A new family of stable mixed finite elements for the 3D Stokes equations

By Shangyou Zhang. Mathematics of Computation, 2004

313 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/49a3eed409e1368d01639efe0cf7aebf7ba4385e

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In Computational Mechanics

In Computational Theory and Mathematics

In Electrical and Electronic Engineering

Citations in the corpus

A Second Elasticity Element Using the Matrix Bubble (2012, background): “However in [26], the 3D version of the HseihClough-Toucher grids is shown to provide a class of meshes without singular vertices and edges, leading us to consider 3D meshes also in the following assumption:”
UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS (2008, background): “Zhang [33] proposed 3D finite elements of P 3 k −Pk−1 type, for k ≥ 3, on macro tetrahedron meshes where each big tetrahedron is subdivided into four subtetrahedra by connecting the barycenter with four vertices.”
Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids (2011, unknown): “But this cannot be done for tetrahedral grids, to remove a singular vertex or a singular edge, without creating a new singular vertex or singular edge, unless we change grids structurally as in [16].”
Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, methodology): “For stable and divergence-free Stokes elements of conforming Pk−Pk−1 types (continuous piecewise Pk for velocity and discontinuous piecewise Pk−1 for pressure), we refer to [56] for a 2D family with any k ≥ 4 on meshes that do not contain singular vertices, to [2] for a 2D element with k = 1 on macr”
On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime (2014, methodology): “This price, which by the way can be avoided completely by divergence-free mixed methods like the classical Scott-Vogelius element[49, 48, 41, 51] or more recent divergence-free discretization approaches[11, 12, 50, 20, 21, 22], is well-known in the finite element community as poor mass conservation ”
ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS (2013, background): “Such a result also hypothetically exists for Scott-Vogelius discretizations on tetrahedra [41], but these discretizations are limited to an extremely restrictive class of macro-element meshes.”
On the convergence rate of grad-div stabilized Taylor–Hood to Scott–Vogelius solutions for incompressible flow problems (2011, background): “(A1) in 2d, k ≥ 4 and the mesh has no singular vertices [19], (A2) in 3d, k ≥ 6 [24], (A3) when k ≥ d and the mesh is a barycenter refinement of a regular mesh [23, 19], or”
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows (2015, methodology): “Indeed, the first robust mixed method on 3D tetrahedral grids was published only in 2005 in [67].”