Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

Abstract

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $$W^{2,p}$$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the $$W^{2,p}$$ strong solution in a discrete $$W^{2,p}$$ -norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken $$W^{1,p}$$ -norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.