The Virtual Element Method (VEM) is a recent numerical technology for the solution of partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. The aim of this work is to develop effective linear solvers for a general order VEM approximation designed to approximate three-dimensional scalar elliptic equations in mixed form. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner allows to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the problem are indefinite, ill-conditioned, and of saddle point nature. The condition number of the resulting positive definite preconditioned system is adaptively controlled by means of deluxe scaling operators and suitable local generalized eigenvalue problems for the selection of optimal primal constraints. Numerical results confirm the theoretical estimates and the reliability of the adaptive procedure, with the experimental condition numbers always very close to the prescribed adaptive tolerance parameter. The scalability and quasi-optimality of the preconditioner are demonstrated, and the performances of the proposed solver are compared with state-of-the-art parallel direct solvers and block preconditioning techniques in a distributed memory setting.

Dassi, F., Zampini, S., Scacchi, S. (2022). Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 391(1 March 2022) [10.1016/j.cma.2022.114620].

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

Dassi, Franco
;
2022

Abstract

The Virtual Element Method (VEM) is a recent numerical technology for the solution of partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. The aim of this work is to develop effective linear solvers for a general order VEM approximation designed to approximate three-dimensional scalar elliptic equations in mixed form. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner allows to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the problem are indefinite, ill-conditioned, and of saddle point nature. The condition number of the resulting positive definite preconditioned system is adaptively controlled by means of deluxe scaling operators and suitable local generalized eigenvalue problems for the selection of optimal primal constraints. Numerical results confirm the theoretical estimates and the reliability of the adaptive procedure, with the experimental condition numbers always very close to the prescribed adaptive tolerance parameter. The scalability and quasi-optimality of the preconditioner are demonstrated, and the performances of the proposed solver are compared with state-of-the-art parallel direct solvers and block preconditioning techniques in a distributed memory setting.
Articolo in rivista - Articolo scientifico
BDDC method; Domain Decomposition; Parallel computing; Saddle-point linear systems; Virtual Element Method;
English
Dassi, F., Zampini, S., Scacchi, S. (2022). Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 391(1 March 2022) [10.1016/j.cma.2022.114620].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/352502
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