\({H^m}\)-Conforming Virtual Elements in Arbitrary Dimension
Chunyu Chen, Xuehai Huang, Huayi Wei
Abstract
.The \(H^m\) -conforming virtual elements of any degree \(k\) on any shape of polytope in \(\mathbb{R}^n\) with \(m, n\geq 1\) 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.Keywords \(H^m\) -conforming virtual elementsWhitney arrayerror analysispolyharmonic equationMSC codes65N1265N1565N2265N30