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A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity
Author(s) -
Cadou J. M.,
PotierFerry M.
Publication year - 2010
Publication title -
international journal for numerical methods in biomedical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.741
H-Index - 63
eISSN - 2040-7947
pISSN - 2040-7939
DOI - 10.1002/cnm.1246
Subject(s) - preconditioner , solver , mathematics , krylov subspace , subspace topology , extrapolation , projection (relational algebra) , linear system , matrix (chemical analysis) , linear subspace , computation , polynomial , convergence (economics) , condition number , iterative method , algorithm , eigenvalues and eigenvectors , mathematical optimization , mathematical analysis , geometry , materials science , physics , quantum mechanics , economics , composite material , economic growth
An iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non‐linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub‐steps. In the first sub‐step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub‐step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well‐known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non‐linear solvers with few matrix factorizations and few iterations. Copyright © 2009 John Wiley & Sons, Ltd.