scholarly journals On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs

2010 ◽  
Vol 31 (5) ◽  
pp. 2261-2290 ◽  
Author(s):  
S. Chandrasekaran ◽  
P. Dewilde ◽  
M. Gu ◽  
N. Somasunderam
Keyword(s):  
Elliptic Pdes ◽  
Numerical Rank ◽  
Schur Complements

scholarly journals A robust solver for elliptic PDEs in 3D complex geometries

2021 ◽  
pp. 110511
Author(s):  
Matthew J. Morse ◽  
Abtin Rahimian ◽  
Denis Zorin
Keyword(s):  
Elliptic Pdes ◽  
Complex Geometries

Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zdeněk Dostál ◽  
Tomáš Brzobohatý ◽  
Oldřich Vlach
Keyword(s):  
Condition Number ◽  
Three Dimensional ◽  
Schur Complement ◽  
Decomposition Methods ◽  
Stiffness Matrices ◽  
Spectral Bounds ◽  
Schur Complements ◽  
Linear Problems ◽  
Primal Method ◽  
Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

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