Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 52 (2018) no. 5, pp. 2003-2035.

We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method.

DOI: 10.1051/m2an/2018011
Classification: 65N25, 65N30, 82D75
Keywords: Diffusion equation, low-regularity solution, mixed formulation, eigenproblem, domain decomposition methods
Ciarlet, P. Jr. 1; Giret, L. 1; Jamelot, E. 1; Kpadonou, F.D. 1

1
@article{M2AN_2018__52_5_2003_0,
     author = {Ciarlet, P.  Jr. and Giret, L. and Jamelot, E. and Kpadonou, F.D.},
     title = {Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2003--2035},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {5},
     year = {2018},
     doi = {10.1051/m2an/2018011},
     zbl = {1460.65137},
     mrnumber = {3891752},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2018011/}
}
TY  - JOUR
AU  - Ciarlet, P.  Jr.
AU  - Giret, L.
AU  - Jamelot, E.
AU  - Kpadonou, F.D.
TI  - Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 2003
EP  - 2035
VL  - 52
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2018011/
DO  - 10.1051/m2an/2018011
LA  - en
ID  - M2AN_2018__52_5_2003_0
ER  - 
%0 Journal Article
%A Ciarlet, P.  Jr.
%A Giret, L.
%A Jamelot, E.
%A Kpadonou, F.D.
%T Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 2003-2035
%V 52
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2018011/
%R 10.1051/m2an/2018011
%G en
%F M2AN_2018__52_5_2003_0
Ciarlet, P.  Jr.; Giret, L.; Jamelot, E.; Kpadonou, F.D. Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 52 (2018) no. 5, pp. 2003-2035. doi : 10.1051/m2an/2018011. http://archive.numdam.org/articles/10.1051/m2an/2018011/

[1] I. Babuska and J. Osborn, Eigenvalue problems, in Vol. II of Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.-L. Lions. North Holland (1991) 641–787. | DOI | MR | Zbl

[2] F. Ben Belgacem and S. Brenner, Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems. Electron. Trans. Numer. Anal. 12 (2001) 134–148. | MR | Zbl

[3] A. Bermudez, P. Gamallo, M.R. Nogueiras and R. Rodriguez, Approximation of a structural acoustic vibration problem by hexahedral finite elements. IMA J. Numer. Anal. 26 (2006) 391–421. | DOI | MR | Zbl

[4] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. | DOI | MR | Zbl

[5] D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4 25 (1997) 131–154. | Numdam | MR | Zbl

[6] D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 (2000) 121–140. | DOI | MR | Zbl

[7] D. Boffi, F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods and Applications. Springer-Verlag (2013). | DOI | MR

[8] D. Boffi, D. Gallistl, F. Gardini and L. Gastaldi, Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form. Math. Comput. 86 (2017) 2213–2237. | DOI | MR | Zbl

[9] A. Bonito, J.-L. Guermond and F. Luddens, Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains. J. Math. Anal. Appl. 408 (2013) 498–512. | DOI | MR | Zbl

[10] D. Braess and R. Verfuerth, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431–2444. | DOI | MR | Zbl

[11] S.C. Brenner, A multigrid algorithm for the lowest-order Raviart-Thomas mixed trangular finite element method. SIAM J. Numer. Anal. 29 (1992) 647–678. | DOI | MR | Zbl

[12] J. Bussac and P. Reuss, Traité de neutronique. Hermann (1985).

[13] P. Ciarlet Jr. E. Jamelot and F.D. Kpadonou, Domain decomposition methods for the diffusion equation with low-regularity solution. Comput. Math. Appl. 74 (2017) 2369–2384. | DOI | MR | Zbl

[14] M. Costabel, M. Dauge and S. Nicaise, Singularities of maxwell interface problems. ESAIM: M2AN 33 (1999) 627–649. | DOI | Numdam | MR | Zbl

[15] M. Dauge, Benchmark Computations for Maxwell Equations. Available at: https://perso.univ-rennes1.fr/monique.dauge/core/ index.html (2018).

[16] J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis. John Wiley & Sons, Inc. (1976).

[17] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag (2004). | DOI | MR | Zbl

[18] R.S. Falk and J.E. Osborn, Error estimates for mixed methods. RAIRO Anal. Numer. 14 (1980) 249–277. | DOI | Numdam | MR | Zbl

[19] P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957–991. | DOI | MR | Zbl

[20] T.-P. Fries and T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84 (2010) 253–304. | DOI | MR | Zbl

[21] L. Giret, Non-conforming Domain Decomposition for the Multigroup Neutron SPN Equations. Ph.D. thesis, EDMH, Université Paris-Saclay (2018).

[22] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR | Zbl

[23] E. Jamelot and P. Ciarlet Jr. Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation.J. Comput. Phys. 241 (2013) 445–463. | DOI | MR | Zbl

[24] E. Jamelot, A.-M. Baudron and J.-J. Lautard, Domain decomposition for the SPN solver MINOS. Transp. Theory Stat. Phys. 41 (2012) 495–512. | DOI | Zbl

[25] E. Jamelot Jr. P. Ciarlet, A.-M. Baudron and J.-J. Lautard, Domain decomposition for the neutron SPN equations, in 21st International Domain Decomposition Conference. Vol. 98 of Lecture Notes in Computational Science and Engineering (2014) 677–685. | MR | Zbl

[26] J.-C. Nédélec, Mixed finite elements in ℝ3. Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

[27] J.E. Osborn, Spectral approximation for compact operators. Math. Comput. 29 (1975) 712–725. | DOI | MR | Zbl

[28] P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods. Vol. 606 of Lecture Notes in Mathematics. Springer (1977) 292–315. | MR | Zbl

[29] A. Sargeni, K.W. Burn and G.B. Bruna, Coupling effects in large reactor cores: the impact of heavy and conventional reflectors on power distribution perturbations, in PHYSOR 2014, Kyoto, Japan, Sept 28–Oct 3, 2014 (2014).

[30] D. Schneider, F. Dolci, F. Gabriel et al., APOLLO3®: CEA/DEN deterministic multi-purpose code for reactor physics analysis, in PHYSOR 2016, Sun Valley ID, USA, May 1–5, 2016 (2016).

[31] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, New York (2008). | DOI | MR | Zbl

[32] M.F. Wheeler and I. Yotov, A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43 (2005) 1021–1042. | DOI | MR | Zbl

Cited by Sources: