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

Dassi, Franco; Zampini, Stefano; Scacchi, Simone

doi:10.1016/j.cma.2022.114620

Cited by 8 publications

(7 citation statements)

References 62 publications

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“…We have also a convergence result Theorem 3.2. Let (u, p) ∈ V × Q be the solution of problem (6) and (u h , p h ) ∈ V h × Q h be the solution of problem (13). Then it holds…”

Section: Verifying the Estimatementioning

confidence: 99%

“…In the VEM literature only a few studies have focused on the construction and analysis of preconditioners for VEM approximations of PDEs; see [1,9,10,12]). BDDC for VEM discretizations of scalar elliptic problems have been first introduced in [4,5] and then extended to mixed formulations of scalar elliptic equations in [13]. To our knowledge, the development of effective non-overlapping domain decomposition preconditioners for VEM discretizations of the Stokes equations is still an open problem.…”

Section: Introductionmentioning

confidence: 99%

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BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations

Bevilacqua¹,

Scacchi²

2022

Preprint

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The Virtual Element Method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints (BDDC) preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

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“…We have also a convergence result Theorem 3.2. Let (u, p) ∈ V × Q be the solution of problem (6) and (u h , p h ) ∈ V h × Q h be the solution of problem (13). Then it holds…”

Section: Verifying the Estimatementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations

Bevilacqua¹,

Scacchi²

2022

Preprint

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show abstract

“…In the VEM literature only a few studies have focused on the construction and analysis of preconditioners for VEM approximations of PDEs; see [1,7,8,10]). BDDC for VEM discretizations of scalar elliptic problems have been first introduced in [2,3] and then extended to mixed formulations of scalar elliptic equations in [11]. To our knowledge, the development of effective non-overlapping domain decomposition preconditioners for VEM discretizations of the Stokes equations is still an open problem.…”

Section: Introductionmentioning

confidence: 99%

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

Bevilacqua

Scacchi

2022

J Sci Comput

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The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

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“…In the last five years, researchers have started developing ad-hoc preconditioners for VEM discretizations of different PDE problems: see [2] for multigrid, [13] for Additive Schwarz and [11] for BDDC and FETI-DP preconditioners for scalar elliptic equations; see [17] for parallel block algebraic multigrid (AMG) preconditioners for different saddle point problems; see [19] and [12] for BDDC preconditioners for mixed formulations of elliptic equations and the Stokes system, respectively. To the best of our knowledge, there are no references yet on effective linear solvers applied to VEM discretizations of Maxwell equations.…”

Section: Introductionmentioning

confidence: 99%

“…(19) 7.15 10 −3 (18) 7.02 10 −3 (15) 6.77 10 −3 (15) 1404 9.44 10 −3 (25) 9.72 10 −3 (24) 9.46 10 −3 (20) 9.43 10 −3 (19) 9156 6.81 10 −2 (32) 6.85 10 −2 (29) 6.40 10 −2 (25) 6.18 10 −2 (24) 72422 6.81 10 −2 (32) 6.85 10 −2 (29) 6.40 10 −2 (25) 6.18 10 −2…”

mentioning

confidence: 99%

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

A.¹,

Dassi²,

Scacchi³

2022

Preprint

Self Cite

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The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been perfomed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

show abstract

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

Cited by 8 publications

References 62 publications

BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations

BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

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