Premium
Preconditioning Mixed Finite Element Saddle‐point Elliptic Problems
Author(s) -
Vassilevski Panayot S.,
Lazarov Raytcho D.
Publication year - 1996
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199601/02)3:1<1::aid-nla67>3.0.co;2-e
Subject(s) - mathematics , preconditioner , saddle point , discretization , finite element method , eigenvalues and eigenvectors , saddle , block matrix , mathematical analysis , elliptic operator , diagonal , matrix (chemical analysis) , diagonal matrix , block (permutation group theory) , linear system , mathematical optimization , combinatorics , geometry , physics , thermodynamics , materials science , quantum mechanics , composite material
We consider saddle‐point problems that typically arise from the mixed finite element discretization of second‐order elliptic problems. By proper equivalent algebraic operations the considered saddle‐point problem is transformed to another saddle‐point problem. The resulting problem can then be efficiently preconditioned by a block‐diagonal matrix or by a factored block‐matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form(δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.