BDDC and FETI-DP for the virtual element method

Abstract

Abstract. We build and analyze Balancing Domain Decomposition by Constraint (BDDC) and Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioners for elliptic problems discretized by the virtual element method (VEM). We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments confirm the theory.

Ill‐conditioning in the virtual element method: Stabilizations and bases

“…where Π ∇ is the matrix representing the action of operator Π ∇ p and S K 3 is defined in (28). In [33], numerical experiments, along with a heuristic motivation, show that choice (28) entails a better behaviour of the method for high p, when approximating a 3D Poisson problem.…”

“…Among the other references, we recall the following: the p and hp version of the method [11][12][13][14][15][16], parabolic problems [17], Cahn-Hilliard, Stokes, Navier-Stokes and Helmoltz equations [18][19][20][21][22], linear and nonlinear elasticity problems [23][24][25], general elliptic problems [26], PDEs on surfaces [27], Domain Decomposition [28], application to discrete fracture networks [29], serendipity VEM [30], VEM on surfaces [27]. The implementation of the method is described in [31], whereas the basic principles of the 3D version of the method are the topic of [32,33].…”

Bricks for the mixed high-order virtual element method: Projectors and differential operators

“…Given f ∈ [L 2 (Ω)] 2 and g ∈ L 2 (Ω), and referring to (14), (15), (23), (28), (25), the virtual elements approximation of the general mixed problem has the form…”

“…The Virtual Element Method has been developed successfully for a large range of mathematical and engineering problems, we mention, as sample, the very brief list of papers [10,28,32,18,46,6,33,7], while for the specific topic of implementation aspects related to the VEM we mention [43,25,14,24,4,8,42,35,45,19]. Concernig the mixed PDEs we refer to [41,30,29,37,31] as a sample of VEM papers dealing with such kind of problem, and to [40,34,26,36] as a representative list of papers treating the same topic with different polytopal technologies.…”

Parallel solvers for virtual element discretizations of elliptic equations in mixed form

“…To our knowledge, in the VEM literature only a few studies have focused on the conditioning of the stiffness matrix resulting from VEM discretizations (see [41,34]) and on the development of preconditioners for VEM approximations of PDEs (see [26,4,31]). Preconditioners for other polyhedral discretizations have been studied in [6,5].…”