We build and analyze a substructuring preconditioner for the Mortar method, applied to elliptic problems, in the - finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [S. Bertoluzza and M. Pennacchio, Appl. Numer. Anal. Comput. Math. 1 (2004) 434–454]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments.
Accepté le :
DOI : 10.1051/m2an/2015065
Mots clés : Domain decomposition methods, iterative substructuring, mortar method, h-pFEM
@article{M2AN_2016__50_4_1057_0, author = {Bertoluzza, Silvia and Pennacchio, Micol and Prud{\textquoteright}homme, Christophe and Samake, Abdoulaye}, title = {Substructuring preconditioners for $h - p$ {Mortar} {FEM}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1057--1082}, publisher = {EDP-Sciences}, volume = {50}, number = {4}, year = {2016}, doi = {10.1051/m2an/2015065}, zbl = {1350.65116}, mrnumber = {3521712}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015065/} }
TY - JOUR AU - Bertoluzza, Silvia AU - Pennacchio, Micol AU - Prud’homme, Christophe AU - Samake, Abdoulaye TI - Substructuring preconditioners for $h - p$ Mortar FEM JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1057 EP - 1082 VL - 50 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015065/ DO - 10.1051/m2an/2015065 LA - en ID - M2AN_2016__50_4_1057_0 ER -
%0 Journal Article %A Bertoluzza, Silvia %A Pennacchio, Micol %A Prud’homme, Christophe %A Samake, Abdoulaye %T Substructuring preconditioners for $h - p$ Mortar FEM %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1057-1082 %V 50 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015065/ %R 10.1051/m2an/2015065 %G en %F M2AN_2016__50_4_1057_0
Bertoluzza, Silvia; Pennacchio, Micol; Prud’homme, Christophe; Samake, Abdoulaye. Substructuring preconditioners for $h - p$ Mortar FEM. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1057-1082. doi : 10.1051/m2an/2015065. http://www.numdam.org/articles/10.1051/m2an/2015065/
A fast solver for Navier stokes equations in the laminar regime using mortar finite element and boundary element methods. SIAM J. Numer. Anal. 32 (1995) 985–1016. | DOI | MR | Zbl
and ,Iterative substructuring preconditioners for mortar element methods in two dimensions. SIAM J. Numer. Anal. 36 (1999) 551–580. | DOI | MR | Zbl
, and ,Substructuring preconditioners for an – domain decomposition method with interior penalty mortaring. Calcolo 52 (2015) 289–316. | DOI | MR | Zbl
, , and ,The role of the inner product in stopping criteria for conjugate gradient iterations. BIT Numer. Math. 41 (2001) 26–52. | DOI | MR | Zbl
, , and ,The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994) 578–632. | DOI | MR | Zbl
and ,Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl
, , , , and ,The mortar spectral element method for fourth-order problems. Comput. Methods Appl. Mech. Eng. 116 (1994) 53–58. | DOI | MR | Zbl
and ,Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems. ESAIM: M2AN 34 (2000) 591–608. | DOI | Numdam | MR | Zbl
, and ,The mortar finite element method for 3d maxwell equations: First results. SIAM J. Numer. Anal. 39 (2001) 880–901. | DOI | MR | Zbl
, and ,C. Bernardi, Y. Maday and A.T. Patera, A new non conforming approach to domain decomposition: The mortar element method. In Collège de France Seminar, edited by H. Brezis and J.-L. Lions. This paper appeared as a technical report about five years earlier. Pitman (1994). | MR | Zbl
Substructuring preconditioners for the three fields domain decomposition method. Math. Comput. 73 (2004) 659–689. | DOI | MR | Zbl
,Analysis of some injection bounds for Sobolev spaces by wavelet decomposition. C. R. Math. 349 (2011) 421–428. | DOI | MR | Zbl
and ,Preconditioning the mortar method by substructuring: the high order case. Appl. Numer. Anal. Comput. Math. 1 (2004) 434–454. | DOI | MR | Zbl
and ,Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1093–1120. | DOI | MR | Zbl
and ,The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. 47 (1986) 103–134. | DOI | MR | Zbl
, and ,The condition number of the schur complement in domain decomposition. Numer. Math. 83 (1999) 127–203. | DOI | MR | Zbl
,C. Canuto, M. Hussaini, A. Quarteroni and T. Zang, Spectral Methods. Scientific Computation. Springer (2006). | MR | Zbl
A FETI-DP preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients. SIAM J. Numer. Anal. 44 (2006) 283–299. | DOI | MR | Zbl
, and ,A capacitance matrix method for Dirichlet problem on polygon region. Numer. Math. 39 (1982) 51–64. | DOI | MR | Zbl
,G.H. Golub and C.F. Van Loan, Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. 3rd edition. Johns Hopkins University Press, Baltimore, MD (1996). | MR | Zbl
A preconditioner for the - version of the finite element method in two dimensions. Numer. Math. 75 (1996) 59–77. | DOI | MR | Zbl
and ,A FETI-DP preconditioner for mortar methods in three dimensions. Electron. Trans. Numer. Anal. 26 (2007) 103–120. | MR | Zbl
,A preconditioner for the FETI-DP formulation with mortar methods in two dimensions. SIAM J. Numer. Anal. 42 (2005) 2159–2175. | DOI | MR | Zbl
and ,Two-level schwarz algorithms with overlapping subregions for mortar finite elements. SIAM J. Numer. Anal. 44 (2006) 1514–1534. | DOI | MR | Zbl
and ,A BDDC method for mortar discretizations using a transformation of basis. SIAM J. Numer. Anal. 4 (2009) 136–157. | DOI | MR | Zbl
, and ,J.L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications. Springer (1972). | MR | Zbl
The mortar finite element method for the cardiac “bidomain” model of extracellular potential. J. Sci. Comput. 20 (2004) 191–210. | DOI | MR | Zbl
,Substructuring preconditioners for mortar discretization of a degenerate evolution problem. J. Sci. Comput. 36 (2008) 391–419. | DOI | MR | Zbl
and ,C. Prud’homme, A strategy for the resolution of the tridimensionnal incompressible Navier-Stokes equations. In vol. 10 of Méthodes itératives de décomposition de domaines et communications en calcul parallèle. Calculateurs Parallèles Réséaux et Systèmes répartis. Hermes (1998) 371–380.
C. Prud’Homme, V. Chabannes, V. Doyeux, M. Ismail, A. Samake and G. Pena, Feel++: A Computational Framework for Galerkin Methods and Advanced Numerical Methods (2012). | MR
C. Prud’homme, V. Chabannes, S. Veys, V. Huber, C. Daversin, A. Ancel, R. Tarabay, V. Doyeux, J.-B. Wahl, C. Trophime, A. Samake, G. Doll and A. Ancel. feelpp v0.98.0. (2014).
A. Samake, S. Bertoluzza, M. Pennacchio and C. Prud’homme, Implementation and numerical results of substructuring preconditioners for the - fem mortar in 2d. In preparation.
C. Schwab, - and -Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Numer. Math. Sci. Comput. Clarendon Press (1998). | MR | Zbl
P. Seshaiyer and M. Suri, Convergence results for non-conforming methods: The mortar finite element method. In vol. 218 of Contemp. Math. AMS (1998) 453–459. | MR | Zbl
Uniform convergence results for the mortar finite element method. Math. Comput. 69 (2000) 521–546. | DOI | MR | Zbl
and ,B. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Vol. 17 of Lect. Notes Comput. Sci. Eng. Springer (2001). | MR | Zbl
Cité par Sources :