Some Estimates of Virtual Element Methods for Fourth Order Problems

TL;DR

This paper extends Virtual Element Method estimates to fourth order problems, specifically the biharmonic equation, by adapting techniques from second order operators, resulting in optimal error bounds and relaxed regularity conditions.

Contribution

It introduces new error estimates for VEM applied to fourth order operators, utilizing shape-regular elements and improved right-hand side discretization.

Findings

Derived new projection and interpolation estimates.

Achieved optimal error bounds for biharmonic problems.

Relaxed regularity requirements for the right-hand side.

Abstract

In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis is based on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates are obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions are provided.