A solver combining reduced basis and convergence acceleration with applications to non-linear elasticity

Jean-Marc Cadou (1) , Michel Potier-Ferry (2)

1 LIMATB - Laboratoire d'Ingénierie des Matériaux de Bretagne
2 LPMM - Laboratoire de physique et mécanique des matériaux

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.

Mots clés: linear solver; reduced basis technique; convergence acceleration technique; non-linear problems

Domaines: Mécanique [physics.med-ph]

Jean-Marc Cadou, Michel Potier-Ferry. A solver combining reduced basis and convergence acceleration with applications to non-linear elasticity. International Journal for Numerical Methods in Biomedical Engineering, 2010, 26 (12), pp.1604-1617. ⟨10.1002/cnm.1246⟩. ⟨hal-00493925⟩