An overlapping Schwarz method for virtual element discretizations in two dimensions

Calvo, Juan G.; Gabriel, J. Rigoberto

doi:10.1016/j.camwa.2018.10.043

Cited by 22 publications

(15 citation statements)

References 11 publications

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“…. , J, be the sequence of finite-dimensional virtual element spaces defined in (6). In order to define the multigrid cycle, we introduce the following intergrid transfer operators.…”

Section: Multigrid Algorithmsmentioning

confidence: 99%

“…However, the design of efficient solvers for the solution of the linear system stemming from the virtual element discretization is still a relatively unexplored field of research. So far, the few existing works in literature have mainly focused on the study of the condition number of the stiffness matrix due to either the increase in the order of the method or to the degradation of the quality of the meshes [4,5] and on the development of preconditioners based on domain decomposition techniques [6,7,8,9,10]. Instead, the analysis of multigrid methods for VEM is much less developed.…”

Section: Introductionmentioning

confidence: 99%

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Agglomeration-based geometric multigrid schemes for the Virtual Element Method

Antonietti¹,

Berrone²,

Busetto³

et al. 2021

Preprint

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In this paper we analyse the convergence properties of two-level, W-cycle and V-cycle agglomeration-based geometric multigrid schemes for the numerical solution of the linear system of equations stemming from the lowest order C 0 -conforming Virtual Element discretization of two-dimensional second-order elliptic partial differential equations. The sequence of agglomerated tessellations are nested, but the corresponding multilevel virtual discrete spaces are generally non-nested thus resulting into non-nested multigrid algorithms. We prove the uniform convergence of the two-level method with respect to the mesh size and the uniform convergence of the W-cycle and the V-cycle multigrid algorithms with respect to the mesh size and the number of levels. Numerical experiments confirm the theoretical findings.

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“…. , J, be the sequence of finite-dimensional virtual element spaces defined in (6). In order to define the multigrid cycle, we introduce the following intergrid transfer operators.…”

Section: Multigrid Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Agglomeration-based geometric multigrid schemes for the Virtual Element Method

Antonietti¹,

Berrone²,

Busetto³

et al. 2021

Preprint

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“…In this section, we present how to construct improved subspace correction preconditioners for the ES-FEM and SSE method. Specifically, we propose some two-level additive Schwarz preconditioners [21] for the ES-FEM and SSE method; note that Schwarz preconditioning is a standard methodology of parallel computing for large-scale finite element problems; see, e.g., [22,23,24]. Although we deal with two-level additive Schwarz preconditioner as descriptive examples, the method of improvement introduced in this section can be applied to various subspace correction preconditioners such as multigrid and domain decomposition preconditioners.…”

Section: Improvement Of Preconditionersmentioning

confidence: 99%

“…This observation guarantees that both the ES-FEM and SSE method can adopt any preconditioner designed for the standard FEM and enjoy the advantages of the preconditioner such as good conditioning or numerical scalability. As a concrete example, we consider overlapping Schwarz preconditioner, which is one of the most broadly used parallel preconditioners for finite element problems [21,22,23,24]. We prove that the standard two-level additive Schwarz preconditioner [21] designed for the standard FEM can be applied to the ES-FEM and SSE method, satisfying the condition number bound C(1 + H/δ), where C is a positive constant independent of the mesh and subdomain sizes, H is the subdomain size, and δ is the overlapping width for the overlapping domain decomposition associated with the additive Schwarz preconditioner.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning for finite element methods with strain smoothing

Lee¹,

Park²

2021

Preprint

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Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element (SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods. First, we analyze the spectrums of the stiffness matrices of the edge-based S-FEM and the SSE method. Then, we propose an improved two-level additive Schwarz preconditioner for the strain smoothing methods by modifying local solvers appropriately. For the sake of convenience of implementation, an alternative form of the preconditioner is also proposed by defining the coarse-scale operation in terms of the standard FEM. We verify our theoretical results through numerical experiments.

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“…Thanks to its flexibility and robustness with respect to mesh design, VEM enjoyed, in recent years, wide success: the theoretical analysis of the method has been extended in different directions [16,17,42,25,7,26,32]. Problems related to the efficiency of the method have been addressed [27,29,5,35], and different model problems have been tackled, see e.g. ( [47,3,4,21,22,44,19,14,39,24,38,6,49,50,33]).…”

Section: Introductionmentioning

confidence: 99%

Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements

Bertoluzza¹,

Pennacchio²,

Prada³

2021

Preprint

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In the framework of virtual element discretizazions, we address the problem of imposing non homogeneous Dirichlet boundary conditions in a weak form, both on polygonal/polyhedral domains and on two/three dimensional domains with curved boundaries. We consider a Nitsche's type method [43,41], and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in [9]. We prove that also for the virtual element method (VEM), provided the stabilization parameter is suitably chosen (large enough for Nitsche's method and small enough for the Barbosa-Hughes Lagrange multiplier method), the resulting discrete problem is well posed, and yields convergence with optimal order on polygonal/polyhedral domains. On smooth two/three dimensional domains, we combine both methods with a projection approach similar to the one of [31]. We prove that, given a polygonal/polyhedral approximation Ω h of the domain Ω, an optimal convergence rate can be achieved by using a suitable correction depending on high order derivatives of the discrete solution along outward directions (not necessarily orthogonal) at the boundary facets of Ω h . Numerical experiments validate the theory.

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An overlapping Schwarz method for virtual element discretizations in two dimensions

Cited by 22 publications

References 11 publications

Agglomeration-based geometric multigrid schemes for the Virtual Element Method

Agglomeration-based geometric multigrid schemes for the Virtual Element Method

Preconditioning for finite element methods with strain smoothing

Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements

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