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

Abstract

The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in ${\mathbb{R}}^n$ with $m,n\ge 1$ and $k\ge m$ 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.

arXiv comment34 pages, 4 figures