H^m-Conforming Virtual Elements in Arbitrary Dimension

05/27/2021

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by   Xuehai Huang, et al.

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The H^m-conforming virtual elements of any degree k on any shape of polytope in ℝ^n with m, n≥1 and k≥ 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.