Hybridizable Discontinuous Galerkin Approximation for Second Order Elliptic Operator in Non-divergence Form and Some Applications
DescriptionThe second order linear elliptic operator in non-divergence form is an important prototype problem. It arises in a fundamental problem often existing in stochastic optimal control - the elliptic Hamilton-Jacobi-Bellman equations. Because of the linearization techniques, it also appears in the research of fully non-linear elliptic problems. In many applications, the coefficient tensor A of elliptic operator in non-divergence form is so rough that it is even not continuous. For instance, the tensor A arising from Hamilton-Jacobi-Bellman equations is only essentially bounded. Thus, it is desirable to develop and analyze robust and efficient numerical approximations for the second order linear elliptic operator in non-divergence form whose coefficient tensor A may be rough.In contrast to the abundant research on elliptic equations in divergence form, there are not many existing works for elliptic operator in non-divergence form. This is because it is usually difficult to de ne weak solutions of suitable variational forms if the coefficient tensors are nonsmooth. Most existing numerical approximations of the elliptic operator in non-divergence form are based on the assumption that the source term's L2-regularity implies the solution is H2-regular. In fact, the above assumption is so restrictive that even Poisson operator in L-shape domain doesn't satisfy.The long-term goal of the P.I. is to develop and study, theoretically as well as computationally, efficient, robust numerical approximation of the second order linear elliptic operator in non-divergence form to avoid the above mentioned shortcoming of most existing numerical methods. Our work may lead to efficient, easy to implement, robust numerical methods for Hamilton-Jacobi-Bellman equations and other fully non-linear elliptic problems. In this proposal, the P.I. will concentrate on the study and development of a new, emerging numerical scheme - hybridizable discontinuous Galerkin (HDG) methods.The HDG methods we propose for the elliptic operator in non-divergence form can release the above assumption on the regularity estimate of the elliptic operator. We can guarantee the convergence even with rough coefficient tensor A. These properties make our numerical methods suitable for approximating partial differential equations having the elliptic operators in non-divergence form.The P.I. is interested in studying and developing HDG methods, which are efficient, easy to implement and robust, for a variety of equations arising in stochastic optimal control, which get involved with second order elliptic operators in non-divergence form.
|Effective start/end date||1/01/20 → …|