A nonconforming virtual element method for the Stokes problem on general meshes

The nonconforming Trefftz virtual element method: general setting, applications, and dispersion analysis for the Helmholtz equation

“…The proof follows along the same lines as that of [38,Theorem 4.3]. For this reason, we only present the modifications that are due to the validity of the weaker Gårding-type inequality (36); see Remark 2. We first observe that, for all…”

“…For the sake of simplicity, write δ h := u H I −u H h . By applying the Gårding-type inequality (36), we deduce…”

“…Notably, the continuity constraint of functions in the approximation spaces are imposed in a weak sense only. Since its inception [6], the nonconforming VEM has received an increasing attention, and has been analysed and applied to several problems: general elliptic problems [14]; Stokes and Navier-Stokes equations [13,35,36,51,52]; eigenvalue problems [23]; the plate bending problem [50]; equations involving the biharmonic and 2m-th operators [4,17]; anisotropic error estimates [15]; the linear elasticity problem [49]; parabolic and fractional-reaction subdiffusion problems [53,34]; the VEM with a SUPG stabilization for advection-diffusion-reaction [11]; fourth order singular perturbation problems [48]; the Kirchhoff plate contact problem [46]; the medius error analysis for the Poisson and biharmonic problem [30]. Its comparison with other skeletal methods such as the hybridized discontinuous Galerkin method (HDG) and the hybrid high-order (HHO) method is investigated in [21].…”

p- and hp- virtual elements for the Stokes problem

“…Proof We begin by proving a discrete "switched" inf-sup condition. Introduce Z C n the complementary space of Z n defined in (34) in V n . In Lemma 6, we proved the existence of a surjective operator n :…”

“…The virtual element method (VEM) is an increasingly popular tool in the approximation to solutions of fluido-static and dynamic problems based on polygonal/polyhedral meshes. In particular we recall: the very first paper on low-order VEM for Stokes [2]; its high-order conforming [11] and nonconforming versions [20,34]; conforming [12] and nonconforming VEM for the Navier-Stokes equation [33]; mixed VEM for the pseudo-stress-velocity formulation of the Stokes problem [17]; mixed VEM for quasi-Newtonian flows [19]; mixed VEM for the Navier-Stokes equation [24]; other variants of the VEM for the Darcy problem [18,45,47]; analysis of the Stokes complex in the VEM framework [9,13]; a stabilized VEM for the unsteady incompressible Navier-Stokes equations [30]; implementation details [23].…”