Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

Abstract

We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d\ge 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $\hat{I}\text{c◯}$ and is rather elementary. We prove stability and error estimates in $H^2(\hat{I}\text{c◯})$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(\hat{I}\text{c◯})$.