Nonconforming Virtual Element Method for 2m2m-th Order Partial Differential Equations in Rn\mathbb R^n

A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.Comment: 33page

Ws,pW^{s,p}-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems

In this work we prove optimal $W^{s,p}$-approximation estimates (with $p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial subspaces. The $W^{s,p}$-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these $W^{s,p}$-estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray--Lions elliptic problems whose weak formulation is classically set in $W^{1,p}(\Omega)$ for some $p\in(1,+\infty)$. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by $h$ the meshsize, we prove that the approximation error measured in a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.Comment: keywords: $W^{s,p}$-approximation properties of elliptic projector on polynomials, Hybrid High-Order methods, nonlinear elliptic equations, $p$-Laplacian, error estimate

Some Error Analysis on Virtual Element Methods

Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with arXiv:1912.0712