H² Conforming Virtual Element Discretization of Nondivergence Form Elliptic Equations
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The numerical discretization of elliptic equations in nondivergence form is notoriously challenging, due to the lack of a notion of weak solutions based on variational principles. In many cases, there still is a well-posed variational formulation for such equations, which has the particularity of being posed in H², and therefore leads to a strong solution. Galerkin discretizations based on this formulation have been studied in the literature. Since H² conforming finite elements tend to be considered impractical, most of these discretizations are of discontinuous Galerkin type. On the other hand, it has been observed in the virtual element literature that the virtual element method provides a practical way to build H² conforming discretizations of variational problems. In this talk, I will describe a virtual element discretization of equations in nondivergence form. I will start with a simple linear model problem, and show how the H² conformity of the method allows for a particularly simple well-posedness and error analysis. I will then discuss the extension to equations with lower-order terms and with Hamilton-Jacobi-Bellman type nonlinearities, and present some numerical results.