Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems

Several physical phenomena are described by systems of partial differential equations (PDEs) that, after space discretization, yield the solution of saddle point algebraic linear systems. In realistic three-dimensional numerical simulations, these linear systems are large scale and ill-conditioned,...

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Bibliographic Details
Published inComputer methods in applied mechanics and engineering Vol. 372; p. 113424
Main Authors Dassi, F., Scacchi, S.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 01.12.2020
Elsevier BV
Subjects
Block preconditioners
Computing time
Discretization
Elliptic functions
Iterative methods
Linear systems
Mathematical analysis
Maxwell's equations
Parallel computing
Partial differential equations
Regularization
Robustness (mathematics)
Saddle points
Saddle-point linear systems
Solvers
Virtual element method
Saddle-point linear systems
Parallel computing
Block preconditioners
Virtual element method
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Summary:Several physical phenomena are described by systems of partial differential equations (PDEs) that, after space discretization, yield the solution of saddle point algebraic linear systems. In realistic three-dimensional numerical simulations, these linear systems are large scale and ill-conditioned, thus they require the development of effective solvers. The aim of this work is the construction and numerical validation of parallel block preconditioners for a set of three-dimensional saddle point problems discretized by the low order virtual element method (VEM). VEM is a recent numerical technology for the approximation of PDEs on polygonal and polyhedral meshes. We focus on the following systems of PDEs: stationary Maxwell equations in the mixed Kikuchi formulation; elliptic equations in mixed form; Stokes system; linear elasticity in the mixed Hellinger–Reissner formulation. We provide two parallel block preconditioners: one based on the approximate Schur complement and the other on a regularization technique. Several numerical experiments are run in parallel on a Linux cluster. We analyze the performance of the iterative solvers in terms of GMRES iterations and computational time. We verify the robustness of the solvers with respect to different polyhedral meshes and the scalability of both the assembling and solution time by varying the number of processors. The performance of the two iterative solvers is also compared with state-of-the-art parallel direct linear solvers.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2020.113424