$H^m$-Conforming Virtual Elements in Arbitrary Dimension

TL;DR

This paper introduces a recursive construction of $H^m$-conforming virtual elements for arbitrary dimensions and degrees, providing theoretical analysis and applying them to solve polyharmonic equations with optimal error estimates.

Contribution

It presents a universal recursive method to construct $H^m$-conforming virtual elements in any dimension and degree, with rigorous analysis and practical application to polyharmonic problems.

Findings

Successfully constructed $H^m$-conforming virtual elements in arbitrary dimensions.

Proved inverse inequalities and norm equivalences for these virtual elements.

Achieved optimal error estimates for the virtual element discretization of polyharmonic equations.

Abstract

The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in $\mathbb R^n$ with $m, n\geq1$ and $k\geq m$ are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case $k=m$, the set of degrees of freedom only involves function values and derivatives up to order $m-1$ at the vertices of the polytope. The inverse inequality and several norm equivalences for the $H^m$-conforming virtual elements are rigorously proved. The $H^m$-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the $H^m$-conforming virtual element method.