SCHUR_PRECOBNN: Solves the sistem (P_B^{NN})^{-1} z=r where P_B^{NN} is the Balancing Neumann-Neumann preconditioner for Schur compl.
(Algorithm 6.4.4., pag. 399 CHQZ3)
Solves the sistem (P_B^{NN})^{-1} z=r
Input: r= r.h.s, column array of length ngamma
ne = number of subdomains
nvli = column array with number of internal nodes of each
subdomain
novg = restriction map from (\cup_i \Omega_i) to \Gamma
set in cosnovg.m
D = diagonal weighting matrix (column array of lenght ngamam)
LGG = structure with maps from \Gamma_i to \Gamma
set in schur_local.m
Am = structure of choleski factors of local matrices
\tilde A_m= A_mm+ M_m/H_m^2 (H_m is the size of the
subdomain of index m, while M_m si the local mass matrix)
Rgamma = matrix of size (ne, ngamma) (defined in CHQZ3, pag. 399)
(R_\Gamma)_ij: = 1/n_j if x_j \in \Gamma_i
= 0 otherwise
PSH = pseudoinverse of Sigma_H, i.e. Rgamma^T A_H^+ Rgamma
AGG = structure of local matrices A_{\Gamma_m,\Gamma_m}
Amm = structure of local matrices (A_{m,m})^{-1}
AGm = structure of local matrices A_{\Gamma_m,m}
Ouput: z = solution, column array of length ngamma
References: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
"Spectral Methods. Fundamentals in Single Domains"
Springer Verlag, Berlin Heidelberg New York, 2006.
CHQZ3 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
"Spectral Methods. Evolution to Complex Geometries
and Applications to Fluid DynamicsSpectral Methods"
Springer Verlag, Berlin Heidelberg New York, 2007.0001 function [z]=schur_precobnn(r,ne,nvli,novg,D,LGG,Am,Rgamma,PSH,AGG,Amm,AGm); 0002 % SCHUR_PRECOBNN: Solves the sistem (P_B^{NN})^{-1} z=r where P_B^{NN} is the Balancing Neumann-Neumann preconditioner for Schur compl. 0003 % (Algorithm 6.4.4., pag. 399 CHQZ3) 0004 % Solves the sistem (P_B^{NN})^{-1} z=r 0005 % 0006 % Input: r= r.h.s, column array of length ngamma 0007 % ne = number of subdomains 0008 % nvli = column array with number of internal nodes of each 0009 % subdomain 0010 % novg = restriction map from (\cup_i \Omega_i) to \Gamma 0011 % set in cosnovg.m 0012 % D = diagonal weighting matrix (column array of lenght ngamam) 0013 % LGG = structure with maps from \Gamma_i to \Gamma 0014 % set in schur_local.m 0015 % Am = structure of choleski factors of local matrices 0016 % \tilde A_m= A_mm+ M_m/H_m^2 (H_m is the size of the 0017 % subdomain of index m, while M_m si the local mass matrix) 0018 % Rgamma = matrix of size (ne, ngamma) (defined in CHQZ3, pag. 399) 0019 % (R_\Gamma)_ij: = 1/n_j if x_j \in \Gamma_i 0020 % = 0 otherwise 0021 % PSH = pseudoinverse of Sigma_H, i.e. Rgamma^T A_H^+ Rgamma 0022 % AGG = structure of local matrices A_{\Gamma_m,\Gamma_m} 0023 % Amm = structure of local matrices (A_{m,m})^{-1} 0024 % AGm = structure of local matrices A_{\Gamma_m,m} 0025 % 0026 % Ouput: z = solution, column array of length ngamma 0027 % 0028 % 0029 % References: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, 0030 % "Spectral Methods. Fundamentals in Single Domains" 0031 % Springer Verlag, Berlin Heidelberg New York, 2006. 0032 % CHQZ3 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, 0033 % "Spectral Methods. Evolution to Complex Geometries 0034 % and Applications to Fluid DynamicsSpectral Methods" 0035 % Springer Verlag, Berlin Heidelberg New York, 2007. 0036 0037 % Written by Paola Gervasio 0038 % $Date: 2007/04/01$ 0039 0040 0041 ra=PSH*r; 0042 0043 y=r-schur_mxv(ra,AGG,Amm,AGm,LGG,novg,ne); 0044 z=schur_preconnl(y,ne,nvli,novg,D,LGG,Am); 0045 y=schur_mxv(z,AGG,Amm,AGm,LGG,novg,ne); 0046 z=z-PSH*y+ra; 0047 return