The conforming virtual element method for polyharmonic problems

TL;DR

This paper develops a conforming virtual element method tailored for polyharmonic boundary value problems, leveraging high-order continuity spaces to achieve convergence in the energy norm.

Contribution

It introduces a novel conforming virtual element approach specifically designed for polyharmonic problems, with theoretical convergence guarantees.

Findings

Method converges in the energy norm.

Applicable to high-order polyharmonic problems.

Provides a theoretical foundation for virtual element approximation.

Abstract

In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form $\Delta^p u =f$, $p\ge1$. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in the energy norm.