Virtual Enriching Operators

We construct bounded linear operators that map $H^1$ conforming Lagrange finite element spaces to $H^2$ conforming virtual element spaces in two and three dimensions. These operators are useful for the analysis of nonstandard finite element methods

Virtual Element Methods on Meshes with Small Edges or Faces

We consider a model Poisson problem in $\R^d$ ($d=2,3$) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges ($d=2$) or small faces ($d=3$).Comment: 36 page

A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation

We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn-Hilliard equaion. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text overlap with arXiv:1709.0400

A quadratic nonconforming vector finite element for H (curl ; Ω) ∩ H (div ; Ω)

We present a quadratic nonconforming vector finite element for problems posed on the space H (curl ; Ω) ∩ H (div ; Ω), where Ω ⊂ R . Generalizations to higher order and higher dimension are also discussed. © 2008 Elsevier Ltd. All rights reserved.

Balancing domain decomposition for nonconforming plate elements

In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by C[1 + ln(H/h)] , where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H, h and the number of subdomains.