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

We propose and analyze an H²-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 ≥ 2. The analysis relies on the continuous Miranda-Talenti estimate for convex domains Ω and is rather elementary. We prove stability and error estimates in H²(Ω), including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in H²(Ω).