, mathematical and numerical models involve simultaneously more than one single phenomenon. In this situation different equations are used in possibly overlapping subregions of the domain in order to approximate the physical model and obtain an efficient reduction of the computational cost. The coupling between the different equations must be carefully handled to guarantee accurate results. However in many cases, since the geometry of the overlapping subdomains is neither given a-priori nor characterized by coupling equations, a matching relation between the different equations is not available; see, e.g. Degond and Jin (SIAM J Numer Anal 42(6):2671–2687, 2005), Gander et al. (Numer Algorithm 73(1):167–195, 2016) and references therein. To overcome this problem, we introduce a new methodology that interprets the (unknown) decomposition of the domain by associating each subdomain to a partition of unity (membership) function. Then, by exploiting the feature of the partition of unity method developed in Babuska and Melenk (Int J Numer Methods Eng 40:727–758, 1996) and Griebel and Schweitzer (SIAM J Sci Comput 22(3):853–890, 2000), we define a new domain-decomposition strategy that can be easily embedded in infinite-dimensional optimization settings. This allows us to develop a new optimal control methodology that is capable to design coupling mechanisms between the different approximate equations. Numerical experiments demonstrate the efficiency of the proposed framework.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This specific approximation is motivated by asymptotic expansion techniques providing in general two problems, one that is uniquely determined and a second one that is determined up to some constants for asymptotic matching [15].
References
Y. Achdou, O. Pironneau, The χ-method for the Navier-Stokes equations. IMA J. Numer. Anal. 13(4), 537–558 (1993)
I. Babuska, J.M. Melenk, The partition of unity method. Int. J. Numer. Methods Eng. 40, 727–758 (1996)
H. Berninger, E. Frénod, M. Gander, M. Liebendorfer, J. Michaud, Derivation of the isotropic diffusion source approximation (idsa) for supernova neutrino transport by asymptotic expansions. SIAM J. Math. Anal. 45(6), 3229–3265 (2013)
A. Borzì, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations (Society for Industrial and Applied Mathematics, Philadelphia, 2012)
A. Borzì, G. Ciaramella, M. Sprengel, Formulation and Numerical Solution of Quantum Control Problems (Society for Industrial and Applied Mathematics, Philadelphia, 2017)
F. Brezzi, C. Canuto, A. Russo, A self-adaptive formulation for the Euler-Navier stokes coupling. Comput. Methods Appl. Mech. Eng. 73, 317–330 (1989)
P.G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications (Society for Industrial and Applied Mathematics, Philadelphia, 2013)
P. Degond, S. Jin, A smooth transition model between kinetic and diffusion equations. SIAM J. Numer. Anal. 42(6), 2671–2687 (2005)
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2002)
M.J. Gander, J. Michaud, Fuzzy domain decomposition: a new perspective on heterogeneous DD methods, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Berlin, 2014), pp. 265–273
M.J. Gander, L. Halpern, V. Martin, A new algorithm based on factorization for heterogeneous domain decomposition. Numer. Algorithm 73(1), 167–195 (2016)
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 1983)
M. Griebel, M.A. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs. SIAM J. Sci. Comput. 22(3), 853–890 (2000)
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24 (Pitman Advanced Publishing Program, Boston, 1985)
M.H. Holmes, Introduction to Perturbation Methods. Texts in Applied Mathematics (Springer, New York, 2013)
W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)
F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Mathematics Graduate Students, vol. 112 (American Mathematical Society, Providence, 2010)
L.A. Zadeh, Fuzzy sets. Inform. Control 8, 338–353 (1965)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Ciaramella, G., Gander, M.J. (2018). Partition of Unity Methods for Heterogeneous Domain Decomposition. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-93873-8_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93872-1
Online ISBN: 978-3-319-93873-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)